S_VaRContributions.m

% this script computes VaR and contributions to VaR from each exposure
% - semi-analytically (under generic elliptical assumption)
% - naively-empirical (VaR is percentile, contributions are numerical derivative
% - refined-empirical (VaR from kernel smoothing, contributions as expectations
% see "Risk and Asset Allocation" - Springer (2005), by A. Meucci

clc; clear; close all;
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% inputs 

N=20; % number of assets
NumSimul=100000; % number of simulations

% elliptical market parameters: 
Mu=rand(N,1); % location
A=rand(N,N)-.5; % scatter
Sigma=A*A';
R = (lognrnd(0,.2,NumSimul,1)-7).^2; % radial distribution

% allocation
a=rand(N,1)-.5;

% VaR confidence
c=.95;

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% generate market scenarios
Z=randn(NumSimul,N);
norm_Z=sqrt(sum(Z.*Z,2));
U=Z./(norm_Z*ones(1,N));  % uniform on sphere
X=U.*(R*ones(1,N));       % elliptical, standard
M=ones(NumSimul,1)*Mu'+X*A'; % elliptical, general

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% semi-analytical approach

% compute quantile of standard marginal numerically
StandardizedQuantile=prctile(X(:,1),(1-c)*100);
% compute VaR and contributions analytically
VaR_an=Mu'*a+StandardizedQuantile*sqrt(a'*Sigma*a)
DVaR_an=Mu+StandardizedQuantile*Sigma*a/sqrt(a'*Sigma*a);
ContrVaR_an=a.*DVaR_an;

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% empirical naive approach

% compute objective
Psi = M*a;

% compute naive VaR as percentile
VaR_simul=prctile(Psi,(1-c)*100);

% compute naive gradient as perturbation
DVaR_simul=[];
for n=1:N
    e=zeros(N,1); 
    e(n)=.01;
    Psi_up=M*(a+e);    
    VaR_up=prctile(Psi_up,(1-c)*100);

    DVaR_simul=[DVaR_simul
        (VaR_up-VaR_simul)/e(n)];
end

% compute marginal contributions
ContrVaR_simul=a.*DVaR_simul;

BadError=mean((ContrVaR_simul-ContrVaR_an).^2);

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% empirical refined approach

% sort market accoring to objective scenarios
[Sort_Psi,Index]=sort(Psi);
Sort_M=M(Index,:);

% define smoothing  kernel for the empirical VaR computation
th=ceil((1-c)*NumSimul); % percentile index
bw=NumSimul/700; % bandwidth
sm=normpdf([1:NumSimul]',th,bw); % smoother
sm=sm/sum(sm); % normalize smoother

% compute VaR as kernel smoohting
VaR_simul_refined=Sort_Psi'*sm

% compute gradient as expectation
DVaR_simul_refined=0*a;
for n=1:N
    DVaR_simul_refined(n)=sm'*Sort_M(:,n);
end
% compute marginal contributions
ContrVaR_simul_refined=a.*DVaR_simul_refined;

GoodError=mean((ContrVaR_simul_refined-ContrVaR_an).^2);

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% plots

figure

subplot(3,1,1)
bar(ContrVaR_an)
xlim([0 N+1])
y_lim=get(gca,'ylim');
grid on
title(['analytical: error = 0'])

subplot(3,1,2)
bar(ContrVaR_simul)
xlim([0 N+1])
set(gca,'ylim',y_lim)
grid on
title(['naive: error = ' num2str(BadError)])

subplot(3,1,3)
bar(ContrVaR_simul_refined)
xlim([0 N+1])
set(gca,'ylim',y_lim)
grid on
title(['refined: error = ' num2str(GoodError)])