S_CornishFisher.m

% this script compares the Cornish-Fisher estiamte of the VaR with the true analytical VaR 
% under the lognormal assumptions
% see "Risk and Asset Allocation" - Springer (2005), by A. Meucci

clc; clear; close all;

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% inputs 

mu=.2; 
sig=.5 ;

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% process data

E_X=exp(mu+sig^2/2);
Sd_X=exp(mu+sig^2/2)*sqrt(exp(sig^2)-1);
Sk_X=sqrt(exp(sig^2)-1)*(exp(sig^2)+2);

c=[.001:.001:.999];
z=norminv(c);

Q_CF = E_X + Sd_X*(  z + Sk_X/6*(z.^2-1)  );
Q_true = logninv(c,mu,sig);

x=Q_true;
f=lognpdf(x,mu,sig);

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% plots
figure
plot(x,f)
grid on
title('pdf')

figure
plot(c,Q_true,'r')
hold on
plot(c,Q_CF);
grid on
legend('true','Cornish-Fisher approx',2)
title('quantile')