% This script projects the distribution of the market invariants for the bond markets
% (i.e. the changes in yield to maturity)
% from the estimation interval to the investment horizon
% Then it computes the distribution of prices at the investment horizon
% see "Risk and Asset Allocation"-Springer (2005), by A. Meucci
clc; clear; close all;
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% inputs
tau=1/52; % time to horizon expressed in years
tau_tilde=1/52; % estimation period expressed in years
FlatCurve=.04;
TimesToMat=[4 5 10 52 520]'/52; % time to maturity of selected bonds expressed in years
% parameters of the distribution of the changes in yield to maturity
u_minus_tau=TimesToMat-tau;
mus=0*u_minus_tau;
sigmas=(20+5/4*u_minus_tau)/10000;
Num_Scenarios=100000;
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% bond market projection to horizon and pricing
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BondCurrent_Prices_Shifted=exp(-FlatCurve*u_minus_tau);
BondCurrent_Prices=exp(-FlatCurve*TimesToMat);
% project bond market to horizon
N=length(TimesToMat); % number of bonds
U=rand(Num_Scenarios,1);
BondMarket_Scenarios=zeros(Num_Scenarios,N);
for n=1:N
% generate co-dependent changes in yield-to-maturity
DY_Scenarios = norminv(U,mus(n)*tau/tau_tilde,sigmas(n)*sqrt(tau/tau_tilde));
% compute the horizon prices, (3.81) in "Risk and Asset Allocation" - Springer
X=-u_minus_tau(n)*DY_Scenarios;
BondMarket_Scenarios(:,n)=BondCurrent_Prices_Shifted(n)*exp(X);
end
% MV inputs - analytical
Exp_Hrzn_DY_Hat=mus*tau/tau_tilde;
SDev_Hrzn_DY_Hat=sigmas*sqrt(tau/tau_tilde);
Corr_Hrzn_DY_Hat=ones(N); % full co-dependence
Cov_Hrzn_DY_Hat=diag(SDev_Hrzn_DY_Hat)*Corr_Hrzn_DY_Hat*diag(SDev_Hrzn_DY_Hat);
[BondExp_Prices,BondCov_Prices]=Dy2Prices(Exp_Hrzn_DY_Hat,Cov_Hrzn_DY_Hat,u_minus_tau,BondCurrent_Prices_Shifted)
% MV inputs - numerical
BondExp_Prices=mean(BondMarket_Scenarios)'
BondCov_Prices=cov(BondMarket_Scenarios)
