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=5. Diakses terakhir tanggal 25 Mei 2009.
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tanggal 17 Januari 2009.
LAMPIRAN A KODE MATLAB
Main Program
clear all;data=xlsread('052509'); data2=plotreal(data); global W x_real y_real nk t x_real=data2(:,1); %data x y_real=data2(:,2); %data y
nk=length(x_real); %banyak data x real %---
% NS %---
beta_nol=[10 -8 -3 1]; %tebakan awal optimasi A=[-1 -1 0 0;-1 0 0 0; 0 0 0 -1]; %constrain eps=[0.001 0.001 0.001]; %syarat berhenti
%weight matrix kk=sum(1/x_real);
for i=1:nk W(i,i)=(1/x_real(i))/kk; end %optimasi [b]=barrier(beta_nol) %error var_NS=leastsqr(b)/(nk-1) SE_NS=sqrt(var_NS)
%producing data for plot x1=x_real(1):0.1:x_real(nk); for i=1:length(x1)
y1(i)=NS(b,x1(i)); end
%plot each segment d=NSSegmenPlot(x1,b);
%--- % Spline %---
nknot=floor(sqrt(nk)); %banyak knot stp=round(nk/nknot); %step index x_real %creating knot for i=1:nknot-1 t(i)=x_real(1+(i-1)*stp); end t(nknot)=x_real(nk);
%pembuatan subselang di luar [a,b] nt=length(t); %banyak data t d1=t(2)-t(1); %distance i=-3,-2,-1 d2=t(nt)-t(nt-1); %distance i=n+1,n+2,n+3
%stretching subsection from 0...n into -3...n-3 for i=nt:-1:1 t(i+3)=t(i); end
for i=1:3
%set equidistance for -3..-1 and n+1..n+3 t(4-i)=t(5-i)-d1; t(nt+3+i)=t(nt+2+i)+d2; end
nt=length(t); %update banyak data t %creating B-spline for i=1:nk for j=1:nt-4 B(i,j)=bspline(x_real(i),j); end end
%finding parameter a by least square LL=transpose(B)*W*B;
a=inv(LL)*transpose(B)*W*y_real; %error measure for price
for j=1:nk y_spln(j)=0; for i=1:(length(t)-4) y_spln(j)=y_spln(j)+bspline(x_real(j),i)*a(i); end end y_spln=transpose(y_spln); P=y_real-y_spln; var_MC=(transpose(P)*P)/(nk-nknot) SE_MC=sqrt(var_MC)
%producing data for plot for j=1:length(x1) y2(j)=0; for i=1:(length(t)-4) y2(j)=y2(j)+bspline(x1(j),i)*a(i); end end %plot figure; plot(x1,y1); hold on; plot(x1,y2,'--k');
xlabel('Time to Maturity in Years'); ylabel('Yield %');
title('Yield Curve'); plot(x_real,y_real,'or');
h = legend('Nelson Siegel','McCulloch','Eksak',4); hold off;
Screenshot input
Berikut adalah contoh tampilan input data pada Microsoft Excel untuk data laporan 6 April 2009 dengan nama file 040609.xls dengan kolom A : maturity time dalam hari; kolom B : yield; kolom C : tanggal pengambilan data; kolom D : waktu jatuh tempo obligasi; kolom E : yield; dan kolom F : penanda apakah data digunakan dalam perhitungan atau tidak. Jika diberi angka 1 maka data tidak digunakan.
plotreal.m
| Fungsi : membuat plot data real dan outlier jika ada. Output dari program ini adalah data yang telah dibersihkan dari outlier (yang diberikan angka 1 pada kolom F) function d=plotreal(data) years=252; x_real=data(:,1)./years; y_real=data(:,2); isOutlier=data(:,6); nk=length(x_real); %plot k=0;ind=1;for i=1:nk
%jika data adalah outlier, masukkan ke list if isOutlier(i)==1
k=k+1;
xOut(k)=x_real(i); yOut(k)=y_real(i); else
%jika bukan, masukkan ke perhitungan pembentukan kurva d(ind,1)=x_real(i); d(ind,2)=y_real(i); ind=ind+1; end end plot(x_real,y_real,'or'); hold on;
xlabel('Time to Maturity in Years'); ylabel('Yield %');
title('Data Yield Asli'); %jika ada outlier, plot outlier if k>0
plot(xOut,yOut,'ok','MarkerEdgeColor','k','MarkerFaceColor','k'); h = legend('Data asli','Data yang Tidak Digunakan',4);
end hold off;
barrier.m
| Fungsi : menghitung nilai optimal parameter Nelson-Siegel dengan menggunakan metode penalti tipe barrier% beta_nol : tebakan awal parameter Nelson Siegel % W : matriks bobot
function k=barrier(beta_nol,W) global x_real y_real lambda lambda=10; step_lambda=0.1; akurasi=10^(-9); sudah=false; iterasi=1; syms b1b2tau g=(1/(b1+b2))+(1/(b1))+(1/tau); %barrier method while (sudah==false) beta_baru= fminsearch(@NSleastsqrW,[beta_nol],[],x_real,y_real); cek=lambda*subs(g,[b1,b2,tau],[beta_baru(1),beta_baru(2),beta_baru(4)]); if (norm(cek) <= akurasi sudah=true; else lambda=lambda*step_lambda; beta_nol=beta_baru; iterasi=iterasi+1; end end banyak_iterasi_barrier=iterasi-1 k=beta_nol;
NSleastsqrW.m
| Fungsi : menghitung jumlah kuadrat selisih berbobot antara data asli dan data hasil model Nelson-Siegel ditambah dengan kendala% b : parameter Nelson Siegel % y : vektor data yield asli
% x : vektor data maturity time yang berpasangan dengan data yield function J = NSleastsqrW(b,x,y)
global lambda W n = length(x); Jtemp = 0; for i=1:n,
Jtemp = Jtemp + W(i,i)*(y(i)-NS(b,x(i)))^2; end
J = Jtemp + lambda * ( (1/(b(1)+b(2))) + (1/(b(1))) + (1/b(4)) );
NS.m
| Fungsi : menghitung nilai yield menggunakan model Nelson-Siegel % x : nilai x (maturity time)% b : parameter Nelson Siegel function K=NS(b,x)
Tt=x/b(4);
K=b(1)+(b(2)+b(3))*(1/Tt)*(1-exp(-Tt))-b(3)*exp(-Tt);
NSleastsqr.m
| Fungsi : menghitung jumlah kuadrat selisih tanpa bobot antara data asli dan data hasil model Nelson-Siegel% b : parameter Nelson Siegel function sieg=NSleastsqr(b) global W x_real y_real nk parsi for i=1:nk Tt=x_real(i)/b(4); parsi(i)=b(1)+(b(2)+b(3))*(1/Tt)*(1-exp(-Tt))-b(3)*exp(-Tt); end; P=y_real-transpose(parsi); sieg=transpose(P)*P;
NSSegmenPlot.m
| Fungsi : membentuk plot setiap segmen parameter Nelson-Siegel% x1 : vektor range x (maturity time) % b : parameter Nelson Siegel
function k=NSSegmenPlot(x1,b) figure; Tt1=x1/b(4); for i=1:length(Tt1) NSseg0(i)=b(1); NSseg1(i)=b(2)*(1/Tt1(i))*(1-exp(-Tt1(i))); NSseg2(i)=b(3)*(1/Tt1(i))*(1-exp(-Tt1(i)))-b(3)*exp(-Tt1(i)); end plot(Tt1,NSseg0); hold on plot(Tt1,NSseg1,'--k','LineWidth',3); xlabel('Time to Maturity in Years'); ylabel('Yield %');
plot(Tt1,NSseg2,'--r');
h = legend('Segmen beta 0','Segmen beta 1','Segmen beta 2',4); hold off
k=1;
bspline.m
| Fungsi : menghitung nilai b-spline kubik % t : matriks titik subselang x% x : nilai x (maturity time) yang dicari nilai splinenya % i : urutan subselang spline
function b = bspline(x,i) global t; if ((x>=t(i))&(x<t(i+1))) b=(x-t(i))^3/((t(i+3)-t(i))*(t(i+2)-t(i))*(t(i+1)-t(i))); elseif ((x>=t(i+1))&(x<t(i+2))) b1=((x-t(i))^2*(t(i+2)-x))/((t(i+3)-t(i))*(t(i+2)-t(i))*(t(i+2)-t(i+1))); b2=((x-t(i))*(x-t(i+1))*(t(i+3)-x))/((t(i+3)-t(i))*(t(i+3)-t(i+1))*(t(i+2)-t(i+1))); b3=((x-t(i+1))^2*(t(i+4)-x))/((t(i+4)-t(i+1))*(t(i+3)-t(i+1))*(t(i+2)-t(i+1))); b=b1+b2+b3; elseif ((x>=t(i+2))&(x<t(i+3))) b1=((x-t(i))*(t(i+3)-x)^2)/((t(i+3)-t(i))*(t(i+3)-t(i+1))*(t(i+3)-t(i+2))); b2=((t(i+4)-x)/(t(i+4)-t(i+1))); b3=((x-t(i+1))*(t(i+3)-x))/((t(i+3)-t(i+1))*(t(i+3)-t(i+2))); b4=((x-t(i+2))*(t(i+4)-x))/((t(i+4)-t(i+2))*(t(i+3)-t(i+2))); b=b1+b2*(b3+b4); elseif ((x>=t(i+3))&(x<t(i+4))) b=(t(i+4)-x)^3/((t(i+4)-t(i+1))*(t(i+4)-t(i+2))*(t(i+4)-t(i+3))); else b=0; end;
