DAFTAR PUSTAKA. Anton, Howard, dan Rorres, C Elementary Linear Algebra : Application Version. Canada : John Wiley & Sans, Inc

DAFTAR PUSTAKA

Anton, Howard, dan Rorres, C. 1994. Elementary Linear Algebra : Application Version. Canada : John Wiley & Sans, Inc. Baki, Isa. Yield Curve Estimation by Spline Based Methods. 2006. Bliss, R. Robert. Testing Term Structure Estimation Methods. Draft 1996. Diebold, Francis X., dan Li, C. Forecasting the Term Structure of Government Bond Yields. Journal of Econometrics, 130 (2005), 337-364 Donohue, Robert J. Introduction to Cashflow Analysis. Regent School Press. Eberly, David. Least-Squares Fitting of Data with B-Spline Curves. February, 2008 Lin, Bing-Huei. Fitting Term Structure of Interest Rates Using B-splines : The Case of Taiwanese Government Bonds. Applied Financial Economics, 12, 2002, 57-75. Lyuu, Yuh-Dauh. 2002. Financial Engineering and Computation : Principles, Mathematics, Algorithms. USA : Cambridge University Press. Mathews, John H. 1987. Numerical Methods fot Mathematics, Science, and Engineering. New Jersey : Prentice-Hall. McDonald, Robert L. 2002. Derivative Markets. USA : Pearson Education. McCulloch, J. Huston. Measuring the Term Structure of Interest Rate. The Journal of Business, 44, No. 1 (1971), 19-31. McCulloch, J. Huston. The Tax-Adjusted Yield Curve. The Journal of Finance, 30, No. 3 (Jun., 1975), 811-830. Nash, Stephen G., dan Sofer, A. 1996. Linier and Nonlinier Programming. USA : McGraw-Hill.

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Nocedal, Jorge, dan Wright, Sephen J. 2006. Numerical Optimization. New York : Springer Verlag. Nelson, Charles R, dan Siegel, Andrew F. Parsimonius Modelling of Yield Curves. The Journal of Business, 60, No. 4(1987), 473-489. Ruf, Johannes. B-Spline of Third Order on a Non-Uniform Grid. Draft 2008. Syamsuddin, M. Catatan Kuliah Teori Ekonometrika. Maret, 2008. Danareksa Debt Research Weekly Reports. http://www.danareksa.com/home/index_news2.cfm?act=riset&categoryId =5. Diakses terakhir tanggal 25 Mei 2009. Riding the Yield Curve. http://www.kiplinger.com/basics/archives/2003/04/bonds4.html. Diakses tanggal 17 Januari 2009.

42

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)); stp=round(nk/nknot);

%banyak knot %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);

43

%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;

44

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;

45

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 b1 b2 tau 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;

46

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 NelsonSiegel % 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 %'); title('Grafik Setiap Segmen Nelson Siegel');

47

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))&(x=t(i+2))&(x=t(i+3))&(x
48

LAMPIRAN B DATA PERIODE 30 MARET - 3 APRIL 2009

49

LAMPIRAN C YIELD CURVE DENGAN MENGGUNAKAN DATA ASLI

Tanggal Pelaporan 02 Maret 09


50




51

Tanggal Pelaporan 13 April 2009



52

Tanggal Pelaporan 11 Mei 2009



53

LAMPIRAN D YIELD CURVE DENGAN MENGGUNAKAN DATA HASIL SARINGAN



54




55




56




57

LAMPIRAN E PARAMETER MODEL DENGAN MENGGUNAKAN DATA ASLI Tanggal Pelaporan

McCulloch

Nelson-Siegel

a1

a2

a3

a4

a5

a6

a7

a8

!0

!1

!2

!

02 Maret 2009

7.8986

9.2427

12.6194

12.2167

12.6376

18.6442

-2.3607

-

13.8577

-4.7678

7.4026E-06

1.7253

10 Maret 2009

-14.5783

13.5034

8.7213

13.5836

10.7028

22.5518

-9.9381

-

12.2062

-7.7486

1.4798E-05

0.1334

16 Maret 2009

94.5127

0.5830

14.5429

10.2646

13.1953

13.2360

16.1487

8.3420

11.6689

3.9073E+05

-3.9077E+05

0.0182

23 Maret 2009

0.2868

15.1222

7.9979

12.6014

9.7475

25.9287

-30.3917

-

14.1541

-2.0205

-7.5608

2.7757

30 Maret 2009

15.7606

6.2185

16.2354

8.9808

11.7020

11.0094

16.5618

-1.6723

11.0158

-5.0241

8.8614

0.6787

06 April 2009

-16.0277

13.7104

9.5611

11.0954

10.7171

12.2056

13.4182

10.4564

11.1216

-4.12E+00

2.6493

0.2812

13 April 2009

48.0726

3.1952

12.1385

10.1990

7.9313

23.5725

-25.0955

-

11.4088

1.8518

-9.2171

0.5338

20 April 2009

-22.4806

15.4735

10.2125

9.7318

9.6856

19.0854

-12.8700

-

10.3211

-3.8871

6.5828

0.6088

27 April 2009

-33.2156

13.2237

9.3130

12.2449

10.6479

15.2991

2.4888

-

11.4675

-10.8421

6.8508

0.3196

11 Mei 2009

-81.6017

26.8599

-1.0237

15.2913

6.9077

16.8372

1.8326

44.5592

9.3324

-68.8532

83.2407

0.0750

18 Mei 2009

-55.8078

21.8681

2.5321

13.1019

-0.2151

41.1290

-80.3651

-

10.7934

-10.7934

-0.0543

0.0951

25 Mei 2009

-29.1671

16.8217

4.0120

12.2455

8.6046

11.0928

19.9544

-14.0641

10.1585

-10.1585

-0.0080

0.0261

58

LAMPIRAN F PARAMETER MODEL DENGAN MENGGUNAKAN DATA HASIL SARINGAN Tanggal Pelaporan

McCulloch

a1

a2

a3

a4

Nelson-Siegel

a5

a6

a7

02 Maret 2009

5.8194

9.8845

12.2897

12.1504

12.9892

17.4154

6.2666

10 Maret 2009

-2.8746

10.6810

9.9692

12.2259

13.3723

16.7237

8.4742

16 Maret 2009

-11.1015

14.6858

6.9742

12.1584

11.3578

15.2804

11.9137

23 Maret 2009

14.3267

7.3153

12.1406

10.6753

14.7389

8.2726

29.6782

30 Maret 2009

10.1761

7.8929

11.3721

10.7720

13.0252

11.9449

14.9916

06 April 2009

-0.2708

9.6808

10.2684

11.3046

11.8142

13.9231

9.0932

13 April 2009

24.3829

6.9241

10.2583

10.7464

12.5621

12.0319

15.0484

20 April 2009

4.6992

8.5119

10.6055

10.9094

12.2278

12.1601

13.8126

27 April 2009

-18.5734

9.4156

10.8461

11.3219

12.1919

12.3736

12.9559

11 Mei 2009

5.2820

8.5286

9.3198

10.4994

9.9872

13.6213

9.2816

18 Mei 2009

2.7023

8.8577

9.6791

9.9667

11.5128

12.5919

12.3824

25 Mei 2009

-19.8952

15.3207

4.8523

11.6911

9.3274

13.6766

8.2700

a8 23.0119 22.3790 26.0445

!0

!1

!2

!

14.0802

-4.8517

1.53E-03

2.1758

14.1243

-5.9499

1.05E-03

1.3963

15.6166

-5.3448

-6.7834

2.3784

13.2034

-4.1129

-3.65E-06

2.9772

12.9788

-4.3591

8.08E-04

2.4870

12.4163

-4.3846

1.28E-03

1.7367

12.7547

-0.1881

-1.15E+01

0.9319

12.5749

-4.5048

4.72E-05

2.4134

12.1114

-8.4950

9.21E-05

0.5811

12.3220

-5.0494

0.0017

2.3799

12.1327

-4.3762

1.41E-04

2.7743

11.4731

-3.76E+00

0.0003

2.0033

!

59

DAFTAR PUSTAKA. Anton, Howard, dan Rorres, C Elementary Linear Algebra : Application Version. Canada : John Wiley & Sans, Inc

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