coupon effecs Fsher Cohen, Kramer and Waugh Ordnary Leas SquaresOLS 3 j τ = a0 a j m a4 log m a5c a6c a7 log C j= τ = a a a [ ] 0 m log m
[ a, b] f Pn E f = max f x P x = f P n ( ) ( ) n ( ) a x b n ξ 3
4 G F ξ ξ ξ = =, ( ) A ( ) ( ) ( ) k k k e A e A e A 3 3,, = = = ( ) O ξ < = 0 ( ) O ξ = ( ) 0 = f ξ ( ) ( ) ( ) 3 6 = f ξ ξ ξ ξ ξ ( ) ( ) G G FG F f ξ ξ ξ ξ = 3 6 6 3 3 ( ) ( ) = 6 f ξ ξ ξ ξ ξ ξ ξ ( ) f n n = = ( ) ( ) ( ) ( ) ( ) = = n A A A A D 3 3 3 3 α α α α ( ) ( ) ( ) ( ) ( ) = A A A O A 3 3 3
( ) 0 lm D = D = lm D [ A, B] ( = 0, n) B k ( ) ξ,..., A = ξ < ξ <... < ξ = n B 0 ( ) ( ξ, ξ k ) B k ( ) B k ( ) ( ) D ' = D j= 0 ( ) ( ) l= 0, l j l j D = lm D ( ) ( ) = k k ξ ξ k k k B ( ) = [ max ( ξ j,0)] 5
[ A, B] ( = 0, n) ξ 0 n,,, n ξ,..., ξ ( = 3, n, n 3) [ B] ξ A, = { ( )} k n Φ φ k =~ α Dˆ = Φα S = CΦ C P = Sα ε α ( ) = α φ ( ) D k = n k k 6
h( ) k ( ) ( ) ( ) h, β = β kφk = φ β k= λ 0 λ h( ) λ λ λ mn GCV T h ( ) d RSS( λ) m n T ) ( p p ) λ h ( ) d = 0 ( λ, θ ) = RSS ( λ ) ( λ ) [ m θk ] 7
k ( λ) θ λ n d j ( m m ), j, Q d j = ml θ L L =,..., n j θ = L Q n( j ) L = [ ] Q m L ( ) z (, T ) 8
( T ) Z (, T ) ( ) Z, τ τ τ f (, τ, T ) INST = lm f (, τ, T ) τ T ( T ) z, z (, T ) dsc f f = (, T ) (, τ, T ) = e = e 00 β0 β β τ β 0 f ( TTM ) β 0 T 365 (, T ) ( T ) Z 00 365 [( T ) z(, T )] [( τ ) z(, τ )] T τ (,τ, T ) = z (, T ) ( T ) z f (, T ) = ( TTM ) INST = T x = β 0 f (, τ, T ) β T e INST TTM τ dx β (, T ) z TTM τ e TTM τ 9
β β ( TTM ) f β τ β β τ f ( TTM ) ( TTM ) 0 z z ( TTM ) 3.0%.5%.0%.5%.0% 0.5% 0.0% -0.5% = β 0 β τ TTM e TTM τ β τ TTM e TTM τ TTM τ z( TTM ) β 0 0 4 6 8 0 f ( TTM ) = β 0 β e TTM τ β TTM τ e TTM τ
( ) TTM f ( ) TTM z ( ) = 3 0 τ τ τ τ β τ β β β TTM TTM TTM e TTM e TTM e TTM f ( ) = 3 0 τ τ β τ τ β τ β β τ τ τ TTM e TTM TTM e TTM e TTM TTM z TTM TTM TTM
( v / v max ) ( n / n max ) ( e ) ( e ) W = W v v max n n max
McCulloch Vascek & Fong Seeley McCulloch Vascek & Fong Seeley Smoohng Nelson & Segel Exend N & S Svensson 3
4
5
6
7
8
9
0
n Mn ( ) w ˆ P P = Mn n = ( ˆ ) w P P = n j = D j D j D j 3
4
5
6
7
8
9
30
3
3
33
34
35
5 6 = 5, 65 36
τ β { x } { x } λ λ λ λ T = [( ) ( )] x x x x λ 4 λ = 0 37
λ λ 38
39
40
4
4
43
. Blss, R.R., 996, Tesng Term Srucure Esmaon Mehods, Federal Reserve Bank of Alana Workng Paper 96-a.. Bolder, D., Srelsk, D., 999, Yeld Curve Modelng a he Bank of Canada, Bank of Canada Techncal Repor No.84. 3. Bank of Inernaonal Selemen, 999, Zero-Coupon Yeld Curves: Techncal Documenaon. 4. Choudhry, M., 004, Analysng & Inerpreng he Yeld Curve, John Wley & SonsAsaPe. Ld. 5. Danel F. W., 997, Splne Mehods for Exracng Rae Curves from Coupon Bond Prces, Federal Reserve Bank of Alana Workng Paper 97-0. 6. Fsher, M., Nychka, D., and Zervos, D., 995, Fng he Term Srucure of Ineres Raes wh Smoohng Splnes, Workng Paper 95-, Fnance and Economcs Dscusson Seres, Federal Reserve Board. 7. McCulloch, J. H., 97, Measure he Term Srucure of Ineres Raes, Journal of Busness, p9-3. 8. McCulloch, J. H., 975, The Tax-Adjused Yeld Curve, Journal of Fnance, Vol. 3, p88-830. 9. Nelson, C. R., Segel, A. F., 987, Parsmonous Modelng of Yeld Curves, Journal of Busness, Vol. 60, p473-489. 0. Seeley, J. M., 99, Esmang he Gl-Edged Term Srucure Bass Splne and Confdence, Journal of Busness Fnance and Accounng, Vol.8, p53-59.. Subramanan, K. V., 00, Term Srucure Esmaon In Illqud Markes, Journal of Fxed Income, Vol., No., p77-86.. Svensson, L. E. O., 994, Esmang and Inerpreng Forward Ineres Raes: 44
Sweden 99-994, NBER Workng Paper Seres 487. 3. Vascek, O., 977, An Equlbrum Characerzaon of he Term Srucure, Journal of Fnancal Economcs, Vol. 5, p77-88. 4. Vascek, O., Fong, G., 98, Term Srucure Esmaon Usng Exponenal Splnes, Journal of Fnance, Vol.38, p339-348. 5. Uned Kngdom Deb Managemen Offce, 000, The DMO s Yeld Curve Model. 6. Yu, I.W., Fung, L., 00, Esmaon of Zero-Coupon Yeld Curve Based on Exchange Fund Blls and Noes n Hong Kong, Hong Kong Moneary Auhory. 45