Steeley [1991]... JAFEE 35 TMU NEEDS E-mail: kentarou.kikuchi@boj.or.jp / /212.7 35
1 1 1 1 2 McCulloch [1971, 1975] Steeley [1991] Tanggaard [1997] Schaefer [1981] Nelson and Siegel [1987] Svensson [1995]... 1 1 23 1 27 1 2 Jonathan Wright Svensson [1995] 36 /212.7
Bank for International Settlements [25] Ioannides [23] Kalev [24] 1989 Oda [1996] 2 22 1989 198 5 198 199 3 1999 21 1999 27 1... 3 198 199 199 37
1 Steeley [1991] 2 3 4 Steeley [1991] 5 1 2 Steeley [1991] Steeley [1991] 1999 1 211 12 4 2 2 1 t T 1 Z (t, T) t T y (t, T) y (t, T) = 1 T t log (Z (t, T)), (1)... 4 212 http://www.imes.boj.or.jp/research/papers/japanese/12-j-3.txt 38 /212.7
t x y (t, t + x), (2) t x y (t, t + x) 2 4 t x Z (t, t + x) Z (t, t + x) x t t r (t) r (t) = lim x y (t, t + x), (3) t S T S < T t T 1 S t S T 1 t T t S T f (t, S, T) Z (t, T) = exp ( f (t, S, T)(T S )) exp ( y (t, S )(S t)), (4) (1) (4) f (t, S, T) f (t, S, T) = 1 ( ) Z (t, T) T S log, (5) Z (t, S ) t t S f (t, S ) 39
( ) 1 Z (t, T) f (t, S ) = lim f (t, S, T) = lim T S T S T S log = Z (t, S ) S log (Z (t, S )), (6) (6) Z (S, S ) = 1 Z (t, t + x) ( x ) Z (t, t + x) = exp f (t, t + s) ds. (7) (1) (7) y (t, t + x) = 1 x x f (t, t + s) ds, (8) 2 1 2 4 /212.7
1 2 A B 9 1 11 A B 2 2 A B B A B 41
3 B 2 B 2 3 3 4 42 /212.7
t = t = Z (, x) y (, x) f (, x) Z (x) y (x) f (x) t = i (i {1,..., n name }) i N i c i i t = T i = {T1 i,..., T i } n i n i cf i cf t = k < l T i k < T i l t = Ti T T = n name i=1 T i := {T 1,..., T ncf }, T j = min i {1,..., n name } 1 k n i cf {T i k ; T i k > T j 1}, T 1 = min i {1,..., n name } 1 k n i cf T i k, t = I = {v 1,..., v ni } P = (P v 1,...,P v n I ) T A = (A v 1,..., A v n I ) T 5 P = P+A P = ( P v 1,..., P v n I ) T T 1 4. v i v i c v i 6 c v i = ( g (c v i, N v i, T 1 ),..., g (c v i, N v i, T j ),...,g(c v i, N v i, T ncf ) ) T, c v i N v i if T j T v i, T j T v i 2 n g (c v i, N v i v i cf, T j ) = c v i N v i + N v i if T j = T v i. 2 n v i cf otherwise c v i n cf 1 c v i j c v i j... 5 P A t = α 6 21 3 21 3 1 4. 43
v i Q v i n cf Q v i = j=1 c v i j Z (T j), (9) 2 4 Z (x) α Z (x) Z (x; α) α (9) α v i Q v i (α) t = Q (α) = (Q v 1 (α),..., Q v n I (α)) T 4 Z (x) Z (x) 3 3 3 3 Z (x) Bank for International Settlements [25] 2 5 1 2 3 McCulloch [1975] Steeley [1991] 2 139 44 /212.7
Fisher, Nychka, and Zervos [1995] Waggoner [1997] Jarrow, Ruppert, and Yu [24] 3 7 u m u m+1 u n 1 u n, m n j l B ( j, x) x [u h, u h+1 ] (m h n 1) (, u m ] [u n, ) l... 7 l l 1 45
McCulloch [1975] McCulloch [1975] Z (x) = u 1 = u = u 1 < u 2 < < u nknot McCulloch [1975] B (k, x) (k =,..., n knot ) (1) x 3 k n knot B (k, x) =, x u k 1, (x u k 1 ) 3 6(u k u k 1 ), u k 1 < x u k, (u k u k 1 ) 2 6 + (u k u k 1 )(x u k ) 2 + (x u k) 2 2 (u k+1 u k 1 )( 2u k+1 u k u k 1 6 (x u k) 3 6(u k+1 u k ), u k < x u k+1, + x u k+1 ), u k+1 < x, 2 k = n knot B (k, x) = x. (1) McCulloch [1975] Z (x) (1) n knot Z (x) = 1 + B (k, x) α k. (11) k= Z () = 1 (1) B (k, ) = (k =,...,n knot ) (11) (11) (9) v i Q v i α = (α,α 1,...,α nknot ) T n cf nknot n cf n Q v i (α) = c v i j + c v i j B (k, T j) α cf k = c v i j + ( cv i ) T Bα. j=1 k= j=1 j=1 (12) B ( j, k) B (k, T j ) n cf (n knot + 1) c v i 46 /212.7
j c v i j n cf 1 α ˆα [( ˆα= arg min P Q (α) ) T ( P Q (α) ) ] α n cf n cf P:= ( P v 1 j=1 c v 1 j,..., P v n I n cf j=1 c v n I j ) T, Q (α) := (Q v 1 (α) c v 1 j,...,q v n I (α) j=1, (13) n cf j=1 c v n I j ) T. Q (α) α (13) α ˆα ˆα = (( cb) T cb) 1 ( cb) T P, (14) c c = ( c v 1,..., c v n I ) T c n I n cf X 1 X McCulloch [1975] Steeley [1991] Steeley [1991] McCulloch [1975] Z (x) Z (x) (15) Z (x) = n knot 1 k= 3 B (k, x) α k. (15) Steeley [1991] B (k, x) McCulloch [1975] B (k, x) u 3 < < u nknot < u nknot +1 < u nknot +2 < u nknot +3 B (k, x) (16) 47
D = 1 1, u k x < u k+1 B (k, x) =B D (k, x) :=, otherwise D > 1 B (k, x) =B D (k, x) = u D+k x x u k B D 1 (k + 1, x) + B D 1 (k, x). u D+k u k+1 u D+k 1 u k (16) (16) B Steeley [1991] D = 4 3 (15) Z () = 1 n knot 1 k= 3 B (k, ) α k = 1, (17) (15) (9) v i Q v i n cf Q v i (α) = j=1 c v i j n knot 1 k= 3 n knot 1 B (k, T j ) α k = k= 3 n cf c v i j B (k, T j) α k = ( c v i ) T Bα. (18) j=1 B ( j, k) B (k, T j ) n cf (n knot + 3) (17) (18) α [ ˆα = arg min ( P Q (α)) T ( P Q (α)) ], α n knot 1 s.t. B (k, ) α k = 1. (19) k= 3 ˆα (2) ˆα = { ( cb) T cb } 1 ( cb) T P + 1 { } 1 BT ( cb)t cb ( cb) T P { } { } 1 ( cb)t cb B. ( cb)t cb 1 B B T (2) 48 /212.7
B = (B ( 3, ),..., B (n knot 1, )) T Steeley [1991] 2 Fisher, Nychka, and Zervos [1995] Fisher, Nychka, and Zervos [1995] Z (x) y (x) f (x) f (x) 8 Fisher, Nychka, and Zervos [1995] f (x) f (x) = n B (k, x) α k. (21) k=m B (k, x) (7) (21) ( x Z (x) = exp n = exp B (k, x) := k=m x ) f (s) ds x = exp n B (k, s) α k ds k=m x n ( B (k, s) ds) α k = exp B (k, x) α k, k=m B (k, s) ds. (22) (22) (9) Q v i n cf n Q v i (α) = c v i j exp B (k, T j ) α k = ( cv i ) T exp ( Bα), j=1 k=m n exp ( Bα) := exp ( B (k, T 1 ) α k ),..., exp ( k=m n B (k, T ncf ) α k )... 8 Fisher, Nychka, and Zervos [1995] 3 Fisher, Nychka, and Zervos [1995] k=m T. (23) 49
B j, k = B (k, T j ) Q v i (α) α α Fisher, Nychka, and Zervos [1995] (23) α = α 1 Q v i (α) 9 ( Q v i (α) ( c v i ) T exp ( Bα ) + ) α exp ( Bα) (α α ) α=α =Q v i (α ) + ( c v i ) T α exp ( Bα) (α α ) α=α =Q v i (α ) ( c v i ) T [ B ( exp ( Bα ) 1 T)] (α α ). (24) 1 1 = (1,..., 1) T (n m + 1) X v i (α ) = ( c v i ) T [ B ( exp ( Bα ) 1 T)], Y v i (α ) = P v i Q v i (α ) + X v i (α ) α, (25) (23) α = α ˆα (α ) [( ˆα (α ) = arg min Y (α ) X (α ) α ) T ( Y (α ) X (α ) α ) ], α X (α ):= (X v 1 (α ),..., X v n I (α )) T, Y (α ):= (Y v 1 (α ),..., Y v n I (α )) T. (26) (26) Z () = 1 (22) n n Z () = exp B (k, ) α k = exp α k = 1, k=m k=m (26) ˆα (α ) = ( X (α ) T X (α ) ) 1 X (α ) T Y (α ), (27) α = α Fisher,... 9 (24) 5 /212.7
Nychka, and Zervos [1995] (27) ˆα (α ) α 1 (24) Q v i (α) (27) ˆα (α 1 ) α 2 α 2 ˆα (α i ) Fisher, Nychka, and Zervos [1995] (21) B (k, x) 3 3 McCulloch [1971] 2 1 B (k, x) McCulloch [1971] Steeley [1991] 2 B (16) D = 3 B (k, x) Steeley [1991] 4. 3 Tanggaard [1997] Tanggaard [1997] 4. 1 11 Q v i (α) n cf Q v i = j=1 c v i j α j = ( c v i ) T α, α j = Z (T j ). (28) Q v i (α) α ˆα (29) [ min ( P Q (α)) T ( P Q (α)) ]. (29) α ˆα ˆα = ( c T c) 1 c T P, (3) c = ( c v 1,..., c v n I ) T... 1 (1) x McCulloch [1971] 11 Tanggaard [1997] Carleton and Cooper [1976] Houglet [198] 51
Schaefer [1981] Schaefer [1981] Z (x) D B D (k, x) D k B D (k, x) = ( 1) j+1 D k xk+ j j k + j, k >, j= D k j := (D k)! (D k j)!j!, B D (, x) =1. (31) Schaefer [1981] Z (x) Z (x) = D B D (k, x) α k. k= Z () = 1 B D (k, ) =, k > α = 1 Z (x) = D B D (k, x) α k = 1 + k= D B D (k, x) α k. (32) k=1 (32) (9) v i Q v i α = (α 1,...,α D ) T n cf D n cf n Q v i (α) = c v i j + c v i j B D (k, T j ) α cf k = j=1 k=1 j=1 j=1 c v i j + ( cv i ) T Bα. (33) B ( j, k) B D (k, T j ) n cf D 52 /212.7
(33) ˆα [( ˆα= arg min P Q (α) ) T ( P Q (α) ) ], (34) α n cf n cf P:= ( P v 1 j=1 c v 1 j,..., P v n I n cf j=1 c v n I j ) T, Q (α) := (Q v 1 (α) c v 1 j,..., Q v n I (α) j=1 n cf j=1 c v n I j ). ˆα ˆα = (( cb) T cb) 1 ( cb) T P, (35) Nelson and Siegel [1987] Nelson and Siegel [1987] f (x) (36) ) ( f (x) = α + α 1 exp ( xα3 x + α 2 exp x ). (36) α 3 α 3 (8) (36) y (x) y (x) = 1 x x ( ) 1 exp ( x/α3 ) f (s) ds = α +α 1 x/α 3 +α 2 ( 1 exp ( x/α3 ) x/α 3 ( exp x )). (37) α 3 v i Q v i (9) Nelson and Siegel [1987] Z (x; α) (37) α := (α,α 1,α 2,α 3 ) T ˆα { ˆα = arg min ( P Q (α)) T ( P Q (α)) }. (38) α 53
Svensson [1995] Svensson [1995] Nelson and Siegel [1987] (36) f (x) f (x) ( f (x) = α + α 1 exp x ) ) ( x + α 2 exp ( xα3 x + α 4 exp x ). (39) α 3 α 3 α 5 α 5 (39) y (x) y (x) = 1 x x ( ) 1 exp ( x/α3 ) f (s) ds = α +α 1 x/α 3 ( 1 exp ( x/α3 ) +α 2 x/α 3 ( 1 exp ( x/α5 ) +α 4 x/α 5 ( exp x α 3 ( exp x α 5 )) )). (4) (4) Nelson and Siegel [1987] v i Q v i 5 2 2 4 4. McCulloch [1975] +1 4. Nelson and Siegel [1987] 4 2 54 /212.7
1 P T v i n v i v i λp v i cf P + λp v i P X ỹ X (x; P) P + λp v i ỹ X (x; P + λp v i ) X T v i n v i cf l X (λ, ε; T v i ) n v i cf l X (λ, ε; T v i ):= n v i cf T v i n v i + cf ε Tncf T v i n v +ε i cf ỹ X (x; P + λp v i ) ỹ X (x; P) 2 dx ỹ X (x; P + λp v i ) ỹ X (x; P) 2 dx. (41) ε λ (41) X v i T v i n v i cf (41) (41) ε λ Nelson and Siegel [1987] 1 4. 55
Steeley [1991] u l = l (l = 3,..., 33) 32 1 4/32 =.125 2 3 1 2/32 =.625 3/32 =.9375 Steeley [1991] Nelson and Siegel [1987] Steeley [1991] McCulloch [1975] McCulloch [1975] u l = l (l =,..., 3), u 1 = 31 1 2/31 =.645 1 2 3/31 =.968 2 3 4/31 =.129 1 Steeley [1991] McCulloch [1975] Steeley [1991] 12 4 4 1 Tanggard [1997] 4... 12 McCulloch [1971] McCulloch [1975] Steeley [1991] 56 /212.7
1 4 8 8 1 2 McCulloch [1971, 1975] Steeley [1991] 4. 4. McCulloch [1971, 1975] Steeley [1991] 4 Vasicek and Fong [1982] McCulloch and Kochin [2] 13 McCulloch [1971, 1975] Steeley [1991] 4 3 2 3 1 3 2 2 2 3 3... 13 Vasicek and Fong [1982] Z (x) x = 1 exp ( αs) Z (s)(= Z (x)) Vasicek and Fong [1982] Z (s) McCulloch and Kochin [2] 57
1 199 2 1 3 21 26 27 5 5 a 2 3 2 4 1 5 5 (a) 21 6 1 (b) 211 9 2 (%) (%) 4.5 4. 4. 3.5 3.5 3. 3. 2.5 2.5 2. 2. 1.5 1.5 1. 1..5.5. 1 4 7 1 13 16 19 22 25 28. 1 4 7 1 13 16 19 22 25 28 n n = 1, 2,..., 1, 15, 2, 3 GJGBn Index n n = 1, 2, 3, 5, 7, 1, 3 2 USGGn Index 1 USGG12M Index 58 /212.7
b 5 a 3 14 2 6 7 6 8 15 2 2 6 29 2 17 2. (%) 1.5 1..5. 1 2 3 4 5 6 7 8 9 1 11 12 13 14 15 16 17 18 19 2 n GJGBn Index... 14 FRB Svensson [1995] Gükaynak, Sack, and Wright [27] FRB 1 2 29 15 7 1 7 28 7 6 8 7 59
2 2 2 1 1 2 2.5 1 1.5 2 2 2 1 1 2 2 1 6 /212.7
.. 2 2...5.5 2 (42) 39 j=2 (y (.5(j + 1)) 2y (.5 j) + y (.5(j 1)) ) 2. (42).. 3... 1 3 2 2 2 5 1 2 3 NEEDS 1999 1 4 21 12 3 7 7 1999 1 15 3 5 2 3 61
7 12 1 8 1 2 6 5 4 2 2 3 1999/1 2/1 1/1 2/1 3/1 4/1 5/1 6/1 7/1 8/1 9/1 1/1 / 2 5 1 2 3 35 3 25 2 15 1 5 1999/1 2/1 1/1 2/1 3/1 4/1 5/1 6/1 7/1 8/1 9/1 1/1 / 2 1989 Oda [1996] 2 22 2 2 62 /212.7
2 4 8 McCulloch [1975] 2 4. 1 3 McCulloch [1975] Steeley [1991] 2 4. 3 1 33 Steeley [1991] Fisher, Nychka, and Zervos [1995] McCulloch [1971] 2 2 4. 1 3 McCulloch [1971] Fisher, Nychka, and Zervos [1995] Steeley [1991] 2 2 4. 2 1 32 Steeley [1991] Tanggaard [1997] 2 4. Schaefer [1981] 2 4. 5 Nelson and Siegel [1987] 2 4. Svensson [1995] 2 4. 2.. 8 63
1.5 1 1.5 2 1 8 8 3 1 1 McCulloch [1975] Steeley [1991] McCulloch [1971] Steeley [1991] Schaefer [1981] Tanggaard [1997] Nelson and Siegel [1987] Svensson [1995] 1,84 3 2,162 619 11,788 4 2,947 8 (%) 3. 2.5 2. 1.5 1..5..5 1. 1 2 3 4 5 6 7 8 9 1 11 12 13 14 15 16 17 18 19 2 Nelson and Siegel [1987] 25 5 2 Svensson [1995] 25 8 29 Tanggaard [1997] 1999 3 19 Schaefer [1981] 26 2 2 64 /212.7
Schaefer [1981] Nelson and Siegel [1987] Svensson [1995] Tanggaard [1997] 2 Schaefer [1981] Nelson 2 McCulloch [1975] Steeley [1991] McCulloch [1971] Steeley [1991] Schaefer [1981] Tanggaard [1997] Nelson and Siegel [1987] Svensson [1995] 1 2 2 36 11 88 2 2 5 3 11 4 15 5 6 7 8 9 1 11 12 13 14 15 16 17 18 19 2 1 2,947 1999 1 4 21 12 3 2 8 ±2 65
9 2.5 (%) 2. 1.5 1..5..5 1 2 3 4 5 6 7 8 9 1 11 12 13 14 15 16 17 18 19 2 Steeley [1991] McCulloch [1975] Svensson [1995] Schaefer [1981] and Siegel [1987] Svensson [1995] 9 Schaefer [1981].5 McCulloch [1975] Steeley [1991] McCulloch [1971] Steeley [1991] 4 4 McCulloch [1975] Steeley [1991] Mc- Culloch [1971] Steeley [1991] 66 /212.7
3 McCulloch [1975] Steeley [1991] McCulloch [1975] Steeley [1991] 29.178.185 3.414 27.58 29.126 29.116.144.198.182 3.93 3.598 3.87 4.728 4.441 4.714 5.41 1 2 4 McCulloch [1975] Steeley [1991] McCulloch [1971] Steeley [1991] 1.587 1 2 1.47 1 2 1.93 1 2 1.991 1 2 1 (42) 2 1 3 4 Steeley [1991] 4 Steeley [1991] 4 4 Steeley [1991] 1 Steeley [1991] 67
1 Steeley [1991] 3. (%) 2.5 2. 1.5 1..5. 1 2 3 4 5 6 7 8 9 1 11 12 13 14 15 16 17 18 19 2 Steeley [1991] 2 2 8 2 2 8 9.5 9 1 9.5 9 1 Steeley [1991] 8 Steeley [1991] 4 Steeley [1991] 3 Steeley [1991] Steeley 1 Steeley Nelson and Siegel [1987] NS Steeley 2 3 Steeley 68 /212.7
1 Nelson and Siegel [1987] 11 Steeley NS 21 12 3 3 1 1 Steeley 12 28 8 2 7 7 6 8 Steeley NS 6 8 Steeley 13 Steeley NS NS 6 Steeley 6 11 21 12 3 2.5 (%) 2. 1.5 1..5. 1 2 3 4 5 6 7 8 9 1 11 12 13 14 15 16 17 18 19 2 Steeley NS 69
12 28 8 2 2.5 (%) 2. 1.5 1..5. 1 2 3 4 5 6 7 8 9 1 11 12 13 14 15 16 17 18 19 2 Steeley NS 13 6 1.5 (%) 1..5..5 1. 1999/1 2/1 1/1 2/1 3/1 4/1 5/1 6/1 7/1 8/1 9/1 1/1 / Steeley NS O/N 16 Steeley NS... 16 Steeley 6 JGB 1 JGB JGB 7 /212.7
Steeley 14 Steeley NS 1 2 1 1bp 2 1bp 1 2 Steeley NS 1. 2 Steeley Fisher, Nychka, and Zervos [1995] Waggoner [1997] Jarrow, Ruppert, and Yu [24] 3 1 Jarrow, Ruppert, and Yu [24] JRY Steeley JRY (21) B 71
14 1 2 3. (%) 1 (%).12 2.5.1 2..8 1.5.6 1..4.5.2.. 1999/1 1/1 3/1 5/1 7/1 9/1 / 2/1 2/1 4/1 6/1 8/1 1/1 Steeley NS 3.5 2 (%) (%).14 3..12 2.5.1 2..8 1.5.6 1..4.5.2. 1999/1 1/1 2/1 2/1 3/1 5/1 7/1 9/1 4/1 6/1 8/1 1/1 Steeley. / NS 72 /212.7
{ 1 Tcf min ( P Q (α)) T ( P Q (α)) + λ α n I } ( f (y) ) 2 dy. (43) (43) (43) λ Jarrow, Ruppert, and Yu [24] Ruppert [1997] 15 Steeley JRY Steeley 15 Steeley JRY 29 2 17 2.5 (%) 2. 1.5 1..5 1 2 3 4 5 6 7 8 9 1 11 12 13 14 15 16 17 18 19 2 JRY Steeley 4 (%) 3 2 1 1 2 3 4 5 6 7 8 9 1 11 12 13 14 15 16 17 18 19 2 JRY Steeley 73
7 JRY Steeley Steeley JRY 15 5 2 1 Steeley [1991] Steeley Steeley 1 74 /212.7
21 Steeley 2 1 199 21 1 2 75
2 23 5 2 22 149 164 14 1989 No. 212-J-3 212 Bank for International Settlements, Zero-coupon Yield Curves: Technical Documentation, BIS paper No.25, 25. Carleton, Willard T., and Ian A. Cooper, Estimation and Uses of the Term Structure of Interest Rates, Journal of Finance, 31(4), 1976, pp.167-183. Fisher, Mark, Douglas W. Nychka, and David Zervos, Fitting the Term Structure of Interest Rates with Smoothing Splines, Federal Reserve System Working Paper No. 95-1, Board of Governors of the Federal Reserve System, 1995. Gürkaynak, Refet S., Brian Sack, and Jonathan H. Wright, The U.S. Treasury Yield Curve: 1961 to the Present, Journal of Monetary Economics, 54(8), 27, pp.2291 234. Houglet, Michel X., Estimating the Term Structure of Interest Rates for Non-homogeneous Bonds, Ph.D Dissertation, University of California Berkeley, 198. Ioannides, Michalis, A Comparison of Yield Curve Estimation Techniques using UK Data, Journal of Banking and Finance, 27(1), 23, pp.1 26. Jarrow, Robert, David Ruppert, and Yan Yu, Estimating the Interest Rate Term Strucuture of Corporate Debt with a Semiparametric Penalized Spline Model, Journal of the American Statistical Association, 99(465), 24, pp.57 66 Kalev, Petko S., Estimating and Interpreting Zero Coupon and Forward Rates: Australia, 1992-21, Working Paper, Monash University, 24. McCulloch, J. Huston, Measuring the Term Structure of Interest Rates, Journal of Business, 44(1), 1971, pp.19 31., The Tax-adjusted Yield Curve, Journal of Finance, 3(3),1975, pp.811 83., and Levis A. Kochin, The Inflation Premium Implicit in the US Real and Nominal Term Structures of Interest Rates, Working Paper #98-12, Ohio State University, 2. Nelson, Charles R., and Andrew F. Siegel, Parsimonious Modeling of Yield Curves, Journal of Business, 6(4), 1987, pp.473 489. 76 /212.7
Oda, Nobuyuki, A Note on the Estimation of Japanese Government Bond Yield Curves, IMES Discussion Paper No.96-E-27, Institute for Monetary and Economic Studies, Bank of Japan, 1996. Ruppert, David, Empirical-bias Bandwidths for Local Polynomial Nonparametric Regression and Density Estimation, Journal of the American Statistical Association, 92(439), 1997, pp.149 162. Schaefer, Stephen M., Measuring a Tax-specific Term Structure of Interest Rates in the Market for British Government Securities, Economic Journal, 91(362), 1981, pp.415 438. Steeley, James M., Estimating the Gilt-edged Term Structure: Basis Splines and Confidence Intervals, Journal of Business Finance and Accounting, 18(4), 1991, pp.513 529. Svensson, Lars E. O., Estimating Forward Interest Rates with the Extended Nelson and Siegel Method, Sveriges Riksbank Quarterly Review, 3(1), 1995, pp.13 26. Tanggaard, Carsten, Nonparametric Smoothing of Yield Curves, Review of Quantitative Finance and Accounting, 9(3), 1997, pp.251 267. Vasicek, Oldrich A., and H. Gifford Fong, Term Structure Modeling Using Exponential Splines, Journal of Finance, 37(2), 1982, pp.339 348. Waggoner, Daniel F., Spline Methods for Extracting Interest Rate Curves from Coupon Bond Prices, Federal Reserve Bank of Atlanta Working Paper 97 1, 1997. 77
1 17 1 6 2 12 2 1 2 5 1 2 3 3 2 9 2 6 2 12 2 18 2 27 9 2 27 1 15 28 6 2 27 12 2... 17 23 18 3 2 2 8 2 5 2 11 2 78 /212.7
3 2 1 1 365 1 1 19 29 6 2 29 6 22 2 3... 19 21 3 3. 2 http://www.mof.go.jp/jgbs/auction/past auction schedule/result22.xls 212 5 1 29 6 2 1 211 212 29 6 22 79
21 3 2 21 1 1 15 1 1 2 23 1 7 2 9 15 3 7 2 9 2 2 21 21 6 22 23 1... 21 3 29 9 21 24 21 23 8 /212.7
1996 9 19 T +7 7 1997 4 1 T + 3 3 1999 1 21 12 4 21 3 g (c v i, N v i, T v i 1 ) 22 d g (c v i, N v i, T v i 1 ) = cv i N v i ( 1 2 + d ). 365 d g (c v i, N v i, T v i 1 ) = cv i N v i d 365.... 22 4.. 81
g (c v i, N v i, T v i 1 ) = cv i N v i, 2 2 21 3 d A v i A v i = c v i N v i d 365. d A v i = c v i N v i d 365 c v i N v i 2 d < 183 d 183, d d 82 /212.7
A v i = c v i N v i d 365 c v i N v i 2 d < 183 d 183. d 83
2 Steeley [1991] 1 3 Steeley [1991] 2 1 1 Steeley [1991] t = 4 1 2 1 15 1 2 1 15 1 1 27 1 2 1 1 1 2 t = V W 1 X Y 1 12 (X V) W +Y +1 n month V W X Y 2 2 t = 1 2 T 1 < T 2 < < T nmonth t = v i (i = 1,...n) V W X Y 2 2 v i 1 4 c v i := ( c v i 1,..., cv i n month ) T c v i 84 /212.7
B B B 2 4. B (k, x) (16) (k, j) B ( j 4, T k ) B = B ( 3, T 1 ) B (n knot 1, T 1 )..... B ( 3, T nmonth ) B (n knot 1, T nmonth ), (A-1) Steeley [1991] α = (α 3,...,α nknot 1) T T j Z (T j ) (15) Bα j α v i Q v i (α) Q v i (α) = ( c v i ) T Bα, (A-2) B B (A-1) 1 (A-1) B (A-2) 4 B 85
86 /212.7