本邦国債価格データを用いたゼロ・クーポン・イールド・カーブ推定手法の比較分析

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1 Steeley [1991]... JAFEE 35 TMU NEEDS / /

2 McCulloch [1971, 1975] Steeley [1991] Tanggaard [1997] Schaefer [1981] Nelson and Siegel [1987] Svensson [1995] Jonathan Wright Svensson [1995] 36 /212.7

3 Bank for International Settlements [25] Ioannides [23] Kalev [24] 1989 Oda [1996]

4 1 Steeley [1991] Steeley [1991] Steeley [1991] Steeley [1991] t T 1 Z (t, T) t T y (t, T) y (t, T) = 1 T t log (Z (t, T)), (1) /212.7

5 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

6 ( ) 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) /212.7

7 1 2 A B A B 2 2 A B B A B 41

8 3 B 2 B /212.7

9 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

10 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) Z (x) Bank for International Settlements [25] McCulloch [1975] Steeley [1991] /212.7

11 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

12 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 ) (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

13 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

14 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

15 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

16 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

17 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] 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

18 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

19 (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

20 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 McCulloch [1975] Nelson and Siegel [1987] /212.7

21 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]

22 Steeley [1991] u l = l (l = 3,..., 33) /32 = /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 = /31 = /31 = Steeley [1991] McCulloch [1975] Steeley [1991] Tanggard [1997] McCulloch [1971] McCulloch [1975] Steeley [1991] 56 /212.7

23 McCulloch [1971, 1975] Steeley [1991] McCulloch [1971, 1975] Steeley [1991] 4 Vasicek and Fong [1982] McCulloch and Kochin [2] 13 McCulloch [1971, 1975] Steeley [1991] 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

24 a (a) (b) (%) (%) 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

25 b 5 a (%) n GJGBn Index FRB Svensson [1995] Gükaynak, Sack, and Wright [27] FRB

26 /212.7

27 (42) 39 j=2 (y (.5(j + 1)) 2y (.5 j) + y (.5(j 1)) ) 2. (42) NEEDS

28 /1 2/1 1/1 2/1 3/1 4/1 5/1 6/1 7/1 8/1 9/1 1/1 / /1 2/1 1/1 2/1 3/1 4/1 5/1 6/1 7/1 8/1 9/1 1/1 / Oda [1996] /212.7

29 2 4 8 McCulloch [1975] McCulloch [1975] Steeley [1991] Steeley [1991] Fisher, Nychka, and Zervos [1995] McCulloch [1971] McCulloch [1971] Fisher, Nychka, and Zervos [1995] Steeley [1991] Steeley [1991] Tanggaard [1997] 2 4. Schaefer [1981] Nelson and Siegel [1987] 2 4. Svensson [1995]

30 McCulloch [1975] Steeley [1991] McCulloch [1971] Steeley [1991] Schaefer [1981] Tanggaard [1997] Nelson and Siegel [1987] Svensson [1995] 1,84 3 2, , ,947 8 (%) Nelson and Siegel [1987] Svensson [1995] Tanggaard [1997] Schaefer [1981] /212.7

31 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] , ±2 65

32 9 2.5 (%) 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

33 3 McCulloch [1975] Steeley [1991] McCulloch [1975] Steeley [1991] McCulloch [1975] Steeley [1991] McCulloch [1971] Steeley [1991] (42) Steeley [1991] 4 Steeley [1991] 4 4 Steeley [1991] 1 Steeley [1991] 67

34 1 Steeley [1991] 3. (%) Steeley [1991] 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

35 1 Nelson and Siegel [1987] 11 Steeley NS Steeley Steeley NS 6 8 Steeley 13 Steeley NS NS 6 Steeley (%) Steeley NS 69

36 (%) Steeley NS (%) /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 Steeley 6 JGB 1 JGB JGB 7 /212.7

37 Steeley 14 Steeley NS bp 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

38 (%) 1 (%) /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 (%) (%) /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

39 { 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 (%) JRY Steeley 4 (%) JRY Steeley 73

40 7 JRY Steeley Steeley JRY Steeley [1991] Steeley Steeley 1 74 /212.7

41 21 Steeley

42 No. 212-J 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 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, 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 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 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 Kalev, Petko S., Estimating and Interpreting Zero Coupon and Forward Rates: Australia, , Working Paper, Monash University, 24. McCulloch, J. Huston, Measuring the Term Structure of Interest Rates, Journal of Business, 44(1), 1971, pp , The Tax-adjusted Yield Curve, Journal of Finance, 3(3),1975, pp , 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 /212.7

43 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, Ruppert, David, Empirical-bias Bandwidths for Local Polynomial Nonparametric Regression and Density Estimation, Journal of the American Statistical Association, 92(439), 1997, pp 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 Steeley, James M., Estimating the Gilt-edged Term Structure: Basis Splines and Confidence Intervals, Journal of Business Finance and Accounting, 18(4), 1991, pp Svensson, Lars E. O., Estimating Forward Interest Rates with the Extended Nelson and Siegel Method, Sveriges Riksbank Quarterly Review, 3(1), 1995, pp Tanggaard, Carsten, Nonparametric Smoothing of Yield Curves, Review of Quantitative Finance and Accounting, 9(3), 1997, pp Vasicek, Oldrich A., and H. Gifford Fong, Term Structure Modeling Using Exponential Splines, Journal of Finance, 37(2), 1982, pp Waggoner, Daniel F., Spline Methods for Extracting Interest Rate Curves from Coupon Bond Prices, Federal Reserve Bank of Atlanta Working Paper 97 1,

44 /212.7

45 auction schedule/result22.xls

46 /212.7

47 T T 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 ( d ). 365 d g (c v i, N v i, T v i 1 ) = cv i N v i d

48 g (c v i, N v i, T v i 1 ) = cv i N v i, 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

49 A v i = c v i N v i d 365 c v i N v i 2 d < 183 d 183. d 83

50 2 Steeley [1991] 1 3 Steeley [1991] Steeley [1991] t = 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

51 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

52 86 /212.7

1 Nelson-Siegel Nelson and Siegel(1987) 3 Nelson-Siegel 3 Nelson-Siegel 2 3 Nelson-Siegel 2 Nelson-Siegel Litterman and Scheinkman(199

1 Nelson-Siegel Nelson and Siegel(1987) 3 Nelson-Siegel 3 Nelson-Siegel 2 3 Nelson-Siegel 2 Nelson-Siegel Litterman and Scheinkman(199 Nelson-Siegel Nelson-Siegel 1992 2007 15 1 Nelson and Siegel(1987) 2 FF VAR 1996 FF B) 1 Nelson-Siegel 15 90 1 Nelson and Siegel(1987) 3 Nelson-Siegel 3 Nelson-Siegel 2 3 Nelson-Siegel 2 Nelson-Siegel