ransacions of he Operaions Research Sociey of Japan 26 49 62-88 ( 24 3 26 ; 26 6 21 ) Morgage Backed Securiies, MBS MBS MBS MBS Collaeralized Morgage Obligaion, CMO CMO : 1. 21 5 [1] [3, 5] MBS (Morgage Backed Securiies) 197 GMNA (Governmen Naional Morgage Associaion, Ginnie Mae) [9] MBS 5 8 MBS [8] 1999 MBS [1] 62
63 MBS MBS MBS [3] Collaeralized Morgage Obligaion, CMO CMO MBS [5] CMO [5] MBS 2. MBS 3. 4. 5. 6. 7. [4]
64 2. 2.1. 1 1 2 +1 3 1 2.2. : ; C ( =1,...,): ; a ( =1,...,): ; v ( =1,..., 1) : ; a =(a 1,...,a ), v =(v 1,...,v ), a = a, v = v, C =(C 1,...,C ) 1: CMO A,B V 1 v = V = V =
65 A (a, v 1, C ) = min{a,c + V 1 (a 1, v 1, C 1 )} ( =1,...,) (1) V (a, v, C ) = min{v,c + V 1 (a 1, v 1, C 1 ) A (a, v 1, C )} (2) ( =1,..., 1) B (a, v, C ) = C + V 1 (a 1, v 1, C 1 ) A (a, v 1, C ) V (a, v, C ) (3) ( =1,...,) A (a, v 1, C ),V (a, v, C ),B (a, v, C ) A,V,B C a,v A,V,B 2.3. a A L (4) (2),(3) L (a, v 1, C ) := a A (4) = a min{a,c + V 1 } = max{,a C V 1 } (5) A = a L (6) V = min{v,c + V 1 + L a }, (7) B = C + V 1 + L a V (8) 2.4. r r (<r ) a =(a 1,...,a ) W (a) := =1 a (1 + r ) (9) 2.5. MBS W (a) W (a) r r ( =1,...,)
66 L v B MBS max a,v s.. 1 W (a) ρ v =1 [ ] E γ L (a, v 1, C ) U L, (1) =1 a, v γ,ρ U L E[X] X 6. γ U L MBS U L (a, v) (3) B (a, v, C ) 3. 3.1. (1) [ ] E γ L (a, v 1, C ) = γ L (a, v 1, c )p(c)dc (11) =1 p(c) C =(C 1,...,C ) C (1) (11) C c R =1
67 : ; r (i) =(r (i) 1,...,r (i) )(i =1,...,): ; c (i) =(c (i) 1,...,c (i) )(i =1,...,): r(i) (11) c (i) =(c (i) 1,...,c (i) ) [ ] E γ L (a, v 1, C ) 1 γ L (a, v 1, c (i) ) (12) =1 r (i), c (i) 3.2. m m Vasicek [7] =1 r + = r + ɛ (m r ) + σ B r ɛ σ B r r Cox-ngersoll-Ross (CR) [2] r + = r + ɛ (m r ) + σ r B (13) r r CR (13) r B r (i) 6. =.1 (13) {r } r r (i) 3.3. 3.3.1. x r y x = =1 y (1 + r ) y = x r (1 + r ) (1 + r ) 1
68 x ( =1,...,) x = k=+1 y (1 + r ) = x k (1 + r ) (1 + r ) (1 + r ) 1 (14) 3.3.2. Schwarz and orous [6] π π Schwarz and orous π π = ˆπ exp {β 1 k + β 2 l + β 3 m }, (15) ˆπ = κ ων(ω)ν 1 1+(ω), ν k = r r s, s, l = (k ) 3, ( ) x m = ln x β 1,β 2,β 3 κ, ω, ν r s s = x k l m MBS MBS 3.3.3. (14) x ( =1,...,) CR r (i) ( =1,...,,,...,) 1 x 1 π 1 x /x 1 x 1 x 1 r x 1 c (i) (i =1,...,) sep1. := 1, x := x,r (i) := r
69 sep2. r (i) 1, x 1,x 1 (15) π 1 sep3. sep4. x := x x 1 (1 π 1 ) x 1 c (i) := (x 1 x )+r x 1 = (1+r ) x 1 x sep5. = := +1 sep2 i =1,..., c (i) 3.4. (12) (5), (7) ( L =(L ) (1) 1,...,L (1),...,L() 1,...,L () ) V =(V (1),...,V (1),...,V(),...,V () ) = = (1) max a,v,l,v s.. 1 W (a) ρ v 1 =1 =1 { =max =min γ U L,, a c (i) } 1, (16) { } v,c (i) + 1 + a, a, v, = = (6) (8) A (i) = ( ) B (i) = B (i) 1,...,B (i) ( ) A (i) 1,...,A (i) A (i) = a, (17) B (i) = c (i) + 1 + a (18)
7 4. (16) (P 1 ) (P 1 ) max a,v,l,v s.. 1 W (a) ρ v 1 =1 =1 γ U L,, a c (i) V 1, (i) = or = a c (i) V 1, (i) (19) v, c (i) + 1 + a, = v or = c (i) + 1 + a, (2) a, v, = = (19),(2) (P 1 ) (19),(2) (24) max a,v,l,v s.. 1 W (a) ρ v 1 =1 =1 γ U L,, a c (i) V 1, (i) (21) v, (22) c (i) + 1 + a, (23), (24) a, v, (25) = = (23),(24) (21) (22),(24) (25)
71 (P 2 ) (P 2 ) max a,v,l,v s.. 1 W (a) ρ v 1 =1 c (i) =1 γ U L, (26) + 1 + a, (27) v,, a, = = (P 1 ) (P 2 ) (P 2 ) [1] 5. 5.1. 2 2 18 16 14 12 Cash Flow 8 6 4 2 5 1 15 2 25 3 ime 2: v
72 5.2. Z Z (15) π x 1 Z := π k x k k=1 v Z v (τ,ξ,z ):=τ + ξ Z ( =1,..., 1) v τ,ξ τ = (τ 1,...,τ ), ξ =(ξ 1,...,ξ ), τ = τ 1, ξ = ξ 1, Z =(Z 1,...,Z ), v (τ, ξ, Z )= (v 1 (τ 1, ξ 1, Z 1 ),...,v (τ, ξ, Z )) max a,τ,ξ s.. 1 W (a) ρ E[v (τ,ξ,z )] =1 [ ( ) ] E γ L a, v 1 (τ 1, ξ 1, Z 1 ), C U L, (28) =1 a, v (τ,ξ,z ) 3. max a,τ,ξ,l,v s.. W (a) 1 1 a, =1 1 ρ (τ + ξ z (i) ) =1 γ U L, { =max, a c (i) { =min τ + ξ z (i),c (i) } 1, (29) + 1 + a }, τ + ξ z (i), = = z (i) i max a,τ,ξ,l,v s.. W (a) 1 1 =1 c (i) 1 ρ (τ + ξ z (i) ) =1 γ U L, (3) + 1 + a, τ + ξ z (i),, a, = =
73 4. (3) (29) 6. 6.1. x = r =.5 =3 ; 5% CR ; r =.4 CR (13) (15) 1 1: CR ɛ.2 κ 1.5 m.5 ω.83 σ.2 ν 1.74 β 1 34.2 β 2 β 3.3 = 3 25 2 amoun ( * 1e+8 ) 15 5 5 1 15 2 25 3 year 3: = SMPLE NUOP Version 6 RedHa8 wih nel Penium 4 CPU 2.8GHz, 1 GB memory = [5] SMPLE, NUOP Mahemaical Sysems, nc.
74 45 6.2. v (1) γ,ρ γ := 1 (1 + r ), ρ := v W (v) ρ (1 + r ) W (v) := 1 =1 v (1 + r ) W (a) ρ W (v) ρ 6.2.1. ρ, U L U L =.1 ρ 1.1 W (a), W (v) 2 U L =.1 3 4 2, 3 W (v) W (a) 2: (U L =.1) ρ W (a) W (v) 1. 665.32912.16.9 67.356264 5.228164.8 67.356334 5.228175.7 675.934293 12.96512.6 675.934359 12.965192.5 687.47828 33.167633.4 7.98763 61.428.3 716.17988 12.53849.2 746.297396 228.257527.1 834.935921 862.815392. 964.916426 > 3: (U L =.1) ρ W (a) W (v) 1. 623.45397.497.9 656.148754 34.2963.8 656.148776 34.2962.7 656.14886 34.298.6 656.148863 34.2979.5 656.148866 34.2999.4 691.417999 18.86996.3 696.513779 125.377859.2 713.76386 2.22955.1 817.4247 96.52624. 951.11381 >
75 85 8 75 W(a) 7 65 U L =.1 U L =.1 6 2 3 4 5 6 7 8 9 W(v) 4: W (v) W (a) ( ) 2,3 W (v) W (a) ρ W (a) W (v) ρ W (a) W (v) ρ = ρ =1 <ρ<1 ρ 4 W (v) W (v) W (a) U L W (v) W (a) U L 6.2.2. 5 (U L,ρ )=(.1,.1) 5 a, v E[Ṽ ] E[Ṽ ] (35), (36) Ṽ (i) E[Ṽ ] := 1 Ṽ (i)
76 5 3 6 B (i) B (i) (18) 7 5 8 5 v E[Ṽ ] 7 4 5 > (U L,ρ )=(.1,.2) 9-12 5 (U L,ρ )=(.1,.1) 13-16 6 5 9 ρ v E[Ṽ ] 8, 12, 16 ( 15) =1 (15 3) 4-6 (U L,ρ )=(.1,.1).5% (U L,ρ )=(.1,.2).7% (U L,ρ )= (.1,.1).2% U L 6, 1, 14 MBS 6.2.3. (17) (18) A (i) B (i) Ṽ (i) 1+r E[W (A)], E[W (B)], E[W (Ṽ )] E[W (A)] := 1 =1 A (i) (1 + r ), E[W (B)] := 1 =1 B (i) (1 + r ), E[W (Ṽ )] := 1 1 Ṽ (i) (1 + r ) =1 E[W (A)] E[W (B)] 17 U L =.1 ρ 1 E[W (A)], E[W (B)], E[W (Ṽ )]
U L =.1, ρ =.1. 77 25 a: Amoun of PAC Bond E[ V ]: Expeced Reserve v: Upper Bound of Reserve L (i) : Loss 25 B (i) : Amoun of Companion Bond 2 2 amoun ( * 1e+8 ) 15 amoun ( * 1e+8 ) 15 5 5 5 1 15 2 25 3 5 1 15 2 25 3 5: a v E[V ], 6: B (i) 25 a: Amoun of PAC Bond L (i) : Loss 25 v: Upper Bound of Reserve E[ V ]: Expeced Reserve L (i) : Loss 2 2 amoun ( * 1e+8 ) 15 amoun ( * 1e+8 ) 15 5 5 5 1 15 2 25 3 5 1 15 2 25 3 7: a, 8: v E[V ], 4: 5 > 1 2 3 4 5 6 7 8 9 1 Probabiliy (%)...1.2.2.1.2.2.1.1 11 12 13 14 15 16 17 18 19 2.3.1.2.2.1.1..1.2.3 21 22 23 24 25 26 27 28 29 3.2.2.1.2.2.1.4.5.4.5
78 U L =.1, ρ =.2. 25 a: Amoun of PAC Bond E[ V ]: Expeced Reserve v: Upper Bound of Reserve L (i) : Loss 25 B (i) : Amoun of Companion Bond 2 2 amoun ( * 1e+8 ) 15 amoun ( * 1e+8 ) 15 5 5 5 1 15 2 25 3 5 1 15 2 25 3 9: a v E[V ], 1: B (i) 25 a: Amoun of PAC Bond L (i) : Loss 25 v: Upper Bound of Reserve E[ V ]: Expeced Reserve L (i) : Loss 2 2 amoun ( * 1e+8 ) 15 amoun ( * 1e+8 ) 15 5 5 5 1 15 2 25 3 5 1 15 2 25 3 11: a, 12: v E[V ], 5: 9 > 1 2 3 4 5 6 7 8 9 1 Probabiliy (%)..2.1.3.3.1.2.2.2.2 11 12 13 14 15 16 17 18 19 2.2.1.3.2.7.3.4.5.5.6 21 22 23 24 25 26 27 28 29 3.6.5.3.4.6.5.5.5.6.5
U L =.1,ρ =.1. 79 25 a: Amoun of PAC Bond E[ V ]: Expeced Reserve v: Upper Bound of Reserve L (i) : Loss 25 B (i) : Amoun of Companion Bond 2 2 amoun ( * 1e+8 ) 15 amoun ( * 1e+8 ) 15 5 5 5 1 15 2 25 3 5 1 15 2 25 3 13: a v E[V ], 14: B (i) 25 a: Amoun of PAC Bond L (i) : Loss 25 v: Upper Bound of Reserve E[ V ]: Expeced Reserve L (i) : Loss 2 2 amoun ( * 1e+8 ) 15 amoun ( * 1e+8 ) 15 5 5 5 1 15 2 25 3 5 1 15 2 25 3 15: a, 16: v E[V ], 6: 13 > 1 2 3 4 5 6 7 8 9 1 Probabiliy (%)...1.1.1.1...1. 11 12 13 14 15 16 17 18 19 2.1..1.1.....1.1 21 22 23 24 25 26 27 28 29 3.1........1.2
8 1 9 8 E[W (A)]+E[W (B)] 7 6 5 4 3 2 E[W (B)] E[W (A)] 2 4 6 8 12 14 16 E[W (V)] 17: E[W (Ṽ )], E[W (A)], E[W (B)] ( ) 17 E[W (Ṽ )] E[W (A)], E[W (B)] E[W (Ṽ )] E[W (A)] E[W (B)] Ṽ (i) (3) A + B = C + Ṽ 1 Ṽ ( =1,...,) r V A + B = C +(1+r V )Ṽ 1 Ṽ ( =1,...,) 1 (1+r ) =1,..., p(c) c R =1 A (1 + r ) p(c)dc + c R = c R =1 =1 B (1 + r ) p(c)dc c (1 + r ) p(c)dc + r V r 1+r c R =1 Ṽ p(c)dc (31) (1 + r ) c R =1 A (1 + r ) p(c)dc E[W (A)], c R =1 B (1 + r ) p(c)dc E[W (B)],
81 c R =1 c (1 + r ) p(c)dc = x, (31) c R =1 Ṽ (1 + r ) p(c)dc E[W (Ṽ )] E[W (A)] + E[W (B)] x + r V r 1+r E[W (Ṽ )] x =,r =.5 r V = r r E[W (A)] + E[W (B)] (32) r V = E[W (A)] + E[W (B)].476 E[W (Ṽ )] (33) E[W (Ṽ )] 6.3. v Z γ,ρ 6.3.1. 18 U L =.1 ρ 1 E[W (A)], E[W (B)], E[W (Ṽ )] 19 E[W (Ṽ )] E[W (A)] 18 E[W (A)] + E[W (B)] (33) (33) 19 E[W (Ṽ )] E[W (Ṽ )] E[W (A)] E[W (Ṽ )] E[W (Ṽ )] E[W (Ṽ )]
82 1 9 9 8 85 E[W (A)]+E[W (B)] 7 6 5 4 3 E[W (A)] 8 75 7 2 E[W (B)] E[W (A)] 2 4 6 8 12 14 16 E[W (V)] 65 Model 1 Model 2 6 2 4 6 8 12 14 16 E[W (V)] 18: E[W (Ṽ )], E[W (A)], E[W (B)] ( ) 19: 7. 1 [5] MBS
83 [1] V. Chváal: Linear Programming (W.H. Freeman, 1983). [2] J.C. Cox, J.E. ngersoll and S.A. Ross: A heory of he erm srucure of ineres raes. Economerica, 53 (1985), 383 48. [3] F.J. Fabozzi: he Handbook of Morgage-Backed Securiies (Probus Publishing Co, 1985). [4] D. Huang, Y. Kai, F.J. Fabozzi, and M. Fukushima: An opimal design of collaeralized morgage obligaion wih PAC-companion srucure using dynamic cash reserve. European Journal of Operaional Research, oappear. [5] : MBS., 1 (23), 2 25. [6] E.S. Schwarz and W.N. orous: Prepaymen and he variaion of morgage-backed securiies. he Journal of Finance, 44 (1989), 375 392. [7] O.A. Vasicek: An equilibrium characerizaion of he erm srucure. Journal of Financial Economics, 5 (1977), 177 188. [8] Federal Reserve Board: hp://www.federalreserve.gov/. [9] Ginnie Mae: hp://www.ginniemae.gov/. [1] : hp://www.jyukou.go.jp/. A 1. (P 2 ) (26) (a, v, L, V ) (P 2 ) 1 =1 γ < U L (34) δ 1 := U L 1 =1 γ >, 1 := 1 + δ 1 γ 1 (i =1,...,), ă 1 := a 1 + δ 1 γ 1
84 δ 1, 1 (i) L 1, ă 1 1, a 1 ( ) γ 1 L(i) 1 + =2 γ L (i) 1, ă 1 (ă, v, L, V ) ( ) = 1 γ 1 1 + δ 1 + γ = 1 (ă, v, L, V ) (26) =1 c (i) γ =2 + δ 1 =U L + 1 + a c (i) + 1 +( + δ 1 γ 1 ) (a + δ 1 γ 1 ), c (i) + (i) 1 + L ă (ă, v, L, V ) (27) (ă, v, L, V ) (P 2 ) W (ă) W (a) = δ 1 γ 1 (1 + r ) > (a, v, L, V ) (ă, v, L, V ) (a, v, L, V ) (P 2 ) (a, v, L, V ) (26) γ >γ +1 ( =1,..., 1) (P 1 ) (P 2 ) 1. (P 2 ) (a, v, L, V ) γ >γ +1 ( =1,..., 1) (a, v, L, Ṽ ) (P (1) 1) Ṽ =(Ṽ,...,Ṽ (1) (),...,Ṽ,...,Ṽ () ) Ṽ (i) 1 := min{v 1, c (i) 1 + 1 a 1 }, (35) Ṽ (i) := min{v, c (i) + Ṽ (i) 1 + a } ( =2,..., 1), (36) Ṽ (i) :=, Ṽ (i) := = (a, v, L, V ) 1 c (i) 1 + 1 a 1, 1 v 1 (35) Ṽ (i) 1 Ṽ (i) 1 Ṽ (i) 1 =2,..., 1 (36) Ṽ (i) 1 {2,..., 1} Ṽ (i) 1 1 Ṽ (i) = min{v,c (i) + Ṽ (i) 1 + a } min{v,c (i) + 1 + a }
85 (a, v, L, V ) (P 2 ) Ṽ (i) 1 (i) 1 Ṽ ( =2,..., 1) (a, v, L, Ṽ ) (P 2 ) (a, v, L, V ) (a, v, L, Ṽ ) (P 2 ) (P 2 ), a + c (i) 1 (37) (a, v, L, Ṽ ) (37) i 1 {1,...,}, 1 {1,...,} L (i 1) 1 > and L (i 1) 1 > a 1 + Ṽ (i 1) 1 c (i 1) 1 Ṽ (i 1) 1 1 δ 2 := min{l (i 1) 1, L (i 1) Ľ (i 1) 1 := L (i 1) 1 δ 2 + δ 2 =max 1 a 1 Ṽ (i 1) 1 + c (i 1) 1 + Ṽ (i 1) 1 1} >, { ( δ2, a 1 + δ ) 2 Ľ (i) 1 := 1 + δ 2 > (i =1,...,, i i 1), ǎ 1 := a 1 + δ 2 > δ 2, Ľ(i) 1, ǎ 1 1, a 1 Ľ(i) 1, ǎ 1 Ľ (i 1) 1 + Ṽ (i 1) 1 c (i 1) 1 Ṽ (i 1) 1 1 } >, (ǎ, v, Ľ, Ṽ ) { } δ2 =max, ǎ + Ṽ (i 1) 1 c (i 1) 1 Ṽ (i 1) 1 1 ǎ 1 + Ṽ (i 1) 1 c (i 1) 1 Ṽ (i 1) 1 1 1 a 1 + Ṽ (i) 1 c (i) 1 Ṽ (i) 1 1 (i =1,...,,i i 1 ) ( 1 + δ ) ( 2 a 1 + δ ) 2 + Ṽ (i) 1 c (i) 1 Ṽ (i) 1 1 (i =1,...,,i i 1 ), Ľ (i) 1 ǎ 1 + Ṽ (i) 1 c (i) 1 Ṽ (i) 1 1 (i =1,...,,i i 1 ) (ǎ, v, Ľ, Ṽ ) (27) Ľ (i) 1 = 1
86 (ǎ, v, Ľ, Ṽ ) (26) (ǎ, v, Ľ, Ṽ ) (P 2 ) W (ǎ) W (a) = δ 2 (1 + r ) 1 > (a, v, L, Ṽ ) (ǎ, v, Ľ, Ṽ ) (a, v, L, Ṽ ) (a, v, L, Ṽ ) (37) i {1,...,} {1,...,} J,K J := K := =max{, a + Ṽ (i) c (i) { j {1,...,} L (j) =, L (j) a + Ṽ (j) c (j) { k {1,...,} L (k), L (k) = a + Ṽ (k) c (k) Ṽ (i) 1} (38) } Ṽ (j) 1, } Ṽ (k) 1 (38) J K = {1,...,} S S c i J c K, {1,..., 1} Ṽ (i) = i 2 J c 2 K 2, 2 {1,..., 1} Ṽ (i 2) 2 > i 2 J c 2 K 2 L (i 2) 2 = a 2 + Ṽ (i 2) 2 c (i 2) 2 Ṽ (i 2) 2 1 > (39) } (i δ 3 := min {Ṽ 2 ) 2, a (i 2) 2 + Ṽ (i 2) 2 c (i 2) 2 Ṽ (i 2) 2 1 >, (4) ˆV (i 2) 2 := Ṽ (i 2) 2 δ 3 { } = max, c (i 2) 2 + Ṽ (i 2) 2 1 a (i 2) 2, ˆL (i 2) 2 := L (i 2) 2 δ 3 { } = max L (i 2) 2 Ṽ (i 2) 2, L (i 2) 2 a (i 2) 2 Ṽ (i 2) 2 + c (i 2) 2 + Ṽ (i 2) 2 1 { } = max L (i 2) 2 Ṽ (i 2) 2, (by (39)), ( ) ˆL (i 2) 2 +1 := L (i 2) 2 +1 + δ 3 > by L (i 2) 2 +1 and (4) δ 3, ˆV (i 2) 2, ˆL (i 2) 2, ˆL (i 2) 2 +1 Ṽ (i 2) 2, L (i 2) 2, L (i 2) 2 +1 ˆV (i 2) 2, ˆL (i 2) 2, ˆL (i 2) 2 +1 (a, v, ˆL, ˆV ) Ṽ (i 2) 2 v 2 ˆV (i 2) 2 <v 2 Ṽ (i 2) 2 c (i 2) 2 + Ṽ (i 2) 2 1 + L (i 2) 2 a 2 ) (i (Ṽ 2 ) 2 δ 3 ( ) c (i 2) 2 + Ṽ (i 2) 2 1 + L (i 2) 2 δ 3 a 2,
87 ˆV (i 2) 2 c (i 2) 2 + Ṽ (i 2) 2 1 + ˆL (i 2) 2 a 2 Ṽ (i 2) 2 +1 c (i 2) 2 +1 + Ṽ (i 2) 2 + L (i 2) 2 +1 a 2 +1 Ṽ (i 2) 2 +1 c (i 2) 2 +1 + (Ṽ (i 2 ) 2 δ 3 ) + = c (i 2) 2 +1 + ˆV (i 2) 2 + ˆL (i 2) 2 +1 a 2 +1 ( L (i 2) 2 +1 + δ 3 ) a 2 +1 (a, v, ˆL, ˆV ) (P 2 ) (a, v, ˆL, ˆV ) (a, v, L, Ṽ ) (a, v, ˆL, ˆV ) (P 2 ) (a, v, L, Ṽ ) (a, v, ˆL, ˆV ) 1 ˆ 1 1 := =1 γ = U L, (41) γ =1 ˆL (i) = U L (42) ((i, ) (i 2, 2 ), (i 2, 2 +1)) (41) (42) 1 ( ) γ 2 L (i 2) 2 + γ 2 +1 L (i 2) 2 +1 1 ( γ 2 ˆL (i 2) 2 + γ 2 +1 ˆL ) (i 2) 2 +1 =, (γ 2 γ 2 +1)δ 3 = (43) γ 2 >γ 2 +1 (4) i J c K, {1,..., 1} Ṽ (i) = (a, v, L, Ṽ ) (19) (35) (36) (a, v, L, Ṽ ) (2) (a, v, L, Ṽ ) (P 1 ) (P 1 ) (P 2 ) X 1,X 2 X 1 X 2 (P 1 ) (P 2 ) ( (P 1 ) ) ( (P 2 ) ) (44) (a, v, L, Ṽ ) (P 2 ) (P 1 ) (44) (P 1 ) 48-1192 41-1 E-mail: kusuna@mosk.ylabs.co.jp
88 ABSRAC OPMAL DESGN OF PAC-COMPANON SRUCURE FOR MORGAGE BACKED SECURES USNG CASH RESERVE akuro Kusuna Yoshiaka Kai Masao Fukushima oyoa Cenral R&D Labs.,nc. Kwansei Gakuin Universiy Kyoo Universiy Morgage backed securiy (MBS) is a produc of morgage securiizaion, which is issued backed by he repaymen cash flow from a loan pool consising of many morgage loans. Collaeralized morgage obligaion (CMO) is a form of MBS, in which repaymen cash flow from a loan pool is reorganized and bonds wih various risks are issued. n his paper, we propose a mehod of designing CMO wih PAC-companion srucure. We divide repaymen cash flow ino wo pars; a par in which principal repaymen schedule mus be saisfied (planned amorizaion class, PAC) and an unsable high-prepaymen risk par (companion). We allow he repaid cash o be reserved in order o repay PAC bondholders in he following periods. We formulae he problem of deermining an opimal PAC-companion srucure as a mahemaical programming problem and use a simulaion-based approach o approximae he problem. We show ha our model can be reformulaed as an equivalen linear programming problem. Furhermore, we propose a modified model which yields a higher performance han he basic model. Finally we conduc numerical experimens and repor he resuls.