RMBS CDO (ABS ABS CDO IMF , , IMF, Global Financial Stability Report 1) Alt-A 4,250 A
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1 ABS CDO ABS (CDO) ABS CDO 6 7 RMBS ABS ABS ABS CDO CDO 1 ABS CDO ABS CDO (B)
2 RMBS CDO (ABS ABS CDO IMF , , IMF, Global Financial Stability Report 1) Alt-A 4,250 ABS ABS CDO 9,800 2/3 72.5% ABS CDO ABS ABS CDO ABS CDO 6 7 RMBS ABS RMBS RMBS CDO CDO 1)
3 ABS CDO ABS CDO 2 CDO ABS CDO CDO CDO CDO CDO CDS 2000 RMBS ABS CDO CDO RMBS 2) ABS CDO CDO ABS RMBS RMBS ABS CDO 1 CDO ABS CDO ABS CDO 2007 ABS CDO 2) ABS CDO Gorton (2009) 3
4 (10 億ドル ) ハイイールドローン投資適格債仕組債その他 Q1-Q Q3-Q Q1-Q Q3-Q Q1-Q Q3-Q Q1-Q Q3-Q4 1: CDO 51% RMBS CMBS CMO ABS CDO CDS SIFMA ABS CDO CDO RMBS RMBS CDO ABS CDO 2.2 RMBS t i L i (t) L i (t) A i i LGD i i τ i i L i (t) = A i LGD i ι {τi t} (1) ι t τ 1 0 3) 3) RMBS 4
5 N L L(t) = N L i (t) = i=1 N A i LGD i ι {τi t} (2) i=1 (2) A i LGD i τ i G(τ 1,, τ N ) G i (τ i ) τ i 4) 5) V i (i = 1,, N) τ 1,, τ N G i (τ i V ) G(τ 1,, τ N V ) = N G i (τ i V ) (3) G i (τ i V ) G(τ 1,, τ N V ) V G V i=1 4) CDO Laurent and Gregory (2005) Li (2008) 5) McNeil et al. (2005) 1 ρ 1 Heitfield (2008) 5
6 G(τ 1,, τ N ) = = i=1 G(τ 1,, τ N V = v)dg V (v) N G i (τ i V = v)dg V (v) (4) X i τ i τ i = G 1 ( i Φ(Xi ) ) Φ V X 1,, X N τ 1,, τ N X i (5) X i = ρ i V + 1 ρ i ε i (i = 1,, N) (5) V V ε i 6) 1 1 ρ i [0, 1] X i i ρ i Corr(X i, X j ) Corr(τ i, τ j ) 7) G i (τ i V ) G i (τ i V = v) = Prob{X i Φ 1( G i (τ i ) ) V = v} { = Prob ε i Φ 1( G i (τ i ) ) } ρ i v 1 ρi = Φ ( Φ 1 ( G i (τ i ) ) ) ρ i v 1 ρi (6) 8) 3 6) G i (τ i ) [0, 1] X i = Φ 1`G i (τ i ) 7) 2 Corr(X i, X j ) = ρ i ρ j ρ Corr(X i, X j ) = ρ ρ i = a 2 i X i = a i V + p 1 a 2 i ε i ρ i 0.1 a i 0.3 8) (6) (4) Li (2000) 6
7 LGD i (i = 1,, N) G i ( ) (i = 1,, N) ρ i (i = 1,, N) ρ i i 2.3 RMBS CDO RMBS RMBS RMBS RMBS 1 RMBS RMBS RMBS RMBS RMBS CDO RMBS RMBS CDO CDO RMBS CDO 1 3 RMBS CDO RMBS RMBS RMBS CDO 3.1 RMBS N RMBS M CDO RMBS 3 RMBS CDO CDO 7
8 RMBS N RMBS M CDO CDO RMBS 3 RMBS CDO 9) 1% 1 CDO 3.2 1,000 N = 1, 000 3% 5 LGD = ρ = 0.1 RMBS CDO 5 10,000 Multiply-With-Carry 10) 3.3 9) 10) Ox 8
9 頻度 14% 12% 1 8% 6% 4% 2% 頻度累積頻度 累積頻度 損失率 2: % 7% 99% 99% 2 RMBS 8 1% 11) ρ 3 ρ % RMBS ) Greenlaw et al. (2008) A 9
10 頻度 5 4 ρ=0 ρ=0.1 ρ= 損失率 3: 頻度 10 8 住宅ローンプール RMBSシニア RMBSメザニン RMBSエクイティ 超過損失率 4: RMBS % 0.2% 2 0.2% 97% 1 10
11 対数頻度 頻度 住宅ローンプール RMBS シニア RMBS メザニン RMBS エクイティ % 損失率 5: RMBS ) RMBS CDO RMBS RMBS 10 CDO CDO 10 M = 10 RMBS RMBS CDO 13) 6 CDO RMBS 99% 7 5 1% 3 12) 13) 11
12 頻度 頻度累積頻度 累積頻度 損失率 6: CDO RMBS 頻度 5 4 RMBS メザニンプール 住宅ローンプール 超過損失率 7: RMBS CDO 1% RMBS CDO RMBS RMBS CDO RMBS 12
13 頻度 4 3 RMBSメザニンプール CDOシニア CDOメザニン CDOエクイティ 超過損失率 8: CDO : 99%VaR 99%ES % 6% 17.8% 19.5% 0.3% RMBS 0.2% RMBS % 77.5% % RMBS % CDO 1.3% 0.7% 20.3% 63.3% 1. CDO 14.1% 5.8% % CDO 24.7% 17.4% % 14) 8 CDO 5 RMBS CDO RMBS 4 CDO 1 RMBS 0.2% CDO 1.3% 14.1% 24.7% 14) CDO CDO Whetten and Adelson (2005) 13
14 % VaR 4 99% ES 99% RMBS CDO RMBS 1 2 RMBS 5.4% RMBS RMBS CDO 1% 99%VaR RMBS RMBS 99% CDO 99% 20.3% RMBS 3 CDO RMBS CDO CDO RMBS 99% 63.3% CDO 99%VaR 17.8% CDO % 4.5% 1 CDO % RMBS CDO RMBS 14
15 2: 99%VaR 99%ES % 8% 23.4% 25.7% 3.4% (±0.) (+3.3%) (+2%) (+5.6%) (+6.2%) (+3.2%) RMBS 3.4% 0.1% 4.2% 7.2% 0. (+3.1%) (+0.1%) (±) (+4.2%) (+7.1%) (+0.) RMBS 47.1% % (+28.1%) (+13.7%) (±) (+22.5%) (+10.) (+21.5%) RMBS % % (0.) (+18.1%) (±) (±0.) (±0.) (+2.6%) CDO % % (+7.7%) (+5.1%) (±) (+79.7%) (+36.7%) (+6.5%) CDO 39.1% 21.4% % (+25.) (+15.6%) (±) (±0.) (±0.) (+21.) CDO 55.3% 44.8% % (+30.7%) (+27.3%) (±) (±0.) (±0.) (+28.7%) CDO % RMBS 1% ( RMBS CDO 9% RMBS CDO CDO 3 ρ RMBS RMBS CDO CDO 99% 10 15
16 3: 99%VaR 99%ES % 7.1% 1% 39.4% 43.2% 10.7% (-5.2%) (+0.1%) (-5%) (+21.7%) (+23.7%) (+10.4%) RMBS 10.6% 1.2% 24.3% % (+10.4%) (+1.2%) (±) (+24.3%) (+28.9%) (+1.8%) RMBS 24.9% 16.9% % (+5.9%) (+11.5%) (±) (+22.5%) (+10.) (+10.6%) RMBS 94.9% 45.1% % (-5.2%) (-20.) (±) (±0.) (±0.) (-37.3%) CDO 13.9% 12.1% % (+12.6%) (+11.4%) (±) (+79.7%) (+36.7%) (+12.3%) CDO 22.8% 17.9% % (+8.6%) (+12.2%) (±) (±0.) (±0.) (+11.1%) CDO % % (+2.4%) (+6.8%) (±) (±0.) (±0.) (+5.6%) 4: RMBS 99%VaR 99%ES 2 CDO 1.6% 0.9% 38.3% 75.6% 1.2% (+0.3%) (+0.2%) () (+17.9%) (+12.3%) (+0.2%) CDO 14.2% 6.1% % (+0.) (+0.3%) () (0.) (0.) (+0.2%) CDO 29.9% 17.8% % (+5.3%) (+0.3%) () (0.) (0.) (+0.5%) 3.5 RMBS RMBS RMBS RMBS CDO CDO RMBS RMBS CDO 4 RMBS RMBS CDO 16
17 15) 4 ABS CDO 2 16) ,000 RMBS 10 CDO RMBS CDO RMBS RMBS RMBS RMBS CDO 2 ABS CDO BIS (2008) Whetten and Adelson (2005) 15) RMBS 16) BIS (2008) ABS CDO ABS CDO ABS CDO RMBS BIS (2008) CDO RMBS 10 1, 000 = 10, 000 RMBS 10 17
18 損失率 ローンプール 損失率 RMBS メザニン 証券化 小 システマティック リスク 大 小 システマティック リスク 大 分散化 損失率 RMBS メザニンプール 損失率 CDO メザニン 小 システマティック リスク 大 再証券化 小 システマティック リスク 大 9: CDO RMBS CDO RMBS CDO V = v 1 Φ(v) CDO CDO
19 損失率 10 8 基準ケース トランシェの規模 : 大 信用補完 : 大 小 システマティック リスク 大 10: CDO ) ) L A B A l l = max{l A, 0} max{l B, 0} B A l dl/dl L < A A < L < B 1/(B A) B A 19
20 損失率 RMBS シニア 損失率 CDO シニア 基準ケースデフォルト確率 : 大デフォルト相関 : 大 小 システマティック リスク 大 小 システマティック リスク 大 損失率 RMBS メザニン 損失率 CDO メザニン 小 システマティック リスク 大 小 システマティック リスク 大 損失率 RMBS エクイティ 損失率 CDO エクイティ 小 システマティック リスク 大 小 システマティック リスク 大 11: 3% % RMBS RMBS CDO 3.4 RMBS CDO RMBS CDO RMBS CDO 20
21 CDO RMBS RMBS CDO RMBS CDO 5 CDO 2007 RMBS (SEC) S&P 3 RMBS CDO RMBS CDO 2002 RMBS CDO 21
22 ABS CDO CDO ABS RMBS RMBS CDO RMBS 5 CDO 3 RMBS 18) RMBS RMBS 5.2 BIS(2008) CDO CDO BIS (2008) 18) RMBS CDO CDO CDO CDO 22
23 II ABS CDO CDO 1 ABS CDO
24 ABS ABS CDO 24
25 1 RMBS 1 i (= 1,, N) τ i τ i G i ρ G i (i = 1,, N) ρ τ i (i = 1,, N) 1. V ε i (i = 1,, N) V N(0, 1), ε i N(0, 1) (i = 1,, N) 2. X i = Φ 1( G i (τ i ) ) Φ X i = ρv + 1 ρε i 3. τ i τ i = G 1 ( i Φ(Xi ) ) t = 1,, T D(t) N(t) N D(t) = ι {τi =t} ι i=1 N(t) = N(t 1) D(t), N(0) = N RMBS j (= 1,, M) RMBS FSj R (t) F Mj R (t) FEj R (t) C R S CR M N j (t) 1 I(t) LGD 25
26 t = 0,, T 1 1. RMBS t+1 N j (t)i(t) RMBS FSj R (t)cr S + FMj R (t)cr M spr R j (t + 1) spr R j (t + 1) = N j (t)i(t) (F R Sj(t)C R S + F R Mj(t)C R M) 2. L R j (t + 1) L R j (t + 1) = (N j (t) N j (t + 1)) 1 LGD Loss R Ej (t + 1) Loss R Ej(t + 1) = min{f R Ej(t) + spr R j (t + 1), L R j (t + 1)} t + 1 F R Ej(t + 1) = F R Ej(t) + spr R j (t + 1) Loss R Ej(t + 1) 3. Loss R Mj (t + 1) { } Loss R Mj(t + 1) = min FMj(t), R max{0, L R j (t + 1) FEj(t) R spr R j (t + 1)} t + 1 F R Mj(t + 1) = F R Mj(t) Loss R Mj(t + 1) 4. Loss R Sj (t + 1) { } Loss R Sj(t + 1) = min FSj(t), R max{0, L R j (t + 1) FEj(t) R spr R j (t + 1) FMj(t)} R t + 1 F R Sj(t + 1) = F R Sj(t) Loss R Sj(t + 1) CDO CDO FS C(t) F M C (t) FE C (t) 26
27 C C S CC M t = 0,, T 1 1. CDO t + 1 RMBS M j=1 F R Mj (t)cr M CDO F C S (t)cc S +F C M (t)cc M spr C (t+1) spr C (t + 1) = M FMj(t)C R M R (FS C (t)cs C + FM(t)C C M) C j=1 2. CDO L C (t + 1) RMBS L C (t + 1) = M Loss R Mj(t + 1) j=1 Loss C E (t + 1) Loss C E(t + 1) = min{f C E (t) + spr C (t + 1), L C (t + 1)} t + 1 F C E (t + 1) = F C E (t) + spr C (t + 1) Loss C E(t + 1) 3. Loss C M (t + 1) { } Loss C M(t + 1) = min FM(t), C max{0, L C (t + 1) FE C (t) spr C (t + 1)} t + 1 F C M(t + 1) = F C M(t) Loss C M(t + 1) 4. Loss C S (t + 1) { } Loss C S (t + 1) = min FS C (t), max{0, L C (t + 1) FE C (t) spr C (t + 1) FM(t)} C t + 1 F C S (t + 1) = F C S (t) Loss C S (t + 1) 27
28 N I(t) G(t) ρ LGD RMBS CDO F S R, F M R, F E R, F S C, F M C, F E C, RMBS CDO CS R, CR M, CC S, CC M, RMBS CDO T 10,000 Multiply-With-Carry 2 ρ ρ τ h h 1 e ht (t 0) G(t) = Pr{τ t} = 0 (t < 0) G G 1 (s) = 1 log(1 s), (0 s < 1) h E[τ] = 1 h, Var(τ) = 1 h 2 τ i τ j h i h j Corr(τ i, τ j ) = E[τ iτ j ] E[τ i ]E[τ j ] Var(τi )Var(τ j ) = E[τ iτ j ] 1 h i h j 1 h i h j = h i h j E[τ i τ j ] 1 28
29 1 ρ X i X j τ i = G 1( Φ(X i ) ), τ j = G 1( Φ(X j ) ) E[τ i τ j ] = E[G 1( Φ(X i ) ) G 1( Φ(X j ) ) ] = 1 h i h j E[log ( 1 Φ(X i ) ) log ( 1 Φ(X j ) ) ] Corr(τ i, τ j ) = E[log ( 1 Φ(X i ) ) log ( 1 Φ(X j ) ) ] 1 = log ( 1 Φ(x) ) log ( 1 Φ(y) ) ϕ(x, y; ρ)dxdy 1 ϕ(x, y; ρ) ρ { } 1 ϕ(x, y; ρ) = 2π 1 ρ exp 1 2 2(1 ρ 2 ) (x2 2ρxy + y 2 ) 4 x 4 4 y 4 dx dy ρ ρ = 0.5 ρ 2 1 数係関相の刻時トルォフデ ρ ( 状態変数の相関係数 ) 12: ρ 45 29
30 [1] Ashcraft, Adam B. and Til Schuermann (2008), Understanding the Securitization of Subprime Mortgage Credit, Federal Reserve Bank of New York Staff Reports. [2] Bank for International Settlements (2008), Credit Risk Transfer. [3] Bank for International Settlements (2009), Consultative Document, Proposed enhancements to the Basel II framework. [4] Das, Satyajit (2005), Credit Derivatives, CDOs & Structured Credit Products 3rd Edition, John Wiley & Sons. [5] Doornik, Jurgen A. (2001), Ox An Object-Oriented Matrix Programming Language 4th ed., Timberlake Press. [6] Gorton, Gary (2009), The Subprime Panic, European Financial Management 15, [7] Greenlaw, David, Jan Hatzius, Anil K. Kashyap and Hyun Song Shin (2008), Leveraged Losses: Lessons from the Mortgage Market Meltdown, presented at the U.S. Monetary Policy Forum [8] Heitfield, Erik (2008), Securitization: a Post Mortem, presented at the Community Development and the Capital Markets. ( [9] International Monetary Fund (2008), Global Financial Stability Report. [10] International Monetary Fund (2009), Global Financial Stability Report Market Update. [11] Laurent, Jean-Paul and Jon Gregory (2005), Basket default swaps, CDOs and factor copulas, Journal of Risk 7, [12] Li, David Xianglin (2000), On Default Correlation: A Copula Function Approach, The Journal of Fixed Income 9, [13] Li, David Xianglin (2008), An Overview on Copula Function Methods in Credit Portfolio Modelling, in The Definitive Guide to CDOs edited by Gunter Meissner, Incisive Media. [14] McNeil, Alexander J., Rudiger Frey, and Paul Embrechts (2005), Quantitative Risk Management: Concepts, Techniques and Tools, Princeton University Press.,,, [15] Standard & Poor s (2002), Global Cash Flow and Synthetic CDO Criteria, Standard & Poor s. 30
31 [16] United States Securities and Exchange Commision (2008), Summary Report of Issues Identified in the Commision Staff s Examinations of Select Credit Rating Agencies. [17] Whetten, Michiko and Mark Adelson (2005), CDOs Squared Demystified, Nomura Fixed Income Reseach. 31
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