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

商品流動性リスクの計量化に関する一考察（その2）―内生的流動性リスクを考慮したストレス・テスト―

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