Risk Analysis and Valuation of Collateralized Debt Obligations

Transcription

1 009 - Rsk Analyss and Valuaon of Collaeralzed Deb Oblgaons (Hsang-Hu Chu) * (Meng-Hsueh Wu) ** Bharah and Shumway (008) Meron Meron Copula (CDO) Gbson(004) Bharah and Shumway(008) Meron Meron Copula * 5454 (049)9-693 #46E-malhhchu@ncnu.edu.w ** E-mal: zfevsnz074@homal.w

2 009 Absrac Ths arcle we nvesgae he valuaon and rsk managemen ssues of collaeralzed deb oblgaons (CDOs). We use he naïve approach proposed by Bharah and Shumway(008) o avod smulaneously solvng he wo nonlnear equaons n Meron s model. And we consruc Copula funcons o descrbe he dependen srucure because he conagon effec of collaeral pool has an mporan mpac on far premum of dfferen ranches. The rsk of CDO ranches can be measured n varous ways, and we presen wo rsk measures by Gbson (004). The smulaed resuls show ha he equy ranche has relavely more rsk han ohers and he uncerany of realzed loss would become nsensve o he maury of CDOs. On he conrary, he proeced levels of he senor ranche would be gradually weak, hus s leverage numbers become more sensve. Fnally, n comparson wh Meron model, we fnd ha he ncreasng amoun of unexpeced loss relave o expeced loss compued by he naïve alernave model sgnfcanly declnes, so hs mples ha he accuracy of esmaed realzed loss ncreases. We conclude ha he undervalued defaul probables would be mproved by he naïve alernave model; ha s, predcng he defaul evens of CDO becomes more accurae. Keywords: CDOs, Meron Model, Copula Funcon, Conagon Effec, Leverage

3 (New Cenury Fnancal) (Fanne Mae) (Fredde Mac) (Lehman Brohers) (Specal Purpose VehcleSPV) (Collaeralzed Deb OblgaonCDO) 008 Senbruck (G7) G () Hull(008) CDO CDO CDO Meron : (00) (004) Meron (Conagon Effec) Bharah and Shumway (008) Meron Copula Meron Meron CDO CDO CDO CDO CDO 648

4 4 009 CDO (Srucural Form Models) (Reduced Form Models) Meron (974) Black and Sholes(973) Black-Scholes N( d ) Meron Black and Cox(976) (Frs Passage Tme Model) Meron model C( ) = C e υ ( T ) Longsaff and Schwarz(995) Leland and Tof(996) ( ) Colln-Dufresne and Goldsen(00) (Hazard Rae) (Posson Dsrbuon)Jarrow and Turnbull(995) (Inensy Model) Lando(998) Jarrow and Yu(00) Davs and Lo(00) Gesecke and Weber(003) (Homogeney) (Symmery) (Defaul Predcable) () Leland (994), Corporae deb value, bond covenans, and opmal capal srucure, Journal of fnance, Vol. 49, 3-5.

5 5 CDO L(000) Gaussan Copula (005) Copula CDO Suden- Copula Meneguzzo and Vecchao (004) (006) Shu-Yng Ln and Gang Shyy (008) (007) Meron (007) Meron CDO CDO 0 (008) CDO (Base Correlaon) CDS (007) CDS 60bps Meron Meron CDO Bharah and Shumway(008) Meron CDO CDO CDO n Sklar(959) F(,, n ) = P( τ, τ,, τ n n ) Copula F(, ) C( F ( ), F ( ), F ( )) n, n n n = Meron { u } * = F ( u ) = nf Fˆ ( ) CDO Gbson(004) CDO

6 6 009 CDO CDO CDO n (Noonal Amoun) A CDO A= = n A (recovery rae) R T (loss gven defaullgd) l = (- R ) A {} Ι (ndcaor funcon) Ι { * < T } T CDO L( ) = l * Ι = { < T} CDO { ϑ, ϑ, ϑ, ϑ } ϑ = 0 = ϑ0 < ϑ < ϑ < ϑ3 = A 0 3 ϑ. ( ],ϑ n ϑ 3 ϑ 06 ϑ ϑ 0. CDO ϑ 0 < L( ) ϑ (. ) (. bar) ϑ < L( ) ϑ (. ) ϑ ϑ0 L( ) ϑ. ( ( ) ) M ( ) = max mn L( ), ϑ ϑ,0 ϑ (aachmen pon)

7 7 ϑ (deachmen pon) ϑ ϑ B(0,) CDO (Defaul T Q Q LegDL) E ( DL) = E B ( 0, ) dm ( ) Q 0 ( ) E CDO T K s κ ( κ =,, TK ) W (Premum Leg W PL) π ( s κ ) π ( s κ ) sκ K ( ( ) ) π ( s ) = ϑ ϑ M ( s ) = mn max ϑ L( s ), 0, ϑ ϑ κ κ κ ϑ 0 < L( sκ ) ϑ sk (. ) CDO (. ) ϑ < L( sκ ) ϑ (. ) ϑ ϑ0. (PL=DL) CDO W W = ( 0, ) ( ) T Q K E B dm 0 ( ( ϑ ) ϑ ϑ ) TK κ = κ κ Q E B(0, s ) mn max L( s ), 0, () (Conagon Effec) Copula Copula Copula

8 8 009 Copula (3) Copula C( ) { n } { C u = u} grounded C ( u,,0,, u ) = 0, u [0,] margn (,,,,,) n,, n U (0,) ( ) C( u, u,, u ) = Pr U u, U u, U u n n n n-ncreasng ( u ) F ( ) Sklar(959) n F (,, n ) F ( ) F (,, n ) F(, ) C( F ( ), F ( ), F ( )) = Copula =, n n n ( ) ( ) f, c u, u f ( ) () = n f ( ) Copula, n,, n (co-movemen) Copula GaussanSuden- Archmedean Copulas Gaussan Copula (Tal Dependence) C G Σ Gaussan Copula Copula ( u,,u ) = Φ ( Φ ( u ),, Φ ( u )) ( ) n Σ n Φ Σ Φ ( ) (, ) λ = lm p( > F ( u ) > F (u )) = 0 U u λ = lm p( F (u ) F (u )) = 0 L u + 0 ρ 0 Gaussan Copula Meneguzzo and Vecchao (004) Gaussan Copula (a) Cholesky ψψ Σ (b) n z = ( z,z, ) ' (c) u ( x ψz) zn = Φ = 3 Cherubn, Umbero, Elsa Lucano, and Waler Vecchao (004), Copula Mehods n Fnance, New York: Wley.

9 9 Suden- Copula Copula CΣ,v ( u,,un ) = Σ,v ( v ( u ),,v ( un )) Σ, ( ) ( ) Suden- Copula v λ U ( v + )( ρ ) 0.5 = λl = v+ [( ) ] Gaussan Copula + ρ Suden- Copula Gaussan Copula(a)(b) z s Suden- Copula u u = ( y = v v s x ) Archmedean Copula C( u, u, u ) = ϕ [ ϕ( u ) + ϕ( u ) + + ϕ( u )] n ϕ( ) Archmedean Copula Clayon Copula ϕ( u) = u α wh α >0 n = α / α ϕ ( ) ( ) = + Clayon Copula C n α C ( u,,u ) = u n + Clayon Copula α α α (al dependence) λ = λ Clayon L U (Condonal Samplng) (,, ) = ( =, = ) C u u u P U u U u U u k k k k k k k k (, ) (, ) C u u C u u = / u u u u u u k k k k k k k n ϕ = ϕ ( k ) ( k ) ( c ) k ( c ) k ( k ) ϕ ( ) ϕ ( ) k (4) ck = ϕ( u ) + ϕ( u) + + ϕ( uk ) ' ( u, u,, un ) (unform dsrbuon) ' ( v, v,, vn ) v k u = = C ( u v ) v ( k) ϕ ( ) k ( α + )( α + ) ( α + k ) 4ϕ ( ) = = ( ) ( + ) k α k α

10 0 009 v ϕ ( ) ( c ) = ( ) ϕ ( c ) u α α α + / α ( v ( v ) + ) = u n α α α ( n ) n ( u n + ) ( vn ) + = / α = Copula (007) Canoncal Maxmum Lkelhood(CML) Copula r, r, û, 0 û, =,,, n =,,, T Gaussan Copula T ˆ ' Σ = ξ ξ = ξ = ( Φ ( û ),, Φ ( ûn )) Suden- Copula T Kendall s au( ˆτ ) ˆΣ Σˆ π, = sn τˆ, CML Suden- Copula ( ) ( ) (,, ) (, ) = n M ˆ = arg max log c uˆ,, uˆ ; v, Σˆ () c ( u,, u ) = n ( ξ,, ξn ) ν ( ξ, ) Σ, ν,, c ( uˆ,, u ˆ ;, ˆ ) ξ Σ = Σˆ n n + n ' ˆ ξ ξ Γ Γ + Σ, n, n + + n ξ, Γ Γ ( + ) = ' ( ξ,,, ξ n, ) = ( û ), =, ξ ( ) Γ Gamma Clayon Copula Suden- Copula > 0 M = C ( (, n, )) ˆ = arg max log c u ˆ,, u ˆ ; α Γ + n c u u α α u u n Γ α n n n α C n α α α,, n ; = + = = ( ˆ, ˆ, ) * u ( ) nf ˆ( ) ˆ { } = F u = F u F ( ) Bharah and Shumway (008) Meron

11 Meron Meron(974) Black and Scholes(973) (V, E ) (V, A ) D Black and Scholes(973) V (0) = V (0) Φ( d ) D e Φ ( d ) (3) r, E, A d ln(v =,A ( 0 ) / D ) + ( r + 0.5σ σ,a,a ) d ln(v,a( 0 ) / D ) + ( r 0.5σ,A ) = = d, A σ σ,a (GBM) Io s Lemma Q dv ( ) = ( f + Φ ( d ) rv f σ V ) d + Φ( d ) σ V dw ( ), f, E, A, A, A, A, A f ( ) f V, A f ( ) σ E f V, A V = Aσ A Φ ( d) (5) V E (3)(5) Newon-Raphson Mehod Fˆ( ) = Φ ( DD ( )) = Φ ( d ) (6) Crobe and Bohn(00) (5) Bharah and Shumway (008) Bharah and Shumway (008) 5% 5% ( σ = σ ), E A E D V, E + D V, E + D D E V D σ = σ + σ (7) ( V, A = V, E + D ) (6) ln( V, A / D ) ( r 0.5 σ, A) F + ( ) = Φ ( ) σ, A

12 009 Hull(008) AAA Gbson(004) CDO Gbson(004) E[ M ( )] E[ M ( )] ϑ ϑ A E[ M ( )] E[ M ( )] M ( ) E[ M ( )] SD = + ( ) SD = E M ( ) E( M ( ) M ( ) M ( ) ϑ ϑ A Meron Gaussan, Suden- Clayon Copula Gbson (004) CDO Meron CDO CDO

13 3 (Inernaonal Index CompanyIIC) (5) BBB CDO CDO 3. CDO CDO 5 0,000. (ϑ - ) (ϑ ) ( ) (Equy) 0 3% 9.6 (Junor) 3% 6% 9.6 (Mezzanae) 6% 0%.8 (Senor) 0% 00% 88 Meron(974) (TEJ) Meron Newon-Raphson Mehod (3)(5) Copula CML GaussanSuden- Clayon Copula( 6) Copula ' ( u, u,, un ) { } ( ) nf ( ) = F u = F u () * ˆ 5CDS Fch/Moody s/s&p BBB-/Baa3/BBB- 6 Clayon Copula α=.54

14 ,89,836, ,49,000, ,596,43, ,747,39, ,45,000, ,44,49, ,357,993, ,98,900, ,947,68, ,700,706, ,058,000, ,86,898,000 49,045,603, ,433,800, ,49,645,000 66,9,54, ,077,500, ,34,606,000 4,37,499, ,750,400, ,45,034,000 68,409,84, ,033,00, ,855,649,500 67,86,803, ,,800, ,049,866,000 57,483,5, ,40,000, ,65,745,500 0,76,039, ,9,00, ,570,943,000 6,499,6, ,383,00, ,5,900 9,48,94, ,457,400, ,30,68,000 0,609,33, ,53,850, ,86,580,500 58,5,508, ,980,900, ,64,549,500 74,33,34, ,77,050, ,500,050,500 96,547,790, ,695,500, ,84,6,000 86,99,88, ,499,50, ,86,0,000 63,66,359, ,6,450, ,84,790,000 63,979,70, ,670,000, ,46,09,000 43,836,733, ,43,000, ,5,084,000,683,784,0, ,634,77,40, ,99,789,500 9,300,906, ,709,550, ,3,470,000 5,46,000, ,96,000, ,6,595,500 3,854,899, ,664,000, ,5,30, ,0,074, ,056,900, ,564,353, ,37,365, ,80,00, ,505,366,000 57,83,97, ,74,50, ,346,94,000,376,064,605, ,0,560,000, ,798,544,000 7,05,6, ,84,000, ,496,9,000 69,64,98, ,96,800, ,079,6,500 90,30,74, ,966,00, ,564,043, KMV / (.85%) TEJ New-Raphson Meron

15 5 CDO CDO 3 GaussanSuden- Clayon 3 Copula 3. Meron Meron Gaussan Suden- Clayon Gaussan Suden- Clayon 4.483% 4.087%.7744% 7.94% 7.355% %.5447%.4386%.343% 8.543% 8.387% % % % % % % 3.80% 0.08% 0.07% % 0.083% 0.74% 0.579% Copula 3 Gaussan Copula Suden- CopulaClayon Copula Clayon Copula Gaussan Copula Clayon Suden- Copula Clayon Copula Gaussan Copula Suden- Clayon Copula (007) (008) Meron Bharah and Shumway (008) Meron 3 3 Meron Copula

16 Meron Meron Gaussan Copula Suden- Copula Clayon Copula

17 7 0.4 Galan(003) Clayon Copula Gbson(004) CDO CDO 4 4 Copula 4 Panel Copula Gaussan Copula Suden- Copula Clayon Copula Clayon Gaussan Copula 4. Panel A: Meron Copula EL UL EL UL EL UL EL UL Gaussan (9.49%) (57.45%) (7.8%) (3.3%) (.64%) (7.6%) (0.06%) (0.79%) Suden (8.6%) (55.9%) (6.69%) (9.85%) (.8%) (7.99%) (0.08%) (0.95%) Clayon (8.7%) (34.46%) (5.8%) (6.58%) (3.75%) (.0%) (0.43%) (3.7%) Panel B: Meron Gaussan (56.04%) (0.6%) (33.67%) (78.5%) (0.69%) (58.69%) (0.99%) (4.7%) Suden (54.48%) (0.7%) (33.8%) (77.67%) (0.66%) (58.75%) (.08%) (4.67%) Clayon (30.36%) (74.35%) (.5%) (60.30%) (6.58%) (5.6%) (.48%) (0.66%) EL(Expeced Loss) UL(Unexpeced Loss) (.)

18 8 009 Meron Meron Meron 4 Panel A B Meron Meron Meron Meron Meron Meron CDO ( 7) 5 Panel A CDO (009) 5 Panel B Hull(008) Bharah and Shumway (008) Meron CDO Copula Gbson (004) CDO CDO Suden- Clayon Copula Gaussan Copula Clayon Copula 7 Meron CDO Meron

19 5. Meron CDO Panel A: 9 Gaussan Suden- Clayon Gaussan Suden- Clayon Gaussan Suden- Clayon Gaussan Suden- Clayon Panel B:

20 0 009 Meron CDO CDO

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