CDO CDO CDO CDO CDO CDO E-mail: kiyotaka_komiya@btm.co.jp
CDOcollateralized debt obligation CDO CDO CDO CDO CDOCDO CDO CDO CDO CDOCDO CDO CDO CDO CDO CDO CDOABSasset backed securities CBOcollateralized bond obligation CLOcollateralized loan obligation CDOCDO
CDO ABSCDO CDO CDO CDO CDO CDO
CDO CDO CDO CDO SPVspecial purpose vehicle SPVCDOCDO BIS CDO CDO CDOCDO SPV CDO Libor + CDO CDO SPVCDO CDO CDO SPV CDO SPV Fabozzi and Goodman CDO
CDO CDOCDO CDO CDOCDOCDO CDO SPV CDScredit default swap TRStotal return swap CDS CDOCDS CDS CDO CDOSPV bankruptcyfailure to payrestructuring ISDAhttp://www.isda.org > CDOCDS
CDOCDO CDO CDS CDO CDO CDO CDO SPV CDSSPV SPV CDS CDS
CDOCDO CDO CDO CDOCDO CBO CDO
CDO CDO CDO CDO CDO CDO CDO CDO CDS CDO CDO CDS CDS CDOCLO CLO CLO CDO
CDO CDO CDO CDS CDSCDO CDO CDO CDO CDS CDO BIS CDO BISGoodman CDO CDSCDO CDS
CDO CDS OECD CDO CDO CDO CDO CDO
CDO p i p j i j p i j a ij p ( p ) ( p i j ) i j = a ij u exp a ij uv + v ( a ij ) dudv uv (.) i j d ij d pi j pi p j ij = p ( p ) p ( p ) i i j j p i p j i j d ij d ij
CDO CDO J. P. Morgan & Co. Nagpal and Bahar CDO
CDO Cifuentes and O ConnorFinger FingerLi Duffie and Gârleanu Cifuentes and O Connorbinomial expansion technique modelcifuentes and O Connor CDO CDO N X i i i pn j N j j N j C p ( p)
Cifuentes and O Connordiversity score MM N i (i = N ) i CDOTL i R i n i L i (T) = ( R i )n i {i T} L (T) N i =L i (T) i T p i N E [ L( T)] = ( R ) n p i = i i i V [L (T)]V [L i (T)] = ( R i ) n i p i ( p i ) N V [ L( T)] = V[ L i( T)] + i = N i j i j d ij V [ L ( T )] V[ L i j ( T)] d ij i j d ij i j p i j d ij = i j p ( p ) i p p pj i i p ( p ) j j. MM M N R p E[ L ( T)] = ( R ) NMp V [ L( T)] = ( R) N M p ( p). {i T} i T
N = N i = n i /M R = N i = n i R i /( NM) N ( Ri) ni p i p = = ( R ) NM i M E[ L( T)] ( R ) ni i = = V[ L( T)] N E[ L( T )] p i = p j d ij = N M = + ( N ) alternative diversity score Finger CDO i {X i (t)} t X i (t) i (t) X i (t) Backman and O Connor
(.)( i (t)) CDO i p i () ( i ())= p i () i () i j d ij Z i () iz i ()< i () CDO CDOCDO i p i () ( i ())= p i () i () i j d ij Z i () i Z i ()< i () Z i () Z i ()Z i () q i () i ()= (q i ()) i Z i ()< i () i t p i (t)(t )t q i (t) (p i (t ) p i (t ))/( p i (t ))
CDO copula X X n F F n F [ F ( X ) F ( x ) F ( X ) F ( x )] F( x xn ) = P n n n n = C F ( x ) F n( x ( n )) C F (X ) F n (X n ) 0 C0 F(t)= P( t) U = F (U) F F n U U n P t t ] = P[ U F ( t ) U F ( t )] C( F t ) F n t )) [ n n n n n = ( ( n = F (U ) n = F n (U n ) Nelsen Li
normal (n) (. ;) ( n) n n n n C( u u ) = P( U u U u ) = ( ( u ) ( u ); ) (.) i j d ij Z (i) U i = (Z (i) ). = F i ( i) i ( Ui ) = Fi ( ( Z )). n i j t () ( (F i (t)) (F j (t)) ;) ij Merton a ij ij = a ijschmidt and Ward
i p i () p i () p i (T) k = T i (k)= (p i (k)) i j d ij Z(i) i (k )< Z (i) i (k) i k Gumbel CDO CDO C C( u u u / p u p ) = exp( {( lnu ) + + ( ln ) } ) n > [0 ] VW F(x) = e x w0 u w = F (u) = ln ( u)
sin(( ) V / ) = W sin( V / ) sin( V ). 0 U U n k = n X X n X k k ln( U k)} = F (exp[ { ]). Marshall and Olkin Marshall and Olkin LT (s)= E [e s ] F( x x x ) = E [ H ( x ) H ( x ) H n = LT ( LT ( F ( x )) + + LT ( x n n ) ( F ( x n ] n ))) LT(s) = exp ( s / )Kanter LT(s) exp ( s / ) / H ( x ) k k = exp( LT ( F ( x k k ))) for k = n H k (X k ) 0 X k XY (X Y ) cov( X Y) ( X Y) = V[ X ] V[ Y] cov(x Y ) XYV[.] Joe
XY XY Duffie and Gârleanudefault intensity t (t) t t Embrechts McNeil and Straumann
(t) ( σ l )basic affine W J(t) t W l t t + s (s) (s)e t t CDO Duffie and Gârleanu Duffie and GârleanuDuffie and Singleton. t t t t t P t + < ) ( ) ( [ ] ) ( ) ( ) ( ) ( ) ( t J t dw t dt t t d + + = ) ( ) ( ) ( exp ) ( t s s s t t u t t e du E s t P + + = = + > s c d c e d c d c b d c s s b ln ) ( ) ( + + + = s l c l d c e d c d c b d c a l s b + + + + ln ) ( s b s b e d c e s ) ( + = + = b + + = = d + c d a = b b = c c = c d d + =. c
X( X σ l X )Y( Y σ l Y )XY X+Y( σ l ) = X + Y l = l X + l Y N N = X + X i C i X C NX i i X C X i ( C σ l C )( i σ l i )X X N X C i ( σ l ) = C + i l = l C + l i t i j jump lc = l lc = l + l C i i j jump = 0 i j X C Duffie and Gârleanu jump i jump ij X C X i X j J C J i J j i j J i = J C +J i J j = J C +J i cov[ J J ] i j jump ( J J ) i = j V[ J ] V [ J ] i j jump( J J ) j. i = l C t l = C l t l t l Schönbucher
Duffie and Gârleanu ( σ l ) TP 0T f T ( ) + ( T) (0) 0 T ) 0 P T = ( T) e + f ( u) ( u du (T )T ( u) = d du P( > u) = e [ ( u) + ( u) (0) ] ( u ) + ( u) (0) ( u) ( u) = ( u) + l ( u) = ( u) + ( u) ( u) CDO i i X ( t) = X ( t t) + ( X ( t t)) t + X ( t t) t J ( t) C C C C C C + X ( t ) = X ( t t) + ( X ( t t)) t + X ( t t) t J ( t ). i i i i i i + i C C i N (0 ) mk Jk( t) = 0 pr = l t pr = l t m k (k = C i ) i X C X i t t i (t) t N + N + [0 ] U0 m k = ln ( U)
CDO CDO CDO CDO CDO p0 p/00
CDO CDO CDO CDO
R&I http://www.r-i.co.jp Nagpal and Bahar
CDO
d ij E[L E ]p 7 00 k 00 k 9k k 00 k C p p C E] = 7 + k p p k= 0 00 ( ) k = 8 00 ( ) 70 E[ L
CDO k 9 k/70 k k MR qe[l E ] M ) k M k k( R E[ L ] = C p ( p) min E M k 0 qm k= ME[L E ]E[L E ]M M > 0 E [L E (M+)] E[L E (M)] 0 Rq pe[l E ]p E[L E (M)]E[L E (M+)] E[L E (M)] E[L E (M)]M E[L E (M+)] E[L E (M)] 0 Mp E[L E (M+)] E[L E (M)] MM
E[L E (M)] E[L E (M + )] E[L E (M)]
p i 55 C i p i ( p ) 55 i CDO CDS CDO Backman and O Connor
CDOCDO
CDS CDO CDO CDOCDO CDO CDO CDS
CDO CDO CDOCDO CDOCDO
CDO CDO
CDO CDO CDO CDO Howard Lee and Mancini CDO
CDO overcollateralization testoccdo CDO CDO CDOCDO minimum net worth testmnwcdo
CDO CDO Moody s Japan K.K CDO Backman A. and G. O Connor Rating Cash Flow Transactions Backed by Corporate Debt 995 Update Moody s Investors Service 995. Cifuentes A. and G. O Connor The Binomial Expansion Method Applied to CBO/CLO Analysis Moody s Investors Service 996. Duffie D. and N. Gârleanu Risk and Valuation of Collateralized Debt Obligations Financial Analyst Journal 57 () 00 pp. 4-59. and K. J. Singleton Credit Risk Pricing Measurement and Management Princeton University Press 003. Embrechts P. A. McNeil and D. Straumann Correlation: Pitfalls and Alternatives Risk (5) May 999 pp. 69-7. Fabozzi F. J. and L. S. Goodman Investing in Collateralized Debt Obligations John Wiley & Sons 00. Finger C. C. A comparison of stochastic default rate models Working Paper The RiskMetrics Group 000. Goodman L. S. Synthetic CDOs: An Introduction The Journal of Derivatives 9 (3) 00 pp. 60-7. Howard D. J. Lee and M. J. Mancini Market Value CBO/CLO Rating Criteria Fitch Ratings 999. Joe H. Multivariate Models and Dependence Concepts Chapman & Hall 997. J. P. Morgan & Co. CreditMetrics TM Technical Document April 997. Kanter M. Stable Densities Under Change of Scale and Total Variation Inequalities The Annals of Probability 3 (4) 975 pp. 697-707.
Li D. X. On Default Correlation: A Copula Function Approach The Journal of Fixed Income 9 (4) 000 pp. 43-54. Marshall A. W. and I. Olkin Families of Multivariate Distributions Journal of the American Statistical Association 83 (403) 988 pp. 834-84. Merton R. On the pricing of corporate debt: the risk structure of interest rates The Journal of Finance 9 974 pp. 449-470. Nagpal K. and R. Bahar Measuring default correlation Risk 4 (3) March 00 pp. 9-3. Nelsen R. B. An Introduction to Copulas Springer New York 999. Schönbucher P. Term Structure Modelling of Defaultable Bonds Review of Derivatives Research (/3) 998 pp. 6-9. Schmidt W. and I. Ward Pricing default baskets Risk 5 () January 00 pp. -4.