risk analysis and valuation of collateralized debt...

�� 2009� 12�� 1-21 �

�

��

Risk Analysis and Valuation of Collateralized Debt Obligations

�� (Hsiang-Hui Chu) *

� � � � � � � � � � � � � �

� � (Meng-Hsueh Wu)**

� � � � � � � � � � � �

��

�� Bharath and Shumway (2008) � � Merton�� ! "# $ % & �' ( ) * +", -. / 012 3 4 5 Merton�� ! 6 7 8 9% : ; < = > : ?@ A 1 Copula B !C D ( ) * + 6 E ", -F G 90H I J K LM N ( ) O P(CDO) 6 Q R S � T U ?��VW Gibson(2004) � � X Y Z [ "\ ] 0X ^ M N ( ) O P_ ` X Y 6 a b c ! ?�' _` X Y 6 d e f g 0h i j a b k l ) m X Y n o a p 0q r s t u , -a b "v w x 9

0yz ) m X Y "a b { | } ! l ~ � � E 6 � � � � j � � �� X Y

0� � � � "� � � � "N � � � � 0l , -� � "� � : � F l �� 0�a b { | }

! s E ¡ > I ¢ £ ?¤ ¥ 0��¦ j 1 Bharath and Shumway(2008) % & � < § 6 7 ¨ �a b / l ¨ � a b / "© ª « � F l ¬ ® Merton �� u 0f g ¯ ° ± ² ³ ´ , -� �0l h i j a b "� � µ w � F l �� ?

� � �� M N ( ) O P�Merton�� Copula B ! �4 ¶ · ¸ �{ | } !

��*� � � � � � � � � � � � �� 54542 � ! " # $ � � % �& ' ( )(049)291-2693

#4622E-mail)[email protected]* **� � � � � � � � �� + , E-mail: [email protected]*

2 ��2009� 12��

�

�

Abstract

This article we investigate the valuation and risk management issues of collateralized debt obligations (CDOs). We use the naïve approach proposed by Bharath and Shumway(2008) to avoid simultaneously solving the two nonlinear equations in Merton’s model. And we construct Copula functions to describe the dependent structure because the contagion effect of collateral pool has an important impact on fair premium of different tranches. The risk of CDO tranches can be measured in various ways, and we present two risk measures by Gibson (2004). The simulated results show that the equity tranche has relatively more risk than others and the uncertainty of realized loss would become insensitive to the maturity of CDOs. On the contrary, the protected levels of the senior tranche would be gradually weak, thus its leverage numbers become more sensitive. Finally, in comparison with Merton model, we find that the increasing amount of unexpected loss relative to expected loss computed by the naïve alternative model significantly declines, so this implies that the accuracy of estimated realized loss increases. We conclude that the undervalued default probabilities would be improved by the naïve alternative model; that is, predicting the default events of CDO becomes more accurate.

Keywords: CDOs, Merton Model, Copula Function, Contagion Effect, Leverage

- . / 0 1 2 3 4 5 6 7 8 9 : � 3�

�

�� 2007 ¹ 4 º 2 »¼ ¬½ ¾ Z [ £ � 0¿ � ÀÁ � � ½ ¾ Q Â Ã Ä Å � Q Â (New

Century Financial) Æ Ç È É 0¿ � Ê Ë ¾ Ì Z [ Í LÎ Ï 0Ð Ñ � � ½ ¾ Z Ò "Ó Ô �Õ2008¹ 7º 13»¿ � Ö ½ ¾ . × 0½ Ø ¿ (Fannie Mae)Ù½ Ú ¿ (Freddie Mac) Û ¦ � �Ü . �Õ 2008¹ 9º 15»Ý Þ Àß � à á â ã ä å æ (Lehman Brothers) Æ ç è é È ÉN � 0ê ë · ¸ ì í î ï "Ý Þ � ð ñ ?Ý Þ � ð ñ o y¤ ò óô � Ý Þ õ ö S �

÷ ø 9ù ú "û ü ý þ 0�� á â � � � ½ ¾ ¡ ¼ PY õ 0� � � � �u � � 9"( Y

� à É ( ¢ � � � � � ". × (Special Purpose Vehicle�SPV)0� � L � @ A � ��j "à � � � ¾ 0� � � � � M N ( ) O P(Collateralized Debt Obligation�CDO)" �yc ! · e � 0� 1 � ö � � 6 ?

Õ 2008 ¹� 0 ! � � " # � Steinbruck ¬ $ % & � (G7) � " # ' ( ¢ g 0¨ DG7y� � ½ ¾ Z Ò � n o "a b - 4000 ) ¿ * � + ' Õ 2008¹ 1º 29»Q Ç 0Î , � É & ¼ ¬ - . � � ½ ¾ F G ù ú 0� / ¨ D a b -Ã Î 0 237.03 ) * (11)? 2 3 QÑ à 4 � 5 6 "! 3 5 D 02008 ¹ 12 ºÎ , � 7 8 Q Â "a b 9 : h ; Ã Î 0 156.5) * 0I < ¹� = ¥ > Ø � � v � ? ) * 0 @ f S � Z [ + � " A ó9?Hull(2008) B ôC D E �F G � à ] " H I J � S � K L " L � 0 M M N O ù ú P ¥ "{ | c ! 0z Q

CDO K T �� " R × v S T U 0A V W à X Y Z P0yz 0[\ µ ] " K T M N ( ) OPo ô ^ _ Ùi � _ A ó" ( ` ?

�� a b ( ) * + 6 E ", -F G 90@ c > d e ¢ Q Ñ "à 4 0R � C � M N ( )

O P6 K T �� 0 f g ° �� CDO Q R S � T U 6 µ w 9�@ A D § �Z [ � � \ ] 0a b CDO _ ` X Y h i "a b { | } ! ?> Y j � S � Z [ �� # $ \ L ® Merton�� "kl v m i � d e 0[:kl à É n o / p � q r X s 0 ! "� � % ; Ù7 8 9%: ; < = > : " t � · ¸ 0Ö u v n o u C , -. / " w ° 9 � �� u 0x y z (2002)Ù { & | �} ÷ ~ Ù j � � (2004) � � � � � & ô � �6 i P# $ ¦ j ® Merton�� 6, -¨ 5 � u ¬ ' D à 4 �� S � K X �� ?ÕS � ÷ ø 9ù ú "�� a b � à * +

�0¤ A ó"V! ô* + � z ", -F G 90 � � , -� � 6 E ~ Õ4 ¶ · ¸ (Contagion Effect)?yz ��VW Bharath and Shumway (2008) � �"Merton�� 0� à É d � Ù,-Z [ � ê d 0@ Ø � Copula B ! � � � ( ) * + 6 E ", -F G 90C § L` � ( �� ", -. / ?

d e f g 0F l ® Merton�� 0� �"Merton�� ¯ ° ± ² ³ ´ , -� � 0l hi j "a b C D ¯ ôµ w 0yz CDO _ ` X Y C § L"S � T U ¯ ô � �0��# $ CDO" K T �� C ¯ ôµ w "� / K T � � ?��" � � � � ô�À�� S � Z [ �

� Ù CDO K T 6 F G # $ �ÀV� ¡ # $ % & �À¢�ô�' d e ÙX ^ { | c ! �À£�9 d # $ d e ?

��

M N ( ) O P CDO � ¤ � ` S � Z [ "à É 0¥ S � Z [ 9à É ", -. / Ùà É6 E "F G 9�¦ x < § "a b ¨ ©0H I ª « CDO " K T ?yz v ¬ � à . × � ®

��;1)< = > ? @ � � A B C D E F G 1648 *

4 ��2009� 12��

�

�

� ) m X Y "¦ â . × I ¯ 0( ) * + "S � Z [ K T ³ � A ó?S � Z [ �� ° ° µ

w Ú ± ² ( ) * + _ Q Â ", -F G 90ô CDO K T d e " C G ± ?

�� S � Z [ �� ò ó¦ ² o Ö ³ ´ � 0 C ôd × ; �� (Structural Form Models)0 µ C

³ ô� � ; �� (Reduced Form Models)? d × ; �� ¶ w · Q Â T � �� 0Merton�� (1974) í ¸ Black and Sholes(1973) ¹ º

¬Q Â T � 0�� » ô C 1Q Â à É ô] "A ¼ â T ô ½ ¾ Q Â ( " ¿ ; À ) ?� ( ) �

� »s Q Â à É T � Á ö C ` Q ø 9" Â Ã � » ô, -0@ ¸ � Black-Scholes Q ; < §Q Â ) m : ( � "T � 0H I < LQ Â , -. / 2( )N d− ?

d × ; �� "¥ � ¦ ² ò ó � Ä Merton �� v m � "kl0Å�� Æ Ç È d e Éi ¨ Ê ? ® �� Ë a b ( ) � � s 0� & ° . , -� � É ø 0¥ Black and Cox(1976)�L Ì � � Í > �� (First Passage Time Model)0 Î U Merton model Ë Ï � � � »", -

. / A ( T � Ð s E I � " V b 0l � Â Ã � ô C \ ! B ! 0 )()( tTeCtC −−

⋅=υ ?

Longstaff and Schwartz(1995) ¦ ² Á y Ñ �� ¬Z [ 9 ( T � 0Ï � Ð Z [ Ø / Ò ¬ .> : A ª � È É o �Ó¬�� 0, - Â Ã Ô x ô Q ø ! A � � Õ m C x r "à �d ×

kl?Leland and Toft(1996) X ^ Q Â ( T � Ù¤ � à �d × Õr C � � klÖ 0Ï � = ×(Ø )Ù È É o � Á A ó ª « y Ñ J K C � ø " , - Â Ã ? Collin-Dufresne and Goldstein(2001)l � � ] � � { | Ù / ô C . > : A Ú Û Q Â T � " Ü 0 w Å d × 9�� Æ ª C Ý i PX ^ ?

� � ; �� ôÜ [ / (Hazard Rate)�� 6 í ¸ 0 � Þ ¬ ß _ Ø � 5 D ! 3 � ¨ 5 àá ¦ ø / " C ³ % & 0I ¥ â � � � ã ä ?¼ ¬, -� � "¦ ø Õ m å æ A ç è w ! "�

� x é 0¥ê ë ì í î X s (Poisson Distribution)0Jarrow and Turnbull(1995)1�ô � � Ì� �L� � ; �� 0¶ · 6 , - ï � �� (Intensity Model)?1� � ; �� , -. / "� ð 0Lando(1998) � Q ñ � Ø / y ò �Jarrow and Yu(2001) ó X ^ v r Q Â E S � Z [ FG 9l , -n o " ª « �Davis and Lo(2001)Ù Giesecke and Weber(2003)�L w Ø � kl QÂ ô r õ 9(Homogeneity)Ùl · 9(Symmetry)" ö � ÷ � ø ù = 0ú F l "z ³ kl�ûb C D 9?

4 5 d × ; �� " ü ýÕ¬ ô . þ � � é A r s w ê d 8 � d e 0V b ó Ð & N P

v ~ Õ � Ø . ' 1ö , - w ¨ 5 9(Default Predictable) � K Ý " � � S � Z [ T U ô ½ "j � 0v Õ d e ¢ i � � � � "(12)�� ; �� " ü ýÕ¬ K T S � ÷ ø 9ù ú ¢ ¯ ��0� c > 9 § ¡ � ! 3 � ô ª « S � T U " ! 0� � à X " � ¯ ô � � A ¯ ô s ·

90ú i � ¢ d e @ 7 T ¿ d e 0 � ù ú " � � � � � 9Z [ "Ò 0¥w ° ~ Õ

C x 6 � Ø � E 0@ A S � Z [ ò ó c > Q ø ¦ x ", -. / Ù � � / � ; ¯ v ô ¡ �

þ � 0¼ ¬, -. / Ú W É & � Q Â S � K L I � 0� É & � Q Â Ð & � � S � . × 6

K L 0 � S � . × K L "à X � Æ Ã 0 v Ð & � w Ú � ¸ Õ K T ¢ ?�

��2�Leland (1994), “Corporate debt value, bond covenants, and optimal capital structure,” Journal of finance, Vol.

49, 1213-1252.

- . / 0 1 2 3 4 5 6 7 8 9 : � 5�

�

��CDO � � � � � � � � � � � � � � � � � � � � Li(2000) Ì � � ~ � s E ", -F G � ! klôà É n o / 6 E "F G � ! 0¡ ¼

Gaussian Copula � ± ² à É ", -s ý?� � � �� (2005) F G v r Copula � l CDOH â K T ÙZ [ 7 + "� � � X ^ 0� à X ô . � � : � � * j � 0 F G Student-t Copulaô¯ ! "C D B ! ?Meneguzzo and Vecchiato (2004) �ê " # (2006) Ù Shu-Ying Lin and Gang Shyy (2008) # $ S � T U l , -F G 9"� � X ^ 0$ ) m �� X Y I ¯ 0S � T U Ù, -F G � ! � � % & ' 0I � � � � X Y 6 S � T U ó Ù, -F G � ! Ò ¬

� % G � ? ( ) * (2007) Ø � Merton�� à É d � Ù, -Z [ � ê d 0v + c > � � ", -

, m -à X Ù( � � " - . , -à X 0� w C § ` � à É ", -. / ?�' X ^ � à X /

X ] "+ Ùl W + 00 P( � � "S � K L ôª « X Y S � T U "ò óy Ñ �� ¦ â u N

O ( � � � z 6 E ", -F G 90_ X Y � × 1 "Q R S � T U �' 2 u C 0I © ª � à

u "Z [ �Õ q r X s "klÖ 0u C , -. / n o _ ` X Y "S � T U É ø 3 4 ?5 6

7(2007) � � Merton�� CDO 6 S � T U 0@ A a b CDO X Y 6 { | · e ?1 10} { | � à X Y s 0- . � "� à n o ' � ( ) * + , -Q Â 8 ! "© ª I � æ �_ `

X Y Ï 9 ( ) * + ", -s ý : ; 0� < � "Ø ¾ < � � n o / � = u �� , - 8 ! =

�0 � � � à u ¬_ ` > ¾ s ýv < � Ø ¾ < m 0 < § � à n o / ? w ° Ï 9 � " ¨

©?@ A B (2008) < § CDO" h i F G � ! (Base Correlation)0ú ¼ ¬Î , d e � . CDS\ ! �0� 1 c > ( ) * (2007)"# $ �' klÖ 0J § ( ) * + "\ ! S � T U ?�i Pd e 0 CDS \ ! "S � T U © ª 0 ÷ � < § " h i F G � ! ` ! : �0 þ � \ !T U � î 60bps1¢ 0C � � . X Y w D § L m � " h i F G � ! 0D E \ ! S � T U "u C 0� ¸ ® Merton S � Z [ �� u C F , -. / ?

2 3 ¢ � "� ð � � 0 G H w 1¦ j Õ ® Merton �� klÖ 0u C , -. /" w ° 9¯ � 0 þ I Ú ª « î CDO _ ` X Y S � T U "µ w � ?yz �� Bharath and Shumway(2008) � �" Merton �� CDO K T �� 0�' CDO ( ) * + ", -sý0t u , -. / u C " w ° 901�£ _ X Y Q R T U 6 µ w � ?

��

��kl CDO ( ) * + �� i n 8 Q Â 0 2 3 Sklar(1959) x � 0( ) * + , -s ý" J m X s ( ) ( )nnn tttPtttF ≤≤≤= τττ ,,,,, 221121 …… w 1¼ ` � Q Â , -X s ö n K "Copula B ! ; o ( ) ( ) ( ) ( )( ) ,,,, 221121 nnn tFtFtFCtttF �� = 0�Ls ® Ù � � " Merton

� � � ' L ( � � ", - s ý ( ) ( ){ }iiiii utFtuFt ≥== − ˆ|inf1* 0<§ CDO "Q R S � T U ?

¤ ¥ Ú W Gibson(2004)0� ' L "¨ � a b ô � � 0x é L CDO "Z [ � � \ ] 0 a b

Ö ` � � P ¥ h i "a b { | } ! ?

6 ��2009 � 12 ��

�

�

��CDO � � �� CDO ( ) * + � M C ] "à É ¦ ø , - � � 0( ) � ¤ G N "ô° O � � æ � 0

�� Ï P CDO "( ) * + i . n ` ] "à É 0� ` � Q � � (Notional Amount)ô Ai�

CDO "¦ â 9 Ï � ô A=∑ =

n

1iAi�, - � � / (recovery rate)ô Ri0�À i 8 Q Â Õ sE ý

T 6 R ¦ ø , - s0l ( ) * + n o " > a b (loss given default�LGD)ô )R-(1 i=il Ai?

¼ ¬ v r ( ) 6 , - s ý v È F r 0��S {}⋅Ι ô C ` \ ] B ! (indicator function)0I { }*it T<

Ι

! � ô 1 g À i 8 Q Â Õ sE ý T 6 R , - 0ó CDO 9 T U a b ô { }*1( )

i

n

ii t TL t l

= <= Ι∑ ?

CDO _ X Y ¦ â L � w Ú � M , - a b " � ¥ � Ó 0 _ x C + a b 9 _ ý

{ }0 1 2 3, , ,ϑ ϑ ϑ ϑ ϑ= � / X 0� � 0 1 2 30 ϑ ϑ ϑ ϑ= < < < = A0IÀ j ³ L � "X Y a b � � � �

ô ( ]j1j ,ϑϑ−

0� a b X s¨ © 1 V 1. g 6 ?

V 1. CDO _ X Y v r s ý 6 a b X sV

1 ) m X Y ô W 0� 0 1( )L tϑ ϑ< ≤ s0 g Õ t s ý " 9 T U a b � X L ) m X Y �

� � � "¢ Y(V 1. � Z [ \ 8 )0 � ¼ ) m X Y "� à u ] < � . a b (V 1. � Z [ bar)0� ^ L � X Y "� à u óT Ý v � ª « 0� � � � � � Ù � � � � X Y " > a b ô ½ ��

1 2( )L tϑ ϑ< ≤ s0 g 9 T U a b h X L ) m X Y a b "¢ Y0ú u ¬ � � � � X Y "¢

Y( _ ¬ V 1. � _ [ Ù Z [ \ 8 E )0) m X Y "� à u ¤ � M "a b � ô 01 ϑϑ − 0ú

? v ë ` a ( ) * + " 9 T U a b 0y z � � � � X Y "� à u � I � ] < b ^ "a b 0

� � M "a b � ô 1( )L t ϑ− 0� � � � X Y � à u 6 > a b 1 V 1. � _ [ þ ö ; 6 0

1 z ´ J w � ! ; ; c C L � X Y " � à u ¬ t s ý � + � � " a b � �

( )( )1( ) max min ( ), ,0j j jM t L t ϑ ϑ−

= − 0� � 1jϑ−ôX Y j 6 a b de ý (attachment point)0I

ϑ1

ϑ0

ϑ2

06

ϑ3

�� 7�

�

jϑ ó ôX Y j 6 a b f e ý (detachment point)0� X Y j ¤ 6 a b 1j jϑ ϑ−

− 0 � X Y j �

� � � " Q � � ? � l B(0,t)ô ts ý " g j y Ñ 0ó À j ³ X Y Õ CDO � � R � � � " � g a b (Default

Leg�DL) j � ô ( ) ( )0

( ) 0,TQ Q

jE DL E B t dM t = ∫ 0� � ( )QE ⋅ ôZ [ � � k l Ö " � g � ?

� , - � � ¦ ø " ¨ © 0_ ` X Y " � à u v ú ] < ( ) * + " a b 0ê r w ° " Ø

¾ < � � Ð & < � 0� � Ë . b Ö � , - " à É � h � > ¾ 0F l � � � � � � � æ 0

y z ' ª « � à u ¥ � < � " i o � ? k l CDO ~ � �E ô T ¹0¦ â u j ¹ > ¾ K�0> ¾ s ý ô κs ( 1, ,κ TK= … )0A 2 3 � à u � � à " X Y v r 0j ¹¸ 1 v r " Q

R S � T U jW ?� à À j ³ X Y " � à u ¬ _ > ¾ s ý � < � " i o � (Premium Leg�

PL)ô ( )jWs

K κπ 0� � ( )sκπ g ¬ > ¾ ý κs ® À À j ³ X Y " � à u Õ > L , - > �

¥ � b ^ " � à � ô�

( )( )1 1( ) ( ) min max ( ), 0 , j j j j j j js M s L s− −

= − − = − −κ κ κπ ϑ ϑ ϑ ϑ ϑ

1 ) m X Y ô W 0� 0 1( )L s< ≤κϑ ϑ s0 g > ¾ s ý ks " 9 T U a b � X L ) m X Y

" f e ý (V 1.� Z [ \ 8 )0) m X Y - . � " # k �� Q � � ] < CDO " a b (V 1.

� Z [ � ö )0y z � b ^ " Q � � ô Z [ \ 8 Ù Z [ � ö 6 E " l ç 0î ¬ � ^ L �

X Y " � � � ó T Ý v � ª « 0�� X Y 1 _ Z Ö [ \ 8 " l ç g 6 0I� � �

� X Y ó ô m _ Ö \ 8 " l ç �� 1 2( )L s< ≤κϑ ϑ s0 g 9 T U a b h X L �� X Y

" de ý 0 ú u ¬ �� X Y " f e ý ( _ ¬ V 1.� _ [ Ù Z [ \ 8 E )0) m X Y " �

à u ¤ � M " a b � ô 01 ϑϑ − 0¥ � � � " � õ ô ½ 0I b ^ " a b ¼ �� X

Y " � � � I � ] < 0�� X Y � à u 6 > � � � 1 V 1.� _ [ \ 8 Ù _ [ � ö

" l ç ; 6 01 Ö ´ J ?

Õ Ð � Ø � E " ö � Ö 0X Y j � à u " i o � 6 j � ¸ L ¬ , - n L 6 j �

(PL=DL)01 C § CDO _ ` X Y Q R " S � T U jW �

( ) ( )( )( )

0

11

0,

(0, ) min max ( ), 0 ,

TQj

j TKQj j j

K E B t dM tW

E B s L s−=

⋅ = ⋅ − − ∫

∑ κ κκϑ ϑ ϑ

(1)

IÕS�÷ø9ùú"��0, - � � 6 E ~ Õ4 ¶ · ¸ (Contagion Effect)0��L s Copula B ! � ± ² ( ) * + 6 E ", - F G 90v r Copula B ! l 4 ¶ · e 6 o ·

ï p . � / � ?

��Copula � ��

8 ��2009 � 12 ��

�

�

Copula B ! (13) ô � � 6 T U . / X s 0Copula B ! )(⋅C q + ê ë ß ` ! � 9�

grounded ( ){ }1, ,0, , 0, [0,1]n iC u u u= ∈… … �margin ( ){ }1,1, , , ,1i iC u u=… … Ù n-increasing?

k l n ` ( ) ", - s ý ô . ! nttt ,, 21 � 0� � . ¡ > . / � , ( )( )i i iF t u= " t � T

U . / X s v p � r C X s )1,0(U 01 ! g ô �

( )1 2 1 1 2 2( , , , ) Pr , ,n n nC u u u U u U u U u= ≤ ≤ ≤… …

� 2 3 Sklar(1959) x � 0l ¬ M ` n K � " J m T U . / X s B ! ( )ntttF ,, 21 … 0� À i

` K � " t � T U X s ô ( )ii tF 0 ( )ntttF ,, 21 … Ù� l ¸ 6 Copula B ! ê ë 1 Ö G � �

( ) ( ) ( ) ( )( ) ,,,, 221121 nnn tFtFtFCtttF �� = 0c > z ; G H ° � C ` � K " t � . / X s 0s

= o t K " t � . / X s ÙF G 9"d × �

( ) ( )1 1 2, , ( )n i if t t c u u f t= ⋅∏… … (2)

f g C ` � K " t � . / X s ( )1, nf t t… w s o Ö # k 0R u ô Copula" t � X s 0�

ô . ! nttt ,, 21 � 6 E "G J 90� � ! � z 6 E " u r � � (co-movement)�¥ u ô

_ ` . ! t v " t � . / c U ?Copula B ! ´ �ò ó X o Gaussian�Student-t Ù

Archimedean Copulas w ß ´ 0� � 1 Gaussian Copula ¤ x � �0I¥ Ö ³ ¯ � �¬

ï y ³ � F Ú (Tail Dependence)"j � � , - 4 ¶ · e ?

Gaussian Copula k l ~ Õ � l · A � x "F G z { |0 ó � Copula B ! x é �

))u(,),u(()u,,u(C n1

11

n1G −−

ΣΣ ΦΦΦ= …… 0� � ( )⋅Φ−Σ

1 ô � � q r X s B ! � ( )⋅Φ−1 ô t

� ] ] q r X s "� B ! ? � c � Ö 8 Q Â ", - F G � ! ( ) 1t,t ji ≠ρ s0� � ´ F Ú

: � ô �

0))u(Ft|)u(Ft(plim 1jj

1ii

1uU =>>= −−

→ −

λ

0))u(Ft|)u(Ft(plim 1jj

1ii

0uL =≤≤= −−

→ +λ

h i À i 8 Q Â ÙÀ j 8 Q Â r s } Õ ³ ´ ~ Ê . / ô 00 � � ( ) * + � c � Ö 8 Q Â

", - ¨ Ê Ùz v Õ 0ó Gaussian Copula @ v � m � � � � ( ) * + ", - F G 9 ?

¡ ¼ Meneguzzo and Vecchiato (2004)" D E 0Gaussian Copula"� ' F � � � 0� /

� ô �(a) � � Cholesky X = � A q r X s "F G � ! ψψ′ = Σ �(b)1] ] q r X s � � '

n ` � � " . ! ( )'n21 z,z,zz �= �(c) ( )i iu x ψz= Φ = ?

��3�� Cherubini, Umberto, Elisa Luciano, and Walter Vecchiato (2004), Copula Methods in Finance, New

York: Wiley.

� � � � � � � � � � � � 9�

�

Student-t Copula k l ~ Õ � l · A � x "F G z { |0ó � Copula B ! x é �

))u(t,),u(t(t)u,,u(C n1

v11

vv,n1t

v,−−

ΣΣ = …… 0� � , ( )tΣ ⋅�ô Þ ¼ � �" � � t X s B ! �I

)(t 1v ⋅− ô Þ ¼ � � 6 t � t X s � B ! ?Student-t Copula � � ´ F Ú : � ô

])1

)1)(1v([(t22 5.0

1vLU ρρλλ

+

−+−== + 0y z F ¯ ¬ Gaussian Copula . ¯ � " � ´ 0w �

� ( ) * + ", - 4 ¶ · ¸ ?

Student-t Copula"� ' Ù Gaussian Copula(a)Ù(b) Ö ` / � C � 0 � � Þ ¼ � �"�

% X s � ' Ù z � � " C ` . ! s0y z ¼ Student-t Copula � C + u1 Ö � ! ;

÷ � �

)xsvy(tu iivi ⋅==

Archimedean Copula * + 01 [ ]11 2 1 2( , , ) ( ) ( ) ( )n nC u u u u u uϕ ϕ ϕ ϕ−= + + +… � g60�

� v r " ( )ϕ ⋅ $ É ø v r " Archimedean Copula B ! ?I Clayton Copula l x

( ) 1 >0u u withαϕ α−

= − 0�� B ! 1 1/( ) ( 1)t t αϕ− −= + 0y z Clayton Copula 1 ! ; ; o �

1

1 11

nCn ii

C ( u , ,u ) u nαα

α

−−

=

= − + ∑… 0 � 1 V © � � w � � � � Clayton Copula B ! 6 � �

. / � ¬ � � 0 � m � � ± ² Î , 1 � � � & ô ò 6( ) * + F G 9 0� � ´ F Ú :

�(tail dependence) X � ô αλ1

L 2−

= Ù αλ1

U 22 − ?Clayton � ' v � � � � X s " % ; 0+

c > ö � � � (Conditional Sampling) � ' �

( ) ( )1 1 1 1 1 1| , , | ,k k k k k k kC u u u P U u U u U u− − −

= ≤ = =� �

( ) ( )1 1

1 1 1 1

1 2 1 1 2 1

, ,/

k kk k k k

k k

C u u C u u

u u u u u u

− −

− −

− −

∂ ∂=∂ ∂ ∂ ∂ ∂ ∂

� �

� �

( )( )1 1

1 11

( )( )

kk

kk

c

c

ϕϕ

− −

− −

−

=

� � ( )1 ( )kϕ − ⋅ g 1( )ϕ− ⋅ " k �� X (14)0I 1 2( ) ( ) ( )k kc u u uϕ ϕ ϕ= + + +� 0y z � ' C +

'),,,( n21 uuu … � � / � ô � Ø � r C X s (uniform distribution) � C + . !

'),,,( n21 vvv … 0A Ô x 11 vu = � k x )v|u(Cv 1222 = 0 � 2 3 ö � � � "! ; � �

��

�4�11

1( ) ( ) ( 1)( 2) ( 1)( ) ( 1) ( 1)

k kk kk

t kt t

tαϕ α α αϕ

α

−− −

− ∂ + + + −= = − +∂

�

10 ��2009 � 12 ��

�

�

( )( )

1 12

2 1 11

( )( )c

vc

ϕϕ

−

−= 0J K ÷ � αα

α

α /11212 )1)1v(v(u −+−

− +−= 0�

α

α

α

α

/1

1)n1(n

1n

1iin 1)1v()2nu(u

−

−−

−

=

−

+−⋅+−= ∑ ?

Õ H â , - s ý "� ' R 0Ì � + C D _ ³ Copula B ! 6 V ! ?�� Ú W ( ) * (2007)

� � "% ; 0Ø � Canonical Maximum Likelihood(CML) C D � + V ! 0� ¤ |��ÙÓ0

@ k l ( ) * + , - s ý "F G 9 w Ø �_ à É n o / "F G 9 C § ?C D Copula B !� "V ! [ Ö �S t,ir g À i 8 Q Â Õ À t � "à É n o / 0@ c > ¡ Z T U X s B ! �

t,ir � , o t,iu 0� � 1u0 t,i ≤≤ 0 n,,2,1i …= A 1, 2, ,t T= … ?Gaussian Copula B ! "V ! C

D w g o '1

1ˆ T

t ttTξ ξ=

Σ = ∑ 0� � ))u(,),u(( n1

11

t−− ΦΦ= …ξ ?Student-t Copula"V ! ó Ø

� Kendall’s tau(τ )C§LFG�! Σ�

=Σ j,ij,i ˆ2

sinˆ τπ

0 � Ø � < L" FG�!1 CML

% & 0< Student-t Copula . / � � B !" ¤ ¹ � CD � � CD � + �( Þ ¼ � )0!

; ; o � ( ) ( )( ), ,12,ˆˆ ˆ ârg max log , , ; ,

=⊂ ∞= Σ∑ …

�

�M t

i t n ttc u u v 0 � � 2 3 ; (2) 0 � �

( ), 1, ,1

,

, ,( , , )

( )ν

ν

ξ ξξ

Σ= ∏

…

…

t n ttn

i t

tc u u

t

< � o ( )2' 1

0.5

, , 2 1, 2

1

1 ˆ12 2

ˆ ˆˆ ˆ, , ; , 1

(1 )2 2

ξ ξ

ξ

+−

−

−

+−

=

+ Γ Γ + Σ Σ = Σ + Γ Γ + ∏…

�

�

� �

�

�

� �

�

nn

t t

ti t n t n

n i t

i

n

c u u

� � 't,nt,1t ),,( ξξξ …= 0 )u(t t,i

1t,i

−

=�

ξ 0I ( )⋅Γ ô Gamma B !?Clayton Copula B !6 V !

ÓCD ´ � Student-t Copula B !0 ¤ ! V !C D � 1 ¤ ¹ � CD � ô � ] 0

( )( ), ,10ˆ ˆ ârg max log , , ; α

=>= ∑ …

�

�M C

i t n ttc u u 0

� � ( )1

1, ,

11

1

ˆ ˆ, , ; 11

nn n αC n α α

i t n t i iii

nαc u u α α u u n

α

− −

− − −

==

Γ + = − + Γ ∑∏… ?

¼ ¢ � � ö "% & �" u ¥0 � � ( ) ( ){ }* 1 înf |i i i i it F u t F t u−= = ≥ � ' _ à É ( )

", - s ý 0� � )( ii tF ��1 Bharath and Shumway (2008) � � Merton � � Ù ® d ×

�� 11�

�

; S � Z [ � � � � 6 ?

�� Merton � ��

Merton(1974) � � í ¸ Black and Scholes(1973) � � L " F G ) K T � ¬ ¬ S � Z [

"� � 0� ) m d � )(V Ei, » ô] "à É ôQ Â à É T � )(V Ai, 0A ¼ â T ôQ Â ( D"

À ) 0 Ú W Black and Scholes(1973)"J ¬ w ; o �

, , 1 2(0) (0) ( ) ( )rti E i A iV V d D e d−= Φ − Φ (3)

t

t)5.0r()D/)0(Vln(d

A,i

2A,iiA,i

1σ

σ++=

tdt

t)5.0r()D/)0(Vln(d A,i1

A,i

2A,iiA,i

2 σσ

σ−=

−+= ?

k l ) m d �Ùà É d �ê ë � \ Ç � � � (GBM)0 � 2 3 Ito’s Lemma < � � �

2 2, 1 1 , 11 , , 1 , ,( ) ( ( ) 0.5 ) ( ) ( )Q

i E i A i A i A i A i AdV t f d rV f V dt d V dW t= +Φ + +Φσ σ

� �0 21 f,f X � ô) m d � B ! ( )⋅f l Ai,V Ù t C � 3 � X 0I 11f ó ( )⋅f B ! l Ai,V "Á

� 3 � X �F ¯ Ö � w ÷ � à É d �� 9 Ù) m d �� 9 6 G � ; �

1( )A AE

E

V σσ dV= Φ (5)

� � (3):(5) ; s m Newton-Raphson Method1< = 7 8 9 J � % : ; 0÷ � à É d �Ù �

� 9 ?h C § L ò ó � � �"V ! �0� w D § À i 8 Q Â Õ Z [ � � . / 5 � Ö ", -

. /ô�

2( ) ( ( )) ( )i iF t DD t d= Φ − = Φ − (6)

Crobie and Bohn(2001) � ö � i � d e " & ' « Ù; (5) U l ³ 0¥ Bharath and

Shumway (2008) 3 � 7 8 9 % : ; < = > : 6 t � · ¸ ª « � , - Z [ "C D 0y z Õ F

r "� � à 4 Ö ¦ ² C � � "% & ?

Bharath and Shumway (2008) k l ( ) " � � /ô � ¸ � E d × � � /" 5%0ª ¢ 8

) � � " 25% � � 6 ( 0.05 0.25D Eσ σ= +� )0y z ¡ ¼ ª ) R r ¥� � à É � � /ô�

,

, ,

i EA E D

i E i E

V Dσ σ σ

V D V D= +

+ +� � (7)

A à É d �1) m d � ª ¢ Â Ã �( , ,i A i EV V D= +� )��0y z ;(6) , - . / Î�ô�

2, ,

,

ln( / ) ( 0.5 )( ) ( )i A i i A

ii A

V D r σ tF t

σ t

+ −= Φ −

� ��

�

?

12 ��2009 � 12 ��

�

�

�� Hull(2008) \ L � à � lP Y õ ö� ÷ ø "S � ù ú 6 � � � v � 0 M M N O P ¥

h i "a b c ! 0 � � � à ] " H Ò ¬ S � K L ¯ ! 6 Õ � � b � ? k � F r Ò ¬ AAAL � "� � � � "X Y 0ú � à * + X � ô() * + � M N () O P 6 à É � 0� � � M

"a b { | } ! . � v r 0� ¸ � Ï 9 "Z [ � u ?� g a b ô C � la b " � 0Ð &

� 1 ��_ ` X Y 6 F lZ [ � 0¥�� 1 R ß � ¡ � ÷ � "d e 0Ú 3 Gibson(2004)� ö"Ö ³ �� CDO Z [ \ ] 0X � ô� g a b "{ | } ! Ù 7 ¨ � a b "{ | } ! 0

1 � ¸ _ ` X Y "Z [ � ¢ ? Gibson(2004) x é � g a b / ô_ ` X Y "¨ � a b � l� � � � " Q � � 6 Ù

W0I ¨ � "a b { | } ! ó 1 _ ` X Y "¨ � a b / £ 1 () * + � g a b / ��6 0

; ¤ X Y j C t � � � � M "a b ô < ` * + C t � � � � M "} ! 0y z X Y j

6 � g a b { | } ! 0� ! ;; �1

[ ( )] [ ( )]j

j j i

E M t E M tAϑ ϑ

−− ∑ 0� � [ ( )]jE M t ô j X Y Õ s

ý t Ö "� g a b 0 [ ( )]E M t ó ô() * + 6 � g a b ?

7 � g a b ó x é ô_ ` X Y " � la b � ª ¢ C ` ] ] U 0h i � A a b ¨ ¥

Ö � + � M " a b � 0 j X Y Õ s ý t Ö " 7 � g a b 1 ! ; ; �

( ) [ ( )]j j jM t E M t SD= + 0 ( )2( ) ( ( )j j jSD E M t E M t= − ô j X Y "a b ] ] U ?r � w D §

L 7 ¨ � a b / 0H I ÷ � 7 ¨ � a b { | } ! 0¥ � j X Y "7 ¨ � a b { | } ! �

1

( ) ( )j

j j i

M t M tAϑ ϑ

−− ∑ ?

��

��Ø � ® Ù � � "Merton � � 0 L s Gaussian, Student-tö Clayton ß ³ Copula

B ! 0K T M N () O P 6 S � T U 0@ A Ù ¯ v r , - � � / l� 6 ª «?¤ ¥ 0Ú W

Gibson (2004) D § L CDO"Z [ ��\ ] 0X ^ Õ ® Ù � � "Merton Ö ` S � Z [ �

� Ö 0l CDO a b { | } ! " õ 0z Q ñ � CDO"~ � � E & ' l� { | } ! . \ ª

«?

�� 13�

�

�� Ú � � \ ! Q Â (International Index Company�IIC)" ¦ § 2 x (15)0�� F L S �

K L L � Õ BBB 1 ¢ "Î , � & 0 � ô�# $ CDO à É � 6 ] "à É 0 � i ¢ d � &

21 8�ù & á â 5 8�P Y ù 3 8 Ù N [ Q Â 3 80 u D 32 8? k l _ 8 Q Â 6 Q � �

v l � ôÃ Î 0 C ¨ © 0¥ CDO() * + � 9 D ô 3.2 ) * 0�� CDO X o � ³

X Y ù ú 0[ 1?z Q CDO ~ � � E l x ô 5 ¹0¦ â . × ª ¹ > ¾ C �0 ¼ â « Ú

� ¬ � ' 20,000�?

1. �# $ M N () X Y ù ú 6 k l

X Y ³ ´ d e ý(ϑi-1) f e ý(ϑi) ¦ â � ( ¨ © )

) m X Y (Equity) 0 3% 9.6 � � � � (Junior) 3% 6% 9.6 �� (Mezzaniae) 6% 10% 12.8 � � � � (Senior) 10% 100% 288

��¸ � Merton(1974) � � 0C D () * + à É ¦ ø , - � � 6 w ° 9 ?Ø � Q Â

2006¹¤ ¥ C ` � » à X 0¨ 5 2007", - . / 0 � � c > Q Â � ¹� à X ¨ 5 Ö C

¹� Q Â ", - . / 0¼ Î , ¡ � Ã n à X (TEJ) � � � d × ;S � Z [ � � "F G

à X 0 � ¤ 8 T � � Í Õ Q " 8 ! � � � (Ù � � (?

¼ Î , ¡ � Ã n ÷ � "Q Ñ à 4 ¥ 0�� Merton � � v w � � ! 0X � ô

] "à É 6 d � Ù � � / ?Ú W Newton-Raphson Method < = (3):(5) Á ö 7 8 9 % : ;0

w ÷ � j 8 � & "à É d � Ù � � � / 0H C / �� j 8 � & ¦ ø , - � � ". / 0� d

e ® ó ¬ 2?

�u 0ô F Ï �() * + , - "4 ¶ · ¸ 0�� c > Copula B ! a ² à É , - 6 F

G 9 0 2 3 CML C D � * Gaussian�Student-t 1 ö Clayton Copula(1 6) B ! "V ! ?�

� V ! C D ¥ 6 _ ` Copula B ! X � � C + . ! '),,,( n21 uuu … 0 � � Ö ;

( ) ( ){ }* 1 ˆinf |i i i i it F u t F t u−= = ≥ J K L * + à É ¦ ø , - "s ý0� ¡ ¼ ;(1) < § L _ `

X Y 6 m � S � T U ?

��

�5�CDS � � � �� Fitch/Moody’s/S&P � � � � BBB-/Baa3/BBB- � � � 6 �� ! Clayton Copula " # �$ % & ' ( α=1.54

14 )*+,-./2009 0 12 1/23 4 5�

�

�

2. ¸�d×;S�Z[��+6à X

� àÉ d � �àÉ � � � ) m d � � ) m � � � ( Â Ã �

A ¯ 88,891,836,805 0.2442 78,492,000,000 0.2765 10,596,143,500 Î ° 368,747,239,125 0.1204 309,452,000,000 0.1435 60,414,492,500 � A 494,357,993,754 0.1425 413,928,900,000 0.1702 81,947,268,500 Î õ 365,700,706,158 0.1148 301,058,000,000 0.1394 65,862,898,000 � ± 429,045,603,397 � 0.1706 � 382,433,800,000 � 0.1914 � 47,491,645,000

) ² 66,219,541,376 0.2627 57,077,500,000 0.3048 9,314,606,000 � R 42,137,499,312 0.4418 38,750,400,000 0.4804 3,451,034,000 ³ E 68,409,841,537 0.1298 43,033,200,000 0.2063 25,855,649,500 ´ µ 167,186,803,312 0.0702 57,212,800,000 0.2053 112,049,866,000 © ð 57,483,251,270 � 0.1723 � 41,140,000,000 � 0.2407 � 16,651,745,500

� ¶ 20,761,039,333 0.3943 19,219,200,000 0.4259 1,570,943,000 Î É H 6,499,162,990 0.2625 6,383,200,000 0.2673 118,151,900 Î ¶ N 29,418,924,796 0.1412 20,457,400,000 0.2031 9,130,682,000 � · á 20,609,331,766 0.1348 9,531,850,000 0.2915 11,286,580,500 © ¸ á 58,512,508,864 � 0.0869 � 24,980,900,000 � 0.2036 � 34,164,549,500

J ¹ á 74,133,314,537 0.0397 16,717,050,000 0.1759 58,500,050,500 � á 96,547,790,807 0.0632 20,695,500,000 0.2950 77,284,162,000 º � P 86,299,288,832 0.1364 32,499,250,000 0.3621 54,816,201,000 * P 163,626,359,942 0.2017 86,262,450,000 0.3826 78,824,790,000 * m P 63,979,702,338 � 0.0789 � 19,670,000,000 � 0.2567 � 45,146,091,000

J » 431,836,733,041 0.2406 387,423,000,000 0.2682 45,252,084,000 Î U » 1,683,784,021,094 0.2664 1,634,717,410,000 0.2744 49,992,789,500 ¼ º 192,300,906,377 0.1567 110,709,550,000 0.2722 83,131,470,000 ½ ¾ 215,461,000,000 0.2712 158,916,000,000 0.3677 57,612,595,500 ) ¿ 32,854,899,274 0.4041 29,664,000,000 0.4476 3,251,130,500

À ; 576,220,074,859 � 0.1825 � 343,056,900,000 � 0.3065 � 237,564,353,500 � ´ » 636,371,365,239 0.1235 585,820,200,000 0.1342 51,505,366,000 Á Â 57,832,297,195 0.3761 45,714,150,000 0.4758 12,346,924,000 � ð 1,376,064,605,539 0.2908 1,201,560,000,000 0.3331 177,798,544,000 ) � 17,105,612,903 0.0456 4,841,000,000 0.1610 12,496,119,000 Ã ù á 69,614,982,405 0.1340 29,296,800,000 0.3183 41,079,262,500 Î Ä 190,130,742,369 � 0.1739 � 168,966,200,000 � 0.1957 � 21,564,043,000

��6 7 � � 8 9 : ; /� 2006 < 0= > ? @ � A B C D E 0F G � � HI � J K L M KMV N O /�

P : I � Q � 1/2 R 5I � � � HS � � T ; L M 4 05� U V W � X T ; (1.85%)YZ [ G \ ] ^

TEJ � � � _ ` /a b New-Raphson c d e Merton � � N f g h � " # /i j W k l ( 9 : ; Y

� � � � � � � � � � � � 15�

�

��CDO � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ) m X Y � . CDO ¦ â "X Y � ¤ � � M ( ) * + a b " # � 0y z ) m X Y "

S�T U � ¬ � Å L � "X Y 0 3 { � X � � � Gaussian�Student-t 1 ö Clayton�'( ) * + X Y "Q R S�T U ?¼ 3 � � � � Ð ¬ � � \ ³ Copula B ! 0) m X Y "S�T U ¤ � 0I � � � � X Y "S�T U ó ¤ u?Õ z k l , - � � / ô ½ ?

3. _ ` X Y S�T U 6�' de

X Y ´ � ® Merton�� " Merton��

Gaussian Student-t Clayton Gaussian Student-t Clayton

) m 4.4283% 4.2087% 1.7744% 17.9422% 17.1355% 7.8794% � � � � 1.5447% 1.4386% 1.1343% 8.5143% 8.3871% 5.0655% � � � � 0.5589% 0.5956% 0.7986% 4.7574% 4.7583% 3.8012% � � � � 0.0128% 0.0171% 0.0897% 0.2083% 0.2274% 0.5279%

v r Copula B ! l 4 ¶ · e 6· ¸ ï p v r ?¼ 3 �' de � � 0) m X Y 1 �

� Gaussian Copula�' "S�T U ¤ � 0� 6ô Student-t Copula0Clayton Copula�' ¤u�I � � �� X Y �' "S�T U 0 ó 1 Clayton Copula ¤ � 0Gaussian Copula¤ u0ò ó y ô ClaytonÙ Student-t Copula B ! �' ¯ ° ± ² Ö % Z[0 Æ � Clayton Copula0 Ï � � � ô . * · ¸ÖuC ( ) a b " w ° 9 « t u?

( ) * + , - F G 9 ¯ s 0� a b ¦ ø ' ¯ � � . * j � 0� . àÉ r s , - �

r s ~ � 6. / � � ?k Ç ¦ ø S�, - � � 0w ° É ø ê ë , - 6³ ´ a b 0Ç I F l

Ú ¦ ø ½ a b ". / � � « ¢ £ 0) m X Y "� g a b � 1 t u0l ¸6S�T U � w

t u?� 60� ( ) * + , - F G 9 ¯ us 0� ¦ ø ½ a b " w ° 9 F l u0ó ) m X

Y "� g a b � ' « � � 0¥ ) m X Y 6S�T U Ù( ) , - F G 9 � % & ' 0w =

È Gaussian Copula�' ) m X Y 6S�T U � ¬ Student-t 1 ö Clayton Copula6 y ?� � � � �� X Y �' de w � 0( ) * + 64 ¶ · e = ï * 0¦ ø r s , - " w

° 9 = � 0ó © ª ê ë , - I É ø "³ ´ a b � 0n o � � �� X Y " - . � Ò

¬ w ° ¦ ø � a b 6Z[Ö0� g a b / F l � � 0¥ + l ¸¯ � "S�T U � ô Z[

a � 0 � � � � �� X Y 6S�T U Ù( ) , - F G 9 � � F G ? ( ) * (2007)Ù @ A B (2008) � � � ö 0Merton �� k x àÉ n o / p � q r X s

Ö0� K C , - . / j ó . uC " ¨ Ê 0y z � � � � Bharath and Shumway (2008)"� � Merton��0� Ù W ¢ � "�' % ;0� _ ` X Y S�T U 6de { � Õ 3?¡ ¼� � � , % ;� � L ", - w ° 9 F l � 0 Ù ¯ 3 " ® Merton ��0! 3 � ¸L

_ ` X Y S�T U � j ¢ £ 6 É ' 0I v r " Copula B ! l S�T U " ª « C Ý ¬ ¢ �

"de ? � � � a b v r , - � � / l _ ` X Y "S�T U " ª « ?# $ ¦ j ) m X Y l , -

� � / F l � � � � X Y ¯ ô � � �þ � ¢ , - � � / = u0X Y "S�T U ó = � 0ú

¼ V 2 � � 60£ � � � � X Y Q 0) m �� Ù� � � � "X Y Õ m , - � � / ©

16 ��2009� 12��

�

�

� 2. � � � � � � � � � � � �� Merton � � �� Merton � � �� ! " # $ % �& ' � ( ) * + � , - . / 0 Gaussian Copula� Student-t

Copula� Clayton Copula0

�� 17��

�

�� 0.4 ! " # $ % & � ' ( Galiani(2003)� � ) * �+ , � � � � � � - , ) . / 0 1 2 �3 4 5 6 7 8 � � (9: � � � � �; < = > �? � � � � � � @ A B C D E ��

� � 0.2 F�� G H I J �K � � � � � � � 0.4 L M N �O P QClayton Copula�RS � � � � � � = > � � � � � � � � � � T U �V W X Y Z ) �

[L �\ ] ^ _ Gibson(2004) ` a b c d e f � g . � � ` a h i �j k l m �n op q r s t . � � � CDO u v w x y Z ` a O z �{ | ) } a �~ � �

��CDO � � � � � � �� R 4 � � � � � w � � � � � w � � � � � � Z � �B C R 4 D E ��

` a h i o � � � � � ��K 3 Copula � � % � a k T �7 8 � � �� [��

�9: � � � � � [U �� S � �T ` a �T � � �Z O � �R 4� Panel � S ��3 )� � % r s � 3 Copula � � � � � � � �= > �� 7 8 (� � � � � � �� Gaussian Copula�� [T �� Z � Student-t Copula�� Clayton Copula � [U � B 9: � � �� o Clayton[��[ � Gaussian Copula�� ) * , ��

R 4. t . � � Z � � � � (� � �

Panel A: ¡ ¢ Merton h i

Copula � � 7 8 � � � � � � � � � � � � � � 9: � � � �

EL UL EL UL EL UL EL UL

19.96 17.02 7.35 9.22 2.70 5.08 0.06 0.23 Gaussian

(19.49%) (57.45%) (7.18%) (31.13%) (2.64%) (17.16%) (0.06%) (0.79%) 19.39 16.15 6.97 8.62 2.93 5.20 0.08 0.28

Student-t (18.61%) (55.91%) (6.69%) (29.85%) (2.81%) (17.99%) (0.08%) (0.95%)

8.56 6.33 5.54 4.88 3.93 4.04 0.45 0.68 Clayton

(8.17%) (34.46%) (5.28%) (26.58%) (3.75%) (22.02%) (0.43%) (3.72%) Panel B: £ ¤�Merton h i

12.51 9.94 7.51 7.57 4.62 5.69 0.22 0.41 Gaussian

(56.04%) (102.62%) (33.67%) (78.15%) (20.69%) (58.69%) (0.99%) (4.27%)

12.10 9.55 7.39 7.33 4.59 5.54 0.24 0.44 Student-t

(54.48%) (101.27%) (33.28%) (77.67%) (20.66%) (58.75%) (1.08%) (4.67%)

6.74 5.09 4.72 4.13 3.68 3.60 0.55 0.73 Clayton

(30.36%) (74.35%) (21.25%) (60.30%) (16.58%) (52.61%) (2.48%) (10.66%)

�� EL(Expected Loss) � � � � � � � � � �� !UL(Unexpected Loss) � � " � # � ��

!(.) � � � � � � � � $ %

18 & ' ( ) * + ,2009- 12.,/0 1 � �

�

$ ¥ �\ ] j k £ ¤ Merton h i (¡ ¢ Merton h i ¦ � � �� § ¨ £ ¤Merton h i � © � � ª � U « n ¬ � Z ' �� R 4� Panel A ( B�� w � � �� ® � � w � � � � � � �o 7 8 � � �� £ ¤ Merton h i « � �� K �¡ ¢ Merton h i U � B � � ¯9: � � � � �� ° � K � k T �± q � � £ ¤ Mertonh i �t . � � � � w � � � K � � w � � � �² �³ J U , ¡ ¢ Merton h i �´ µ £¤Merton h i ¬ ¶ · 6 ¸ ¹ � � �º » �¼ ½ � « ¾ ¿ � � k � À Á �� S £ ¤Mertonh i % Â Ã Ä 7 Å Æ �Ç � À Á J �³ { T �

[L �r s CDO u v w x y � � � � �È É (Ê 7)�+ R 5� Panel A�� CDO u vw x y �Ë Ä 7 Ì Í Î Ï � � Ð Ñ 4 �7 8 � � � 9Ò Ó � � �Ô Õ 7 8 � � ��

Ö 4 x � �± × Ø , ¾ ¿ �� & U Î Ï � � �� Á Ù � �Ú+ , 7 8 � � Û Â ��

� Ü Ý �� (9: � � �� Ö 4 x Þ ß �à 5 6 �Ã á â ã �ä * � � � ,

Î Ï � � �� Á Ù � �³ { T �å � � � � � Ö 4 x ² �� " # � ' �æ . ' {

| ) } a ç è �é �ê ë � � Î Ï �u v w x Z Fw �7 8 � � �ì í ¥ �î ï ð K Ë �

� � � � �\ ñ n ¬ ¾ É � ò Ü �Ú� � Î Ï ó � �L w k T �� Z Fà ô D ��

õ ö n o ÷ ø 7 8 � � �ê ë � � �X ± � , 7 8 � � �ù Ü ú �< �u v w x Z Fw ^

ð û } ` a ü ý �~ � �3 þ � � 9: � � � � �ì í ¥ �� û } ` a , u v w x L w

�� n ¬ � �± � � ÷ � ¯ � � � ( � (2009) ) * � R 5� Panel B � S 7 8 � � �� w � � � � � � È U � � Ö 4 x Þ ß � � �9

: � � � � �� w � � � � � � È T �3 � Ö 4 x Þ ß � " # �7 8 � � �� K

� ² ��³ J U , � � � � �� t . � � � ± Z x �� w � � � � � � â �¼ ½ �

� w �¸ ¹ Ð Ñ Î Ï 4 � � � � 9: � � �� õ n ¬ � 5 V ��

��

ñ � � � É Ô Õ � Ï � � � � � � � � �� A o � � � Ï � � � � ¤c Õ ! [� "

#�Â Ã Ä 7 Å Æ - , � $ % Æ � É Z ) � & ß O ' ( �ª ) * � � ` a + ! , ì í �

-�Ú� . L ´ µ �� ; < / 0 0 1 2 � �Hull(2008) ] A { � ì í ¥ 3 4 5 ? � � Z� � Ç 6 6 � � 7 8 � � � � 9 : �\ ] ; W r s & ß Bharath and Shumway (2008)�£¤ Merton h i _ < CDO�� Ç � � = > Copula � � « � Ä 7 Ì Í Z x �� K? � � @ A B C � � ª � U « � D E �Ü F Ç � t � � � � � � Z À Á J �± q � G

Gibson (2004)�` a b c d e ��H CDO . L ´ µ � � � � � � � � I § t . �� J K

` a � G H I J �{ | ) } a � ç è �+ , 7 8 �� CDO � 9 Ò Ó � � � � Z �� Ä 7 � � K ? � Z M L = > e � M 3 4 u N ® � � Z n ¬ � �X ± 7 8 ��

( � � K ? � � O �K � P c Q $ Ì R ' ( ¸ ¹ Ð Ñ Î Ï � Student-t � Clayton Copula�7 8 �� o � � Gaussian Copula h S � � � � � [ T �Ú 7 8 �� Û 5 � � �

Ü Ý ��" � � 9 : � � �� Á Ù � k � G H �X ± � � � � � ( � � K ?

� - , 3 O T U �� o Clayton Copula h S � [ T �

�� 7�2 3 4 5 6 7 8 9 : ; < Merton = > ? @ A B C D CDO E F � G � H I � J ,K L � M N 3 O P

Merton = > Q R � > S %

�� 19��

�

� 5. � � � Merton � � � � � � CDO � � � � � � �

Panel A: � � � � � �

� � � � � � � � � ! " � � # ! " � � $ % ! " � �

& Gaussian &Student-t &Clayton Gaussian &Student-t &Clayton Gaussian &Student-t &Clayton Gaussian &Student-t &Clayton

0.5 31.392 30.562 26.975 1.886 2.158 3.871 0.038 0.401 1.312 0.000 0.002 0.024 1 27.061 25.566 17.993 4.816 5.300 6.421 0.935 1.393 3.057 0.005 0.018 0.158

1.5 23.153 22.380 13.736 6.518 6.494 6.512 2.026 2.284 3.764 0.028 0.043 0.263 2 20.266 19.715 11.310 7.462 7.270 6.196 2.745 2.945 4.063 0.058 0.074 0.340

2.5 18.355 17.796 10.046 7.795 7.660 5.883 3.238 3.458 4.035 0.087 0.100 0.392 3 16.640 16.133 8.935 7.899 7.812 5.579 3.777 3.855 3.984 0.114 0.130 0.440

3.5 15.357 14.851 8.203 7.863 7.757 5.323 4.085 4.138 3.911 0.143 0.161 0.475 4 14.237 13.768 7.641 7.746 7.652 5.079 4.334 4.401 3.820 0.171 0.187 0.505

4.5 13.268 12.894 7.161 7.690 7.521 4.913 4.511 4.498 3.744 0.195 0.214 0.529 5 12.507 12.098 6.740 7.514 7.391 4.719 4.617 4.589 3.682 0.220 0.239 0.552

Panel B: ' � � � � � �

0.5 31.138 29.568 24.195 4.206 4.918 7.486 0.243 2.759 4.603 0.000 0.038 0.125 1 25.285 23.186 14.216 8.621 8.767 7.659 3.352 4.119 5.044 0.065 0.117 0.396

1.5 20.627 19.549 10.512 9.403 9.091 6.687 4.676 4.897 4.853 0.131 0.178 0.513 2 17.517 16.752 8.630 9.485 9.078 5.987 5.139 5.236 4.665 0.199 0.240 0.573

2.5 15.522 14.912 7.620 9.245 8.946 5.507 5.372 5.483 4.407 0.249 0.273 0.615 3 13.861 13.335 6.795 8.904 8.659 5.112 5.643 5.606 4.198 0.286 0.312 0.647

3.5 12.611 12.098 6.223 8.551 8.285 4.799 5.720 5.639 4.014 0.322 0.351 0.673 4 11.549 11.106 5.776 8.180 7.950 4.527 5.751 5.687 3.847 0.355 0.381 0.696

4.5 10.679 10.274 5.408 7.906 7.621 4.327 5.745 5.610 3.713 0.383 0.413 0.714 5 9.942 9.553 5.088 7.572 7.326 4.127 5.686 5.542 3.600 0.414 0.441 0.730

20 ��2009 � 12 ��

�

��

� � ! � � � � �" # $ % & ' ( ) * + , - . ! � / 0 1 � 2 3 � � �

��4 * 5 6 �7 8 9 + : ; � � � � < = > ! �- . � / ? @ A < = >

! �� B , CD E F G HI J K L � � � � ! / 0 � M N O � P �

��4 * 5 6 �Q R S T U < = > ! �� B � V� � � W X Y > 5 Z >

� [ \ � � � � � F G HI J K L � �] ^ _ ` a b c d J K L � � �� X Y

> 5 e > �� f ? @ V g h i j k l Merton �m j � � n o � p q r s t u v w x 2 3 P < = > !

y � � = > ! y �z { N | � , � � � � � ! �} ~ � f � 0 : ; + , u v � y

�, } � p � 2 3 CDO � � � } ~ � �m j � � � f � � g h � � CDO�� no O � � � f � 0 �� V

� � � � � � � � � � � � � � � 21�

�

�� H� � � �� (2009)� � x � � ¡ 9 ¢ & o £ ] ¤ � ¥ ¦ � � � �§ � ¨

��© ª « ¬ © � > V ® ¯ (2006)/ ° ± ² ³ m j n o 9 ¤ � £ ] ¦ � � � -Copula ´ µ ¶ � · ¸ M ¹ º

» ¼ ½ � � b ¾ ¿ À h V Á Â Ã HÄ Å Æ �Ç È É (2004)� Ê Ë Ì Â º » Í � � = Î Ïm j � �µ �Ð Ñ � � � µ

Ò Ó Ô ¨ � � �� © Õ ¬ © Ö > 155-179V × Ø � (2008)£ ] ¤ � ¥ ¦ Ù Ú u v � Û � � � � - c Ê Ë f Ü ¶ � Ê Ë M ¹ º » Ý Þ

¹ ß ¾ ¿ À h V à á â (2007)� £ ] ¤ � ¥ ¦ � � � -Copula ã ��ä j ¨ � � � © å ª æ ¬ © � >

21-52V ç è é Hê ë ì (2005)� £ ] ¤ � ¥ ¦ � � � -Copula�µ ¨ � � �� © Õ

í © Ö > 53-84V î ï ð (2007)£ ] ¤ � ¥ ¦ � � � - ñ ò ó F ¦ � ��ô õ ¶ � Ê Ë M ¹ º » Ý Þ ¹

ß ¾ ¿ À h V

Bharath, S. T., and Tyler Shumway (2008), “Forecasting Default with the Merton Distance to Default Model,” Review of Financial Studies, Vol. 21, No. 3, 1339-1369.

Galiani, Stefano S. (2003), “Copula Functions and their Application in Pricing and Risk Managing Multiname Credit Derivative Products,” Working Paper, King’s College London.

Gibson, Michael S. (2004), “Understanding the Risk of Synthetic CDOs,” FEDS Discussion

Papers, no. 2004-36, Board of Governors of the Federal Reserve System. Giesecke, Kay, and Lisa Goldberg (2004), “Sequential Defaults and Incomplete Information,”

Journal of Risk, Vol. 7, No. 1, 1-26. Hull, John C. (2008), “The Credit Crunch of 2007: What Went Wrong? Why? What Lessons

Can Be Learned,” Working Paper, University of Toronto. Li, D. X. (2002), “Valuing Synthetic CDO Tranches Using Copula Function Approach,”

RiskMetrics Group Working Paper. Lin, Shu-Ying, and Gang Shyy (2008), “Credit Spreads, Default Correlations and CDO

Tranching: New Evidence from CDS Quotes,” Working paper. Menguzzo, Davide, and Walter Vecchiato (2004), “Copula Sensitivity in Collateralized Debt

Obligations and Basket Default Swaps,” Journal of Futures Markets, Vol. 24, No. 1, 37-70. Merton, R. C. (1974), “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates,”

Journal of Finance, Vol. 29, No. 2, 449-470.

risk analysis and valuation of collateralized debt...

Documents