Investigating the Sources of Default Risk: Lessons from Empirically Evaluating Credit Risk Models
Abstract
From a credit risk perspective, little is known about the distress factors -- economy-wide or firm-specific -- that are important in explaining variations in defaultable coupon yields. This paper proposes and empirically tests a family of credit risk models. Empirically, we find that firm-specific distress factors play a role (beyond treasuries) in explaining defaultable coupon bond yields. Credit risk models that take into consideration leverage and book-to-market are found to reduce out-of-sample yield fitting errors (for the majority of firms). Moreover, the empirical evidence suggests that interest rate risk may be of first-order prominence for pricing and hedging. Measured by both out-of-sample pricing and hedging errors, the credit risk models perform relatively better for high grade bonds. Controlling for credit rating, the model performance is generally superior for longer maturity bonds compared to its shorter maturity counterparts. Using equity as an instrument reduces hedging errors. This paper provides an empirical investigation of credit risk models using observable economic factors.
M S W a W B e U O o d I n v e s t i g a t i n g t h e S o u r c e s o f D e f a u l t R i s k : L e s s o n s f r o m E m p i r i c a l l y E v a l u a t i n g C r e d i t R i s k M o d e l s (cid:3) G u r d i p B a k s h i , D i l i p M a d a n , F r a n k Z h a n g F e b r u a r y 2 8 , 2 0 0 1 (cid:3) B a k s h i a n d M a d a n a r e b o t h a t D e p a r t m e n t o f F in a n c e , R o b e r t H . S m it h S c h o o l o f B u s in e s s , U n iv e r s it y o f a r y la n d , C o lle g e P a r k , M D 2 0 7 4 2 , a n d Z h a n g is a t F e d e r a l R e s e r v e B o a r d , D iv is io n o f R e s e a r c h a n d S t a t is t ic s , M a il t o p 9 1 , W a s h in g t o n D .C . 2 0 5 5 1 . B a k s h i c a n b e r e a c h e d a t T e l: 3 0 1 - 4 0 5 - 2 2 6 1 , E m a il: g b a k s h i@ r h s m it h .u m d .e d u , e b s it e : w w w .r h s m it h .u m d .e d u / (cid:12) n a n c e / g b a k s h i/ ; M a d a n a t T e l: 3 0 1 - 4 0 5 - 2 1 2 7 , E m a il: d b m @ r h s m it h .u m d .e d u , n d W e b s it e : w w w .r h s m it h .u m d .e d u / (cid:12) n a n c e / d m a d a n / ; a n d Z h a n g a t T e l: 2 0 2 - 4 5 2 - 3 7 6 0 , E m a il: x z h a n g @ fr b .g o v . e t h a n k R o g e r B e t a n c o u r t , E lt o n D a a l, M ic h a e l F u , M a r k G r in b la t t , L e v e n t G u n t a y , N e n g jiu J u , C h u c k L a H a ie , ia o L u , Y u k i N a k a , L e m m a S e n b e t , K e n S in g le t o n , H a lu k U n a l, a n d G e r a ld W h it n e y fo r h e lp fu l d is c u s s io n s . A n a r lie r v e r s io n o f t h is p a p e r w a s p r e s e n t e d a t F a n n ie M a e , F e d e r a l R e s e r v e B o a r d , G e o r g ia I n s t it u t e o f T e c h n o lo g y , n iv e r s it y o f N e w O r le a n s , a n d S U N Y B in g h a m t o n . C o n fe r e n c e p a r t ic ip a n t s a t t h e 2 0 0 1 A F A m e e t in g s in N e w r le a n s a n d t h e A c c o u n t in g a n d F in a n c e C o n fe r e n c e a t M ic h ig a n p r o v id e d s e v e r a l u s e fu l c o m m e n t s . T h e s u g g e s t io n s f G r e g D u (cid:11) e e a n d T y le r S h u m w a y h a v e im p r o v e d t h is p a p e r . T h e v ie w s e x p r e s s e d h e r e in a r e t h e a u t h o r s o w n a n d o n o t n e c e s s a r ily r e (cid:13) e c t t h o s e o f t h e F e d e r a l R e s e r v e B o a r d o r it s s t a (cid:11) . 0
(cid:12) p (cid:12) b f e a m p c e I n v e s t i g a t i n g t h e S o u r c e s o f D L e s s o n s f r o m E m p i r i c a l l y E v a l u a t i n g A b s t r a c t F r o m a c r e d it r is k p e r s p e c t iv e , lit t le is k n o w n a b o u t t h e r m - s p e c i(cid:12) c - t h a t a r e im p o r t a n t in e x p la in in g v a r ia t io n s a p e r p r o p o s e s a n d e m p ir ic a lly t e s t s a f a m ily o f c r e d it r is k r m - s p e c i(cid:12) c d is t r e s s f a c t o r s p la y a r o le ( b e y o n d t r e a s u r ie s o n d y ie ld s . C r e d it r is k m o d e ls t h a t t a k e in t o c o n s id e r a t io o u n d t o r e d u c e o u t - o f - s a m p le y ie ld (cid:12) t t in g e r r o r s ( f o r t h e m p ir ic a l e v id e n c e s u g g e s t s t h a t in t e r e s t r a t e r is k m a y b e o n d h e d g in g . M e a s u r e d b y b o t h o u t - o f - s a m p le p r ic in g a o d e ls p e r f o r m r e la t iv e ly b e t t e r f o r h ig h g r a d e b o n d s . C o n e r f o r m a n c e is g e n e r a lly s u p e r io r f o r lo n g e r m a t u r it y b o n d o u n t e r p a r t s . U s in g e q u it y a s a n in s t r u m e n t r e d u c e s h e d g m p ir ic a l in v e s t ig a t io n o f c r e d it r is k m o d e ls u s in g o b s e r v a b 1 e f a u l t R i s k : C r e d i t R i s k M o d e l s d is t r e s s f a c t o r s { e c o n o m y - w id e o r in d e f a u lt a b le c o u p o n y ie ld s . T h is m o d e ls . E m p ir ic a lly , w e (cid:12) n d t h a t ) in e x p la in in g d e f a u lt a b le c o u p o n n le v e r a g e a n d b o o k - t o - m a r k e t a r e m a j o r it y o f (cid:12) r m s ) . M o r e o v e r , t h e f (cid:12) r s t - o r d e r p r o m in e n c e f o r p r ic in g n d h e d g in g e r r o r s , t h e c r e d it r is k t r o llin g f o r c r e d it r a t in g , t h e m o d e l s c o m p a r e d t o it s s h o r t e r m a t u r it y in g e r r o r s . T h is p a p e r p r o v id e s a n le e c o n o m ic f a c t o r s .
B u ild in g c r e d it r is k m o d e ls a s t h e b a s is f o r e v a lu a t in g d e f a u lt e x p o s u r e s is o f f u n d a m e n t a l im p o r t a n c e t o (cid:12) n a n c ia l e c o n o m is t s . C o n s is t e n t w it h t h is o b j e c t iv e , t h e o r e t ic a l r e s e a r c h c o n t in u e s t o s h e d lig h t o n t h e q u a lit a t iv e n a t u r e o f c r e d it s p r e a d s a n d t h e ir d e p e n d e n c ie s o n e s s e n t ia l f e a t u r e s o f t h e d e f a u lt a b le c o n t r a c t ( i. e . , c r e d it r a t in g o f t h e p a r t ic ip a t in g p a r t ie s a n d (cid:12) r m - s p e c i(cid:12) c / s y s t e m a t ic d e f a u lt c h a r a c t e r is t ic s ) . F o r in s t a n c e , J a r r o w a n d T u r n b u ll ( 1 9 9 5 ) p r o p o s e a v a lu a t io n f r a m e w o r k w h e r e t h e u n d e r ly in g a s s e t o r t h e d e r iv a t iv e c o u n t e r p a r t y m a y d e f a u lt . D u (cid:14) e a n d S in g le t o n ( 1 9 9 7 , 1 9 9 9 ) t r e a t d e f a u lt a s a n u n p r e d ic t a b le e v e n t g o v e r n e d b y t h e in s t a n t a n e o u s p r o b a b ilit y o f d e f a u lt . O n t h e o t h e r h a n d , M a d a n a n d U n a l ( 1 9 9 8 ) a n a ly t ic a lly d e c o m p o s e t h e r is k o f d e f a u lt in t o c o m p o n e n t s r e la t e d t o t im in g a n d r e c o v e r y r is k s . E a c h o f t h e a f o r e m e n t io n e d a p p r o a c h e s v ie w d e f a u lt a s o c c u r r in g a t a s u r p r is e s t o p p in g t im e . I n a r e la t e d w o r k , J a r r o w , L a n d o , a n d T u r n b u ll ( 1 9 9 7 ) d e v e lo p a t h e o r e t ic a l m o d e l w h e r e t h e b a n k r u p t c y p r o c e s s o b e y s a d is c r e t e s t a t e s p a c e M a r k o v c h a in in c r e d it r a t in g ( s e e a ls o t h e g e n e r a liz a t io n in L a n d o ( 1 9 9 8 ) ) . F in a lly , in M e r t o n ( 1 9 7 4 ) , L o n g s t a (cid:11) a n d S c h w a r t z ( 1 9 9 5 ) a n d C o llin - D u f r e s n e a n d G o ld s t e in ( 1 9 9 9 ) , d e f a u lt is m o d e le d u s in g a p r e d ic t a b le s t o p p in g t im e ; n a m e ly , d e f a u lt o c c u r s w h e n a c o n t in u o u s p r o c e s s lik e t h e (cid:12) r m v a lu e r e a c h e s a d e f a u lt b o u n d a r y . E a c h c o n t r ib u t io n p r o v id e s a r ic h p a r a m e t e r iz a t io n o f t h e p r ic e o f c r e d it s e n s it iv e s e c u r it ie s . W h ile s ig n i(cid:12) c a n t a d v a n c e s h a v e b e e n m a d e in in t e r p r e t in g c r e d it r is k s , t h e r e is a r e la t iv e p a u c it y o f e m p ir ic a l s t u d ie s t h a t in v e s t ig a t e t h e s o u r c e s o f c r e d it r is k s u s in g o b s e r v a b le e c o n o m ic f a c t o r s ( t h e e x c e p t io n s w ill b e n o t e d s h o r t ly ) : W h ic h f u n d a m e n t a l e c o n o m ic f a c t o r s - e c o n o m y w id e o r (cid:12) r m - s p e c i(cid:12) c - c a p t u r e v a r ia t io n s in d e f a u lt r is k ? W h ic h c r e d it r is k m o d e l is s u it a b le f o r p r ic in g a n d m a r k in g - t o - m a r k e t d e f a u lt c o n t in g e n t s e c u r it ie s ? W h ic h m o d e l p e r f o r m s t h e b e s t in h e d g in g c r e d it e x p o s u r e s ( in a ll r e le v a n t d im e n s io n s ) ? E m p ir ic a l in v e s t ig a t io n s o f c r e d it r is k m o d e ls a t t e m p t in g t o a n a ly t ic a lly c a p t u r e p a t t e r n s o f s t r u c t u r a l d e p e n d e n c ie s o n t h e o r e t ic a lly in t e r p r e t a b le g r o u n d s a r e c le a r ly d e s ir a b le f r o m s e v e r a l p e r s p e c t iv e s ( f o r e x a m p le in im p le m e n t in g t h e B a s le c o m m it t e e r e c o m m e n d a t io n s o n m a n a g in g d e f a u lt r is k ) . E v e n w h e n a p a r t ic u la r s e t o f m o d e ls is t h e o r e t ic a lly a p p e a lin g a n d r e a s o n a b le o n n o r m a t iv e g r o u n d s , t h e s e le c t io n o f t h e p r e f e r r e d c r e d it r is k m o d e l h a s b e e n h a m p e r e d b y a f e w c o n s id e r a t io n s . F ir s t , t h e r e is o f t e n a m is m a t c h b e t w e e n t h e o r e t ic a l c o n s t r u c t s a n d t r a d e d s e c u r it ie s : t h e m a j o r it y o f t h e t r a d e d d e b t in s t r u m e n t s a r e c o u p o n p a y in g , w h ile e x t a n t m o d e ls f o c u s a t t e n t io n o n v a lu in g d e f a u lt a b le z e r o s . W h e n d e f a u lt is a f a c t o r , t h e c o u p o n b o n d s a r e n o t a p o r t f o lio o f z e r o s ( a ll r e m a in in g c o u p o n s s h a r e t h e s a m e d e f a u lt t im e ) . S e c o n d , a lt h o u g h c o n c e p t u a lly e le g a n t , t h e m o d e ls in t h e p r e d ic t a b le s t o p p in g t im e c la s s a r e d i(cid:14) c u lt t o im p le m e n t . A s a r g u e d e ls e w h e r e , t h e c a p it a l s t r u c t u r e o f t h e (cid:12) r m is g e n e r a lly f a r t o o c o m p le x t o s p e c if y r e c o v e r y ( t o a ll c la im a n t s ) in t h e e v e n t o f d e f a u lt . M o r e o v e r , a t t h e e m p ir ic a l le v e l, t h e s e m o d e ls g e n e r a t e a c o u n t e r f a c t u a lly lo w s h o r t - t e r m y ie ld s p r e a d f o r h ig h q u a lit y b o r r o w e r s ( s e e t h e r e le v a n t d is c u s s io n in B r iy s a n d 1
d ( c d r t t d a (cid:12) { o T c f w a C c 4 t t e s s e V e r e n n e ( 1 9 9 7 ) , C o llin s - D u f r e s n e a n d G o ld s t e in ( 1 9 9 9 ) , D u (cid:11) e e ( 1 9 9 9 ) , D u (cid:14) e a n d S in g le t o n 1 9 9 9 ) , a n d S h u m w a y ( 2 0 0 1 ) ) . T h e c la s s o f m o d e ls w e t e s t e m p ir ic a lly s h a r e a n u m b e r o f f e a t u r e s in c o m m o n . O n e , o u r h a r a c t e r iz a t io n o f c r e d it r is k r e lie s o n t h e s u r p r is e s t o p p in g t im e a p p r o a c h . I n p a r t ic u la r , w e e v e lo p a c la s s o f c r e d it r is k m o d e ls t h a t in c o r p o r a t e t h e D u (cid:14) e - S in g le t o n ( 1 9 9 9 ) a s s u m p t io n t h a t e c o v e r y ( in d e f a u lt ) is p r o p o r t io n a l t o t h e p r e - d e f a u lt a b le m a r k e t v a lu e o f d e b t . T w o , it is s h o w n h a t t h e p r ic e o f t h e d e f a u lt a b le c o u p o n b o n d c a n b e in t e r p r e t e d a s t h e m a r t in g a le e x p e c t a t io n o f h e p r o m is e d f a c e v a lu e a n d c o u p o n s , w h e n e a c h p a y o (cid:11) is d is c o u n t e d b y t h e r e le v a n t d e f a u lt a b le is c o u n t r a t e ; t h is r a t e e m b o d ie s t im e v a lu e , lo s s a r r iv a l r a t e s a n d r e c o v e r y . T h r e e , w e d e v e lo p in n a ly t ic a l c lo s e d - f o r m , a s e t o f t h r e e - f a c t o r c r e d it r is k m o d e ls t h a t d e p e n d o n s y s t e m a t ic a s w e ll a s r m - s p e c i(cid:12) c d is t r e s s v a r ia b le s . T o b e c o n s is t e n t w it h t h e e x is t in g lit e r a t u r e , w e p o s it t w o f a c t o r s t h e r is k - f r e e in t e r e s t r a t e a n d it s s t o c h a s t ic lo n g - r u n m e a n { t o c a p t u r e m a c r o e c o n o m ic e (cid:11) e c t s n t h e in s t a n t a n e o u s lik e lih o o d o f d e f a u lt ( D u (cid:11) e e ( 1 9 9 8 ) a n d L it t e r m a n a n d S c h e in k m a n ( 1 9 9 1 ) ) . h e (cid:12) r m - s p e c i(cid:12) c d is t r e s s f a c t o r s w e c o n s id e r in c lu d e le v e r a g e , b o o k - t o - m a r k e t , p r o (cid:12) t a b ilit y , la g g e d r e d it s p r e a d , a n d s c a le d e q u it y p r ic e . T h e s o u r c e s o f d e f a u lt r is k a r e a p p a r e n t in o u r e m p ir ic a l r a m e w o r k . O u r e m p ir ic a l s t u d y t h a t e x a m in e s t h e p r ic in g a n d h e d g in g a c c u r a c y o f c r e d it r is k m o d e ls it h o b s e r v a b le e c o n o m ic f a c t o r s ( s ix d is t in c t m o d e ls ) is b a s e d o n a p a n e l o f 9 3 c o r p o r a t io n s n d t r e a s u r y S T R I P S . T h is d a t a s e t is m e r g e d f r o m L e h m a n B r o t h e r s F ix e d I n c o m e D a t a b a s e , O M P U S T A T a n d C R S P . F r o m t h e o r ig in a l d a t a , w e o m it t e d b o n d s t h a t a r e p u t a b le , c a lla b le , o n v e r t ib le , o r h a v e a s in k in g f u n d p r o v is io n . O v e r a ll, t h e (cid:12) lt e r e d d a t a b a s e c o n s is t s o f m o r e t h a n 6 , 0 0 0 c o r p o r a t e c o u p o n b o n d p r ic e s a n d 2 0 , 0 0 0 t r e a s u r y S T R I P S . A t w o - s t e p p r o c e d u r e is f o llo w e d t o im p le m e n t e v e r y c r e d it r is k m o d e l, a n d t o in f e r t h e s t r u c u r a l p a r a m e t e r s . I n t h e (cid:12) r s t s t e p , w e e s t im a t e t h e r is k - n e u t r a liz e d p a r a m e t e r s o f t h e t e r m s t r u c u r e u s in g o n ly t r e a s u r y S T R I P S . K e e p in g t h e t r e a s u r y p a r a m e t e r s c o n s t a n t a c r o s s a ll (cid:12) r m s , w e s t im a t e t h e r is k - n e u t r a liz e d p a r a m e t e r s o f t h e (cid:12) r m - s p e c i(cid:12) c d is t r e s s f a c t o r . W e e m p lo y e d a t w o t e p p r o c e d u r e b e c a u s e t h e j o in t e s t im a t io n is c o m p u t a t io n a lly in f e a s ib le . O u r e m p ir ic a l e x e r c is e s u p p o r t t h e f o llo w in g g e n e r a l (cid:12) n d in g s : (cid:15) F ir m s p e c i(cid:12) c d is t r e s s f a c t o r s s u c h a s le v e r a g e a n d b o o k - t o - m a r k e t a r e p o s it iv e ly r e la t e d t o y ie ld s in t h e (cid:12) r m c r o s s - s e c t io n . T h e r e la t io n s h ip r e m a in s s ig n i(cid:12) c a n t , e v e n a f t e r c o n t r o llin g f o r c r e d it r a t in g ; (cid:15) I n c o r p o r a t in g a s t o c h a s t ic m e a n in t e r e s t r a t e f a c t o r t o t h e c r e d it r is k m o d e l e n h a n c e s it s p e r f o r m a n c e . O u r a n a ly s is in d ic a t e s t h a t t h e s e n s it iv it y o f t h e d e f a u lt a b le d is c o u n t r a t e t o t h e in t e r e s t r a t e is p o s it iv e b u t le s s t h a n u n it y . C o n s is t e n t w it h o u r p r e d ic t io n s , t h e 2
e s t im a t e d u n c o n d it io n a l c r e d it y ie ld is h ig h e r f o r le s s c r e d it w o r t h y (cid:12) r m s ; (cid:15) M e a s u r e d b y b o t h t h e a b s o lu t e p e r c e n t a g e p r ic in g e r r o r s a n d t h e a b s o lu t e y ie ld b a s is p o in t e r r o r s , t h e le v e r a g e r a t io m o d e l a n d t h e b o o k - t o - m a r k e t r a t io m o d e l c o n s is t e n t ly o u t p e r f o r m t h e in t e r e s t r a t e o n ly m o d e l ( t h e c r e d it r is k m o d e l w it h n o (cid:12) r m - s p e c i(cid:12) c d is t r e s s ) . L e v e r a g e a n d b o o k - t o - m a r k e t c o n s id e r a t io n s a r e r e le v a n t f o r a b o u t 7 0 % o f t h e (cid:12) r m s . T h e m o d e l im p r o v e m e n t is m o s t p r o n o u n c e d a m o n g lo n g - t e r m b o n d s ; (cid:15) F o r o u r b o n d s a m p le , in t e r e s t r a t e r is k c a p t u r e s t h e (cid:12) r s t - o r d e r e (cid:11) e c t o f d e f a u lt . O n c e in t e r e s t r a t e c o n s id e r a t io n s a r e t a k e n in t o a c c o u n t , t h e p r ic in g im p r o v e m e n t is m a r g in a l. T h is r e s u lt is r o b u s t a c r o s s s u b s a m p le s r e s t r ic t e d b y c r e d it r a t in g , m a t u r it y a n d in d u s t r ie s ; (cid:15) T h e c r e d it r is k m o d e ls e x h ib it d is t in c t a v e r a g e m is p r ic in g p a t t e r n s : t h e le v e r a g e a n d b o o k t o - m a r k e t r a t io m o d e ls o v e r p r ic e , w h ile t h e in t e r e s t r a t e m o d e l u n d e r p r ic e , lo n g - t e r m b o n d s . T h e e c o n o m e t r ic a n a ly s is o f v a lu a t io n e r r o r s s h o w s t h a t if o n e w e r e t o c o n s id e r e x p a n d in g o n t h e s y s t e m a t ic d e f a u lt f a c t o r s , t h e d e f a u lt p r e m iu m w o u ld b e a g o o d c a n d id a t e ; (cid:15) O u r p r ic in g e r r o r m e t r ic s e s t a b lis h t h a t c r e d it r is k m o d e ls p e r f o r m b e t t e r f o r h ig h g r a d e b o n d s t h a n f o r lo w g r a d e o n e s ; lik e w is e , t h e e m p ir ic a l p e r f o r m a n c e is s u p e r io r f o r lo n g e r m a t u r it y b o n d s . O v e r a ll, o u r in v e s t ig a t io n s u p p o r t s t h e v ie w t h a t t h e c r e d it r is k m o d e ls h a v e p r ic in g p e r f o r m a n c e a d e q u a t e f o r m a r k in g r is k e x p o s u r e s . W e a ls o e v a lu a t e t h e e r r o r s f r o m d e lt a - h e d g in g a s h o r t p o s it io n in t h e d e f a u lt a b le c o u p o n b o n d . I n t h is h e d g in g s t r a t e g y , w e c o n s t r u c t a r e p lic a t in g p o r t f o lio t h a t c o n s is t s o f a p o s it io n in g in t w o z e r o - c o u p o n b o n d s a n d t h e u n d e r ly in g (cid:12) r m ’ s e q u it y . T h is r e s e a r c h e s t a b lis h e s s e v e r a l k e y in s ig h t s . F ir s t , u s in g e q u it y a s a h e d g in g in s t r u m e n t r e d u c e s d y n a m ic h e d g in g e r r o r s . S e c o n d , t h e h e d g in g e (cid:11) e c t iv e n e s s o f t h e s t o c k p r ic e m o d e l is s u p e r io r r e la t iv e t o a lt e r n a t iv e c r e d it r is k m o d e ls ( b a s e d o n a b s o lu t e a n d m e a n p e r c e n t a g e h e d g in g e r r o r s ) . T h ir d , t h e a n a ly s is r e v e a ls t h a t in t e r e s t r a t e r is k is c r u c ia l t o t h e p r o (cid:12) t / lo s s a c c o u n t s o f t h e h e d g e r . A ll c r e d it r is k m o d e ls o v e r - h e d g e t h e t a r g e t . T h is p a p e r is o r g a n iz e d a s f o llo w s . I n S e c t io n 1 , w e p r e s e n t a g e n e r ic f r a m e w o r k t o p r ic e d e f a u lt a b le c o u p o n b o n d s . S e c t io n 2 p r o p o s e s a c la s s o f t h r e e - f a c t o r c r e d it r is k m o d e ls . T h e d e f a u lt a b le c o u p o n b o n d d a t a is d e s c r ib e d in S e c t io n 3 . I n S e c t io n 4 , w e e x a m in e t h e c r o s s - s e c t io n a l r e la t io n s h ip b e t w e e n m a r k e t y ie ld s a n d (cid:12) r m - s p e c i(cid:12) c d is t r e s s f a c t o r s . S e c t io n 5 d e s c r ib e s t h e e s t im a t io n p r o c e d u r e a n d t h e in - s a m p le r e s u lt s . T h e o u t - o f - s a m p le p r ic in g e x e r c is e s a r e c o n d u c t e d in S e c t io n 6 . S e c t io n 7 o u t lin e s t h e h e d g in g s t r a t e g y a n d e x a m in e s t h e h e d g in g p e r f o r m a n c e o f c r e d it r is k m o d e ls . C o n c lu s io n s a r e p r o v id e d in S e c t io n 8 . A ll t e c h n ic a l d e t a ils a r e in t h e A p p e n d ix . 3
1 D (cid:13) T a c c r t p e T p s a o u t P w a t a P r i c i n g D e f a u l t a b l e C o u p o n B o n d s e f a u lt a b le c o u p o n b o n d s , lik e o t h e r d e b t c o n t r a c t s , a r e d e (cid:12) n e d b y t h e ir p r o m is e d s t r e a m o f c a s h o w s t h r o u g h t im e . T y p ic a lly t h e s e c o n s is t o f a p r o m is e d f a c e v a lu e , F , t o b e p a id a t m a t u r it y , a n d a s t r e a m o f c o u p o n p a y m e n t s t o b e p a id in t h e in t e r im . T o a c c o m m o d a t e b o t h d is c r e t e n d c o n t in u o u s c o u p o n p a y m e n t s , w e d e n o t e b y C ( t ) t h e n o n - d e c r e a s in g f u n c t io n o f c u m u la t e d t o u p o n p a y m e n t s u n t il t im e . I n a ll g e n e r a lit y , t h e f u n c t io n C ( t ) n e e d n o t b e d e t e r m in is t ic a n d o u ld d e p e n d o n e c o n o m ic in f o r m a t io n a s it b e c o m e s a v a ila b le . I n a d d it io n t o s p e c if y in g t h e p r o m is e d p a y m e n t s , d e f a u lt a b le d e b t r e c o g n iz e s t h a t t h e r e is a T y T a n d o m t im e a t w h ic h d e f a u lt o c c u r s . A t t h is t im e , a p a y m e n t ( ) is m a d e in f u l(cid:12) llm e n t o f y T h e d e b t o b lig a t io n . T h e r e c o v e r y , ( ) , if a n y , is g e n e r a lly f a r b e lo w t h e v a lu e o f t h e r e m a in in g T (cid:31) t r o m is e d p a y m e n t s . W e a s s o c ia t e w it h t h e r a n d o m t im e , t h e u n it s t e p f u n c t io n ( ) , a s m a d e x a c t b e lo w : t (cid:21) T 1 8 (cid:31) t ( ) = ( 1 ) < 0 O t h e r w is e . : (cid:31) t h e d e f a u lt a b le d e b t c o n t r a c t c a n n o w b e d e (cid:12) n e d b y t h e e n t it ie s : ( F , T , C ( t ) , ( ) , y ( t ) ) t h a t a r e = ; (cid:20) t (cid:20) T ; = ; P r e s u m e d a d a p t e d t o t h e in f o r m a t io n (cid:12) lt r a t io n 0 o f a p r o b a b ilit y s p a c e ( (cid:10) ) , t a t is f y in g t h e u s u a l t e c h n ic a l c o n d it io n s . t b t (cid:17) r s d s W e s u p p o s e t h a t t h e s p o t in t e r e s t r a t e is g iv e n b y r ( t ) a n d ( ) e x p ( ) is t h e 0 (cid:16) (cid:17) R s s o c ia t e d a c c u m u la t io n o f t h e m o n e y m a r k e t a c c o u n t . A c c o r d in g t o D u (cid:14) e ( 1 9 9 6 ) , t h e a b s e n c e f a r b it r a g e o p p o r t u n it ie s is e n s u r e d b y t h e e x is t e n c e o f a p r o b a b ilit y m e a s u r e Q e q u iv a le n t t o P n d e r w h ic h t h e m o n e y m a r k e t d is c o u n t e d g a in s p r o c e s s e s f o r a ll a s s e t s a r e m a r t in g a le s . I t f o llo w s t (cid:28) t h a t t h e t im e p r ic e o f t h e d e f a u lt a b le c o u p o n b o n d w it h m a t u r it y p e r io d s f r o m t im e , d e n o t e d t ; (cid:28) ( ) , is g iv e n b y : t (cid:28) + b t b t ( ) ( ) Q P t ; (cid:28) E (cid:0) (cid:31) u d C u (cid:0) (cid:31) t (cid:28) F ( ) = ( 1 ( ) ) ( ) + ( 1 ( + ) ) t (cid:26) Z b u b t (cid:28) ( ) ( + ) t t (cid:28) + b t ( ) (cid:0) (cid:31) u y u d (cid:31) u j = ; + ( 1 ( ) ) ( ) ( ) ( 2 ) t (cid:27) Z b u ( ) t Q E h e r e is t h e e x p e c t a t io n o p e r a t o r u n d e r t h e p r o b a b ilit y m e a s u r e Q . t T h e (cid:12) r s t in t e g r a l in e q u a t io n ( 2 ) a c c o u n t s f o r t h e s t r e a m o f c o u p o n p a y m e n t s r e c e iv e d a s lo n g s t h e r e is n o d e f a u lt a n d s t o p p e d a t t h e d e f a u lt t im e . T h e s e c o n d t e r m a c c o u n t s f o r t h e r e c e ip t o f h e p r o m is e d f a c e v a lu e g iv e n n o d e f a u lt . F in a lly , t h e la s t in t e g r a l a c c o u n t s f o r t h e s in g le r e c o v e r y d (cid:31) u u (cid:28) t t h e d e f a u lt t im e ( o n n o t in g t h a t ( ) e q u a ls t h e K r o n e c k e r d e lt a f u n c t io n a t = , a n d is 4
z e r o a t t im e s o t h e r t h a n t h e d e f a u lt t im e w h e n it is o n e ) . A s a r t ic u la t e d in D u (cid:14) e a n d S in g le t o n ( 1 9 9 7 , 1 9 9 9 ) a n d M a d a n a n d U n a l ( 1 9 9 8 ) , t h e d i(cid:14) c u lt y in e v a lu a t in g e q u a t io n ( 2 ) c o m e s f r o m (cid:31) t h a v in g t o a d d r e s s t h e d is c o n t in u o u s r a n d o m s t e p p r o c e s s ( ) . M o s t d e f a u lt m o d e ls e m p lo y a s t o p p in g t im e t o c h a r a c t e r iz e d e f a u lt t im e . S p e c i(cid:12) c a lly in t h e c la s s o f m o d e ls t h a t u s e p r e d i c t a b le s t o p p in g t im e , d e f a u lt o c c u r s w h e n a c o n t in u o u s p r o c e s s lik e t h e (cid:12) r m v a lu e r e a c h e s a d e f a u lt b o u n d a r y ( i. e . , M e r t o n ( 1 9 7 4 ) , L o n g s t a (cid:11) a n d S c h w a r t z ( 1 9 9 5 ) a n d C o llin - D u f r e s n e a n d G o ld s t e in ( 1 9 9 9 ) ) . T h e s e c o n d c la s s o f m o d e ls v ie w d e f a u lt a s o c c u r r in g a t a s u r p r i s e s t o p p in g t im e ( D u (cid:14) e a n d S in g le t o n ( 1 9 9 7 , 1 9 9 9 ) , J a r r o w a n d T u r n b u ll ( 1 9 9 5 ) , a n d M a d a n a n d U n a l ( 1 9 9 8 ) ) . T h e s e m o d e ls , t h o u g h s ile n t o n t h e d e (cid:12) n it io n o f t h e d e f a u lt e v e n t , f o c u s t h e ir a t t e n t io n o n t h e in s t a n t a n e o u s lik e lih o o d o f d e f a u lt . O u r f o r m u la t io n o f d e f a u lt in ( 2 ) is c o n s is t e n t w it h t h e s u r p r is e s t o p p in g t im e a p p r o a c h . h t F o r a s u r p r is e d e f a u lt t im e t h a t is a s t o p p in g t im e , t h e r e e x is t s a p o s it iv e p r o c e s s ( ) , c a lle d t h e h a z a r d r a t e p r o c e s s , s u c h t h a t t (cid:31) t (cid:0) (cid:0) (cid:31) u h u d u ( ) ( 1 ( ) ) ( ) ( 3 ) Z 0 Q h t is a m a r t in g a le . W h e n w e h a v e a m a r t in g a le u n d e r t h e p r o b a b ilit y , t h e n w e r e f e r t o ( ) a s t h e h t d t r is k - n e u t r a l h a z a r d r a t e p r o c e s s . H e u r is t ic a lly s p e a k in g , ( ) m o d e ls t h e p r o b a b ilit y o f d e f a u lt h t in t h e in t e r v a l ( t , t + d t ) , t h e in s t a n t a n e o u s lik e lih o o d o f d e f a u lt . T h e p r o c e s s ( ) is a d a p t e d t o G a s u b (cid:12) lt r a t io n o f c o n t in u o u s e v o lv in g in f o r m a t io n . I n t h is c a s e , M a d a n a n d U n a l ( 1 9 9 8 ) s h o w t t h a t : u Q E (cid:0) (cid:31) u j G (cid:0) h s d s : [( 1 ( ) ) ] = e x p ( ) ( 4 ) u t (cid:18) (cid:19) Z t U s in g ( 4 ) a n d it e r a t e d e x p e c t a t io n s , it im m e d ia t e ly f o llo w s t h a t t (cid:28) u t (cid:28) + + b t b t ( ) ( ) Q P t ; (cid:28) E (cid:0) h s d s d C u (cid:0) h s d s F ( ) = e x p ( ) ( ) + e x p ( ) t (cid:26) (cid:18) (cid:19) (cid:18) (cid:19) Z Z Z b u b t (cid:28) ( ) ( + ) t t t t (cid:28) u + b t ( ) (cid:0) h s d s y u h u d u j G : + e x p ( ) ( ) ( ) ( 5 ) t (cid:18) (cid:19) (cid:27) Z Z b u ( ) t t U n lik e e q u a t io n ( 2 ) , t h e p r ic in g e q u a t io n ( 5 ) e lim in a t e s a ll r e f e r e n c e t o t h e d is c o n t in u o u s p r o c e s s (cid:31) t ( ) . E q u a t io n ( 5 ) r e d u c e s t h e p r o b le m o f p r ic in g d e f a u lt a b le c o u p o n d e b t t o t h a t o f p r ic in g n o n - d e f a u lt a b le d e b t w it h a n a lt e r e d d is c o u n t r a t e a n d c a s h (cid:13) o w c la im . (cid:21) t I f w e n o w f o llo w D u (cid:14) e a n d S in g le t o n ( 1 9 9 9 ) a n d d e (cid:12) n e r e c o v e r y a s a p r o p o r t io n , ( ) , o f t h e p r e - d e f a u lt v a lu e o f t h e d e f a u lt a b le d e b t s o t h a t y t (cid:21) t P t ; (cid:28) ; ( ) = ( ) ( ) ( 6 ) 5
t h e n a n a p p lic a t io n o f I t o ’ s le m m a s h o w s t h a t t h e p r ic e o f t h e d e f a u lt a b le d e b t m a y b e w r it t e n a s ( s e e t h e A p p e n d ix f o r in t e r m e d ia t e s t e p s ) : t (cid:28) t u + + Q P t ; (cid:28) E (cid:0) r s h s (cid:0) (cid:21) s d s d C u ( ) = e x p [ ( ) + ( ) ( 1 ( ) ) ] ( ) + t (cid:26) (cid:18) (cid:19) Z Z t t t (cid:28) + : F (cid:0) r s h s (cid:0) (cid:21) s d s j G ( 7 ) e x p [ ( ) + ( ) ( 1 ( ) ) ] t (cid:27) (cid:18) (cid:19) Z t B e c a u s e it is n o t g e n e r a lly p o s s ib le ( in t h e D u (cid:14) e - S in g le t o n a p p r o a c h ) t o s e p a r a t e t h e e (cid:11) e c t s o f h t (cid:0) (cid:21) t h a z a r d r a t e p r o c e s s ( ) f r o m t h a t o f t h e lo s s p r o c e s s ( 1 ( ) ) , d e (cid:12) n e t h e a g g r e g a t e d e f a u lt a b le d is c o u n t r a t e a s : R t (cid:17) r t h t (cid:0) (cid:21) t : ( ) ( ) + ( ) [ 1 ( ) ] ( 8 ) T h is d is c o u n t r a t e c o n s o lid a t e s t im e v a lu e , lo s s a r r iv a l r a t e s a n d r e c o v e r y c o n s id e r a t io n s . T h e n , c t in t h e c a s e o f d e t e r m in is t ic a n d c o n t in u o u s c o u p o n r a t e ( ) , t h e d e f a u lt a b le d e b t e q u a t io n c a n b e s im p li(cid:12) e d a s : (cid:28) (cid:3) (cid:3) P t ; (cid:28) c t u P t ; u d u F P t ; (cid:28) ; ( ) = ( + ) ( ) + ( ) ( 9 ) Z 0 w h e r e t u + (cid:3) Q P t ; u E (cid:0) R s d s j G ; ( ) = e x p ( ) ( 1 0 ) t t (cid:26) (cid:18) (cid:19) (cid:27) Z t t u is t h e p r ic e o f t h e u n it f a c e d e f a u lt a b le z e r o - c o u p o n b o n d w it h m a t u r it y + . R t I n t h e n e x t s e c t io n , w e d e r iv e m o d e ls f o r ( ) t h a t le a d t o e m p ir ic a lly t e s t a b le c lo s e d - f o r m m o d e ls f o r t h e p r ic e o f d e f a u lt a b le c o u p o n d e b t . I n e a c h c r e d it r is k m o d e l, t h e p r ic e o f d e f a u lt a b le d is c o u n t b o n d s is e x p o n e n t ia l a (cid:14) n e in t h e s t a t e o f t h e e c o n o m y . 2 A C l a s s o f C r e d i t R i s k M o d e l s C o n s id e r t h e f a m ily o f a g g r e g a t e d e f a u lt a b le d is c o u n t r a t e m o d e ls s h o w n b e lo w ( e a c h (cid:12) r m is n in d e x e d b y ) : R t r t X t ; n ; (cid:1) (cid:1) (cid:1) ; N ; ( ) = (cid:3) + (cid:3) ( ) + (cid:3) ( ) = 1 ( 1 1 ) n ;n r ;n x ;n n 0 X t w h e r e r ( t ) is t h e s p o t in t e r e s t r a t e a n d ( ) s u r r o g a t e s (cid:12) r m - s p e c i(cid:12) c d is t r e s s . T h is s p e c i(cid:12) c a t io n n is t h e o r e t ic a lly r e a s o n a b le , a s it in c o r p o r a t e s b o t h a n e c o n o m y - w id e v a r ia b le a n d a (cid:12) r m - s p e c i(cid:12) c v a r ia b le . T o k e e p a p a r s im o n io u s f a c t o r s t r u c t u r e , w e h a v e a s s u m e d t h a t d is t r e s s is d r iv e n b y X t R t a s in g le - f a c t o r ( ) a n d a s in g le s y s t e m a t ic r is k f a c t o r r ( t ) . O u r c h a r a c t e r iz a t io n o f ( ) is n n X t s u (cid:14) c ie n t ly v e r s a t ile t o a c c o m m o d a t e a K - f a c t o r m o d e l o f ( ) , h o w e v e r . L ik e it s d e f a u lt - f r e e n 6
p r e d e c e s s o r , t h e lin e a r it y o f R ( t ) in r ( t ) a n d X ( t ) is e m p lo y e d f o r a n a ly t ic a l t r a c t a b ilit y . E q u a t io n ( 1 1 ) f o r m s a c o n v e n ie n t b a s is f o r t h e e m p ir ic a l a n a ly s is o f c r e d it r is k m o d e ls . A lt h o u g h n o t y e t d e r iv e d in c lo s e d - f o r m , t h e c o e (cid:14) c ie n t s (cid:3) , (cid:3) a n d (cid:3) a r e k e y t o u n d e r s t a n d in g r x 0 v a r ia t io n s in d e f a u lt a b le y ie ld s . (cid:3) m e a s u r e s t h e le v e l o f t h e u n c o n d it io n a l in s t a n t a n e o u s c r e d it 0 y ie ld . I f (cid:3) is p o s it iv e , d e f a u lt a b le b o n d y ie ld s a r e p o s it iv e ly r e la t e d t o in t e r e s t r a t e s . G iv e n t h e r e x is t in g e v id e n c e o n c o - m o v e m e n t s b e t w e e n t r e a s u r y a n d c o r p o r a t e y ie ld c u r v e s ( D u (cid:11) e e ( 1 9 9 8 ) ) , it is e x p e c t e d t h a t (cid:3) is p o s it iv e . S im ila r ly , (cid:3) a s s e s s e s t h e s ig n i(cid:12) c a n c e o f t h e p a r t ic u la r (cid:12) r m - s p e c i(cid:12) c r x d is t r e s s v a r ia b le . A s s u m in g t h a t X ( t ) is p o s it iv e ly a s s o c ia t e d w it h (cid:12) r m - s p e c i(cid:12) c d is t r e s s , t h e c r e d it > q u a lit y o f t h e (cid:12) r m d e t e r io r a t e s w h e n d is t r e s s r is e s ( p r o v id e d (cid:3) 0 ) . x ;n C A S E 1 : T h r e e s p e c ia l c a s e s o f ( 1 1 ) a r e o f r e le v a n c e t o t h e lit e r a t u r e . S e t t in g (cid:3) = (cid:3) = 0 a n d x 0 C A S E 2 : (cid:3) = 1 g iv e s t h e t e r m s t r u c t u r e o f d e f a u lt - f r e e b o n d s . U n d e r t h e p a r a m e t r ic r e s t r ic t io n r (cid:3) = 0 , o n e o b t a in s t h e c la s s o f c r e d it r is k m o d e ls c o n s id e r e d b y D u (cid:14) e a n d S in g le t o n ( 1 9 9 7 ) . x ;n X t W h e n ( ) is a h id d e n M a r k o v v a r ia b le , o u r f r a m e w o r k a d m it s t h e s p e c i(cid:12) c a t io n a d o p t e d in D u (cid:11) e e C A S E 3 : ( 1 9 9 9 ) . T h e r e s t r ic t io n (cid:3) = (cid:3) = 0 r e d u c e s t o t h e m o d e l o f J a r r o w a n d T u r n b u ll r x ( 1 9 9 5 ) . E q u a t io n ( 1 1 ) in d ic a t e s t h a t c r o s s - s e c t io n a l v a r ia t io n s in c r e d it r is k a r e p r im a r ily d u e t o X c r o s s - s e c t io n a l v a r ia t io n s in (cid:3) , (cid:3) , (cid:3) , a n d ( t h e in t e r e s t r a t e is c o m m o n t o a ll (cid:12) r m s ) . ;n r ;n x ;n n 0 A la t e r a n a ly s is s h o w s t h a t t h e p a r a m e t e r s o f t h e d e f a u lt a b le d is c o u n t r a t e v a r y s y s t e m a t ic a lly w it h c r e d it r a t in g , a s m o d e le d a ls o b y J a r r o w , L a n d o , a n d T u r n b u ll ( 1 9 9 7 ) . T o b e s p e c i(cid:12) c , t h e m a g n it u d e o f t h e u n c o n d it io n a l in s t a n t a n e o u s c r e d it y ie ld , (cid:3) , is lo w e r f o r h ig h e r q u a lit y (cid:12) r m s ;n 0 r e la t iv e t o it s m o r e d is t r e s s e d c o u n t e r p a r t s . T o o b t a in a c la s s o f t r a c t a b le c r e d it r is k m o d e ls , t w o m o d e lin g d e c is io n s a r e m a d e f o r t h e r e m a in d e r o f t h e p a p e r . W e (cid:12) r s t s p e c if y t h e d y n a m ic s o f t h e n o m in a l in t e r e s t r a t e a n d t h e (cid:12) r m - s p e c i(cid:12) c d is t r e s s f a c t o r . N e x t , w e j u s t if y o u r c h o ic e f o r t h e d is t r e s s f a c t o r s . A t t h e o u t s e t , r t a s s u m e t h a t t h e in t e r e s t r a t e , ( ) , e v o lv e s a c c o r d in g t o a t w o - f a c t o r m o d e l ( u n d e r t h e e q u iv a le n t m a r t in g a le m e a s u r e ) : d r t (cid:20) z t (cid:0) r t d t (cid:27) d ! t ; ( ) = [ ( ) ( ) ] + ( ) ( 1 2 ) r r r d z t (cid:20) (cid:22) (cid:0) z t d t (cid:27) d ! t ; ( ) = [ ( ) ] + ( ) ( 1 3 ) z z z z z t ! t ! t w h e r e t h e lo n g - r u n m e a n o f t h e s h o r t r a t e is r e p r e s e n t e d b y ( ) . ( ) a n d ( ) a r e s t a n d a r d r z (cid:26) (cid:20) (cid:20) B r o w n ia n m o t io n s , w it h c o r r e la t io n . A s is c o n v e n t io n a lly in t e r p r e t e d , ( ) a r e t h e r a t e s r ;z r z r t z t (cid:27) (cid:27) o f m e a n - r e v e r s io n f o r ( ) ( ( ) ) . T h e d i(cid:11) u s io n c o e (cid:14) c ie n t s a n d a r e c o n s t a n t s . z ( t ) is r z u n o b s e r v a b le a n d w ill b e in f e r r e d f r o m t h e t r e a s u r y y ie ld c u r v e . (cid:28) U n d e r t h e s t a t e d a s s u m p t io n s ( 1 2 ) - ( 1 3 ) , t h e p r ic e o f d e f a u lt - f r e e d is c o u n t b o n d m a t u r in g in 7
B t ; (cid:28) p e r io d s f r o m t im e t , ( ) , is : t (cid:28) + Q B t ; (cid:28) (cid:17) E (cid:0) r s d s (cid:0) (cid:11) (cid:28) (cid:0) (cid:12) (cid:28) r t (cid:0) (cid:13) (cid:28) z t ; ( ) e x p ( ) = e x p ( ) ( ) ( ) ( ) ( ) ( 1 4 ) t (cid:26) (cid:18) (cid:19) (cid:27) Z t h i (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) 2 k (cid:28) (cid:20) (cid:28) (cid:20) (cid:28) (cid:20) (cid:28) 1 e x p ( ) 1 e x p ( ) (cid:28) r z e x p ( ) e x p ( ) z r 1 2 (cid:12) (cid:28) (cid:17) (cid:13) (cid:28) (cid:17) (cid:11) (cid:28) (cid:17) (cid:0) (cid:27) (cid:12) s d s (cid:0) w h e r e ( ) ( ) ( ) ( ) , + a n d (cid:0) r 0 k (cid:20) (cid:20) (cid:20) 2 r z z r (cid:28) (cid:28) (cid:28) R 1 2 2 (cid:27) (cid:13) s d s (cid:20) (cid:22) (cid:13) s d s (cid:0) (cid:26) (cid:27) (cid:27) (cid:12) s (cid:13) s d s ( ) + ( ) ( ) ( ) . T h e t w o - f a c t o r m o d e l is a d o p t e d f o r z z r ;z r z z 0 0 0 2 R R R s e v e r a l r e a s o n s . F ir s t , w h e n (cid:12) t t e d t o t h e t r e a s u r y y ie ld c u r v e , it is f o u n d t o r e d u c e t h e e m p ir ic a l (cid:12) t t in g e r r o r s r e la t iv e t o t h e o n e - f a c t o r c o u n t e r p a r t ( s e e a ls o , a m o n g o t h e r s , B u h le r , U h r ig , W a lt e r , a n d W e b b e r ( 1 9 9 9 ) a n d D a i a n d S in g le t o n ( 2 0 0 0 ) ) . S t a t is t ic a lly , t h e o n e - f a c t o r m o d e l is r e j e c t e d in f a v o r o f a t w o - f a c t o r m o d e l o f in t e r e s t r a t e s . S e c o n d , L it t e r m a n a n d S c h e in k m a n ( 1 9 9 1 ) , C h e n a n d S c o t t ( 1 9 9 3 ) a n d D u (cid:11) e e ( 1 9 9 8 ) h a v e s h o w n t h a t m o v e m e n t s in t h e le v e l a n d s lo p e o f t h e y ie ld c u r v e c a p t u r e a la r g e f r a c t io n o f t r e a s u r y t e r m s t r u c t u r e v a r ia t io n s . T h e s e c o n s id e r a t io n s s u g g e s t t h a t a t w o - f a c t o r m o d e l is d e s ir a b le o n e m p ir ic a l a n d t h e o r e t ic a l g r o u n d s . F in a lly , t h o u g h m o r e c o m p le x in t e r e s t r a t e m o d e ls c o u ld b e e m p lo y e d , t h is c o m e s a t t h e c o s t o f a lo s s o f p a r s im o n y a n d im p le m e n t a b ilit y . N e x t , f o r it s a n a ly t ic a l t r a c t a b ilit y , w e a s s u m e t h a t t h e u n d e r ly in g d is t r e s s f a c t o r o b e y s a p r o c e s s o f t h e t y p e : d X t (cid:20) (cid:22) (cid:0) X t d t (cid:27) d ! t ; ( ) = [ ( ) ] + ( ) ( 1 5 ) n x ;n x ;n n x ;n x ;n ! (cid:26) (cid:17) ! ; ! X t w h e r e is a s t a n d a r d B r o w n ia n m o t io n . L e t C o v ( ) . F o r e a c h ( ) p r o x y , a s s u m p - x r ;x t r x n t io n s ( 1 2 ) - ( 1 3 ) a n d ( 1 5 ) le a d t o a d is t in c t m o d e l o f c r e d it r is k . S in c e t h e s t r u c t u r a l p a r a m e t e r s h a v e a s t a n d a r d in t e r p r e t a t io n , u n n e c e s s a r y r e p e t it io n is a v o id e d . N o t ic e t h a t t h e f o r m o f ( 1 5 ) is a r o b u s t t h r e e p a r a m e t e r s p e c i(cid:12) c a t io n f o r t h e p r o c e s s o f t h e (cid:12) r m - s p e c i(cid:12) c d is t r e s s f a c t o r . (cid:28) C o n s id e r n o w t h e p r ic e o f a u n it f a c e d e f a u lt a b le d is c o u n t b o n d w it h p e r io d s le f t t o m a t u r it y . r X n U s in g t h e d y n a m ic s o f a n d a n d s o lv in g ( 1 0 ) , w e h a v e ( t h e s u b s c r ip t is s u p p r e s s e d ) : (cid:3) P t ; (cid:28) (cid:0) (cid:11) (cid:28) (cid:0) (cid:12) (cid:28) r t (cid:0) (cid:13) (cid:28) z t (cid:0) (cid:18) (cid:28) X t ; ( ) = e x p [ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ] ( 1 6 ) w h e r e : (cid:0) (cid:0) k (cid:28) (cid:3) [ 1 e x p ( ) ] r r (cid:12) (cid:28) (cid:17) ; ( ) ( 1 7 ) k r (cid:0) (cid:0) (cid:20) (cid:28) (cid:0) (cid:20) (cid:28) (cid:0) (cid:0) (cid:20) (cid:28) (cid:3) [ 1 e x p ( ) ] (cid:3) [ e x p ( ) e x p ( ) ] r z r z r (cid:13) (cid:28) (cid:17) ; ( ) + ( 1 8 ) (cid:20) (cid:20) (cid:0) (cid:20) ( ) z z r (cid:0) (cid:0) (cid:20) (cid:28) (cid:3) [ 1 e x p ( ) ] x x (cid:18) (cid:28) (cid:17) ; ( ) ( 1 9 ) (cid:20) x 8
a n d (cid:28) (cid:28) (cid:28) (cid:28) 1 1 1 2 2 2 2 2 2 (cid:11) (cid:28) (cid:17) (cid:28) (cid:0) (cid:27) (cid:27) (cid:27) (cid:12) s d s (cid:20) (cid:22) (cid:18) s d s (cid:0) (cid:13) s d s (cid:0) (cid:18) s d s ( ) (cid:3) ( ) + ( ) ( ) ( ) x x 0 r z x Z Z Z Z 2 2 2 0 0 0 0 (cid:28) (cid:28) (cid:28) (cid:13) s d s (cid:0) (cid:26) (cid:27) (cid:27) (cid:12) s (cid:18) s d s (cid:0) (cid:26) (cid:27) (cid:27) (cid:12) s (cid:13) s d s : (cid:20) (cid:22) ( ) ( ) ( ) ( ) ( ) ( 2 0 ) + r ;x r x r ;z r z z z Z Z Z 0 0 0 W e (cid:12) r s t o b s e r v e t h a t t h e p r ic e o f t h e d e f a u lt a b le z e r o is e x p o n e n t ia l a (cid:14) n e in t h r e e s t a t e v a r ia b le s : t h e in t e r e s t r a t e , t h e s t o c h a s t ic lo n g - r u n m e a n in t e r e s t r a t e a n d t h e (cid:12) r m - s p e c i(cid:12) c d is t r e s s f a c t o r . T h e m o d e l h a s 1 3 s t r u c t u r a l p a r a m e t e r s . S p e c i(cid:12) c a lly , t h e r e a r e 3 p a r a m e t e r s in t h e d e f a u lt a b le d is c o u n t r a t e s p e c i(cid:12) c a t io n , 6 in t h e in t e r e s t r a t e p r o c e s s a n d 4 a s s o c ia t e d w it h t h e d y n a m ic s o f t h e (cid:12) r m - s p e c i(cid:12) c d is t r e s s f a c t o r . S e c o n d , u n d e r p o s it iv it y o f (cid:3) a n d t h e s p e e d o f a d j u s t m e n t s in t h e in t e r e s t r a t e p r o c e s s , t h e r r t z t d e f a u lt a b le d is c o u n t b o n d p r ic e is n e g a t iv e ly r e la t e d t o ( ) a n d ( ) . M o r e p r e c is e ly , w e h a v e (cid:3) @ P t ; (cid:28) ( ) (cid:3) (cid:3) t ; (cid:28) (cid:17) (cid:0) (cid:12) (cid:28) P t ; (cid:28) < ; (cid:1) ( ) = ( ) ( ) 0 ( 2 1 ) r @ r (cid:3) @ P t ; (cid:28) ( ) (cid:3) (cid:3) (cid:0) (cid:13) (cid:28) P t ; (cid:28) < : t ; (cid:28) (cid:17) = ( ) ( ) 0 ( 2 2 ) (cid:1) ( ) z @ z F u r t h e r m o r e , t h e b o n d p r ic e is a ls o n e g a t iv e ly a s s o c ia t e d w it h t h e d is t r e s s f a c t o r , a s s e e n b y (cid:3) @ P t ; (cid:28) ( ) (cid:3) (cid:3) t ; (cid:28) (cid:17) (cid:0) (cid:18) (cid:28) P t ; (cid:28) < ; (cid:1) ( ) = ( ) ( ) 0 ( 2 3 ) x @ X > p r o v id e d (cid:3) 0 . T h e s e e x p r e s s io n s f o r t h e lo c a l r is k e x p o s u r e s a r e la t e r e m p lo y e d t o d e v e lo p x d e lt a - n e u t r a l h e d g e s f o r m a r k e d - t o - m a r k e t r is k s . (cid:28) T h ir d , t h e y ie ld t o m a t u r it y o f t h e d e f a u lt a b le d is c o u n t b o n d , f o r m a t u r it y , is (cid:3) P t ; (cid:28) (cid:11) (cid:28) (cid:12) (cid:28) r t (cid:13) (cid:28) z t (cid:18) (cid:28) X t lo g [ ( ) ] ( ) + ( ) ( ) + ( ) ( ) + ( ) ( ) (cid:3) Y t ; (cid:28) (cid:17) (cid:0) : ( ) = ( 2 4 ) (cid:28) (cid:28) F r o m e x p r e s s io n s ( 1 7 ) - ( 1 9 ) , w e c a n s e e t h a t t h e s e n s it iv it y o f y ie ld t o m a t u r it y w it h r e s p e c t t o e a c h o f t h e t h r e e s t a t e v a r ia b le s is d e c r e a s in g in m a t u r it y . B a s e d o n t h is f e a t u r e o f t h e m o d e l, t h e r is k e x p o s u r e s o f lo n g - t e r m d e f a u lt a b le d is c o u n t b o n d s a r e lo w e r a n d it m a y b e p o s s ib le t o ig n o r e t h e s e r is k s in d e v e lo p in g h e d g e s . A lb e it w it h d i(cid:11) e r e n t r is k e x p o s u r e s , e q u a t io n ( 2 4 ) d e c o m p o s e s t h e c r e d it y ie ld in t o a s y s t e m a t ic r is k c o m p o n e n t a n d a (cid:12) r m - s p e c i(cid:12) c r is k c o m p o n e n t . (cid:3) ! Y t ; (cid:28) r t X t (cid:28) ! 1 W h e n lim ( ) = (cid:3) + (cid:3) ( ) + (cid:3) ( ) . L e t t in g , w e o b t a in t h e a s y m p - (cid:28) ;n r ;n x ;n n 0 0 2 2 2 2 2 2 2 (cid:3) (cid:26) (cid:27) (cid:27) (cid:27) r ;z r z (cid:3) (cid:27) (cid:27) (cid:3) (cid:3) (cid:3) r r r z r x x 2 2 2 Y t ; 1 (cid:0) (cid:22) (cid:0) (cid:0) (cid:22) (cid:0) (cid:0) t o t ic d e f a u lt a b le y ie ld a s : ( ) = (cid:3) + (cid:3) + (cid:3) z r x x 0 (cid:20) (cid:20) (cid:20) r z (cid:20) (cid:20) 2 2 2 r z x (cid:26) (cid:27) (cid:27) r ;x r x (cid:3) x (cid:3) r . O u r t h r e e - f a c t o r m o d e l o f d e f a u lt a b le b o n d s o (cid:11) e r s t h e (cid:13) e x ib ilit y t o p r o d u c e v a r io u s (cid:20) (cid:20) r x d e f a u lt a b le y ie ld c u r v e s h a p e s in c lu d in g d o u b le h u m p e d y ie ld c u r v e . 9
I s w f d s d h p f T h e p r ic e o f t h e d e f a u lt a b le c o u p o n b o n d c a n n o w b e c o m p u t e d b y in s e r t in g ( 1 6 ) in t o ( 9 ) . Y t ; (cid:28) n p a r t ic u la r , t h e y ie ld t o m a t u r it y o n a d e f a u lt a b le c o u p o n b o n d , ( ) , c a n b e r e c o v e r e d b y o lv in g t h e f o llo w in g n o n - lin e a r e q u a t io n : (cid:28) P t ; (cid:28) (cid:0) c t u (cid:0) Y t ; (cid:28) u d u (cid:0) F (cid:0) Y t ; (cid:28) (cid:28) ; 0 = ( ) ( + ) e x p [ ( ) ] e x p [ ( ) ] ( 2 5 ) Z 0 h ic h is , in p r in c ip le , s o lv a b le g iv e n c o u p o n b o n d p r ic e a n d t h e p r o m is e d c a s h (cid:13) o w s t r e a m ( o n e (cid:28) o r e a c h ) . T h e y ie ld c u r v e o f t h e d e f a u lt a b le c o u p o n b o n d in h e r it s t h e s a m e s t r u c t u r e a s t h a t is p la y e d b y t h e z e r o s . B e f o r e c lo s in g t h is s e c t io n , w e d is c u s s p r o x ie s f o r t h e (cid:12) r m - s p e c i(cid:12) c d is t r e s s f a c t o r u s e d in t h e X t t u d y . F iv e c a n d id a t e s f o r ( ) a r e s e le c t e d f o r t h e ir e m p ir ic a l p la u s ib ilit y . E a c h c h o ic e is a n im e n s io n le s s q u a n t it y a n d le a d s t o a d is t in c t t e s t a b le m o d e l o f c r e d it r is k . W e la t e r s u m m a r iz e o w w e ll t h e s e v a r ia b le s e x p la in t h e c r o s s - s e c t io n o f c o r p o r a t e y ie ld s . T h is a n a ly s is s h o w s t h a t t h e r o p o s e d v a r ia b le s h a v e in c r e m e n t a l in f o r m a t io n o n c o r p o r a t e y ie ld s . E a c h (cid:12) r m - s p e c i(cid:12) c d is t r e s s a c t o r is d is c u s s e d in t u r n : X l e v e r a g e 1 . A s s u m e t h a t is (cid:12) r m . S t a n d a r d c o r p o r a t e (cid:12) n a n c e t h e o r y s u g g e s t s t h a t le v e r a g e n c a p t u r e s (cid:12) r m - le v e l d is t r e s s . L e v e r a g e is a ls o a k e y in g r e d ie n t in t h e s t r u c t u r a l m o d e ls o f M e r t o n ( 1 9 7 4 ) , L o n g s t a (cid:11) a n d S c h w a r t z ( 1 9 9 5 ) a n d C o llin - D u f r e s n e a n d G o ld s t e in ( 1 9 9 9 ) . L e v e r a g e R a t i o M o d e l W e w ill r e f e r t o t h is m o d e l a s t h e . X B o o k - t o - 2 . L e t b e t h e r a t io o f a (cid:12) r m ’ s b o o k v a lu e o f e q u it y t o it s m a r k e t v a lu e ( i. e . , n M a r k e t ) . A c c o r d in g t o F a m a a n d F r e n c h ( 1 9 9 2 ) , (cid:12) r m s w it h h ig h b o o k - t o - m a r k e t a r e r e la t iv e ly m o r e d is t r e s s e d . I n c o n t r a s t , (cid:12) r m s w it h lo w b o o k - t o - m a r k e t a r e s t r o n g e r (cid:12) r m s w it h B / M R a t i o M o d e l g o o d c a s h (cid:13) o w p r o s p e c t s . T h is m o d e l w ill b e r e f e r r e d t o a s t h e . X P r o (cid:12) t a b i l i t y 3 . T h e n e x t c a n d id a t e f o r is t h e o f a (cid:12) r m . T h is v a r ia b le r e (cid:13) e c t s t h e a b ilit y n o f a (cid:12) r m t o h o n o r d e b t o b lig a t io n s o u t o f it s o p e r a t in g in c o m e ( T it m a n a n d W e s s e ls ( 1 9 8 8 ) ) . I n t h is m o d e l, d e f a u lt p r o b a b ilit y is in v e r s e ly r e la t e d t o p r o (cid:12) t a b ilit y . C r e d it r is k m o d e ls t h a t P r o (cid:12) t a b i l i t y M o d e l in c o r p o r a t e p r o (cid:12) t a b ilit y c o n c e r n s w ill r e f e r r e d t o a s t h e . X 4 . F o r t h e f o u r t h m o d e l, w e a s s u m e t h a t is d r iv e n b y t h e la g g e d c r e d it s p r e a d ( in t h e s p ir it n L a g g e d S p r e a d M o d e l o f D u (cid:14) e a n d S in g le t o n ( 1 9 9 7 ) ) . W e w ill r e f e r t o t h is m o d e l a s t h e . X s t (cid:17) S t = b t 5 . F in a lly , is t h e s c a le d s t o c k p r ic e s ( t ) , i. e . , ( ) lo g ( ( ) ( ) ) . I n t h is c a s e , w e p o s it n 1 2 d s t (cid:0) (cid:27) d t (cid:27) d ! t ( ) = + ( ) , w h ic h is a p a r a m e t r ic s p e c ia l c a s e o f ( 1 5 ) . T h e r e s u lt in g c r e d it x x x 2 r is k m o d e l h a s 1 1 p a r a m e t e r s . T h e u s e o f e q u it y p r ic e s is e s p e c ia lly a t t r a c t iv e f r o m t h e 1 0
h e d g in g p e r s p e c t iv e ( M a d a n a n d U n a l ( 1 9 9 8 ) ) . W e w ill r e f e r t o t h is c r e d it r is k m o d e l a s t h e S t o c k P r i c e M o d e l . A s a lr e a d y m e n t io n e d , w h e n (cid:3) is s e t e q u a l t o z e r o , t h e c r e d it r is k m o d e l r e d u c e s t o a t w o - x f a c t o r m o d e l w h e r e m o v e m e n t s in t h e t r e a s u r y c u r v e a r e t h e s o le s o u r c e o f c r e d it r is k . B e c a u s e o f t h is p r o p e r t y , t h e m o d e l w it h (cid:3) = 0 w ill b e u s e d t o b e n c h m a r k t h e p e r f o r m a n c e o f t h e a b o v e x a lt e r n a t iv e c r e d it r is k m o d e ls . T h e g e n e r a l m o d e l ( 1 1 ) is c o m p a r a b le t o D u (cid:11) e e ( 1 9 9 9 ) . B u t in t h e in t e r e s t o f e v a lu a t in g t h e o u t - o f - s a m p le p r ic in g a n d h e d g in g p e r f o r m a n c e o f c r e d it r is k m o d e ls , w e r e p la c e h id d e n f a c t o r s b y id e n t i(cid:12) a b le f a c t o r s . T o lim it t h e s c o p e o f o u r in v e s t ig a t io n , w e r e s t r ic t o u r a t t e n t io n t o t h e c la s s o f r e d u c e d - f o r m d e f a u lt a b le d is c o u n t r a t e m o d e ls . F ir s t , a s d e f a u lt is t r ig g e r e d o n ly a t m a t u r it y , t h e M e r t o n ( 1 9 7 4 ) m o d e l c a n n o t b e e a s ily a d a p t e d t o p r ic e d e f a u lt a b le c o u p o n b o n d s . S e c o n d , u n le s s j u m p s a r e a d d e d , t h e s e m o d e ls im p ly a c o u n t e r f a c t u a lly lo w c r e d it s p r e a d s f o r s h o r t - m a t u r it y d e f a u lt a b le b o n d s . O n t h e o t h e r h a n d , C o llin s - D u f r e s n e a n d G o ld s t e in ( 1 9 9 9 ) h a v e s h o w n t h a t t h e s t r u c t u r a l m o d e ls in t h e o n e - f a c t o r c la s s h a v e u n d e s ir a b le lo n g - r u n y ie ld p r o p e r t ie s . F u r t h e r m o r e , J o n e s , M a s o n , a n d R o s e n f e ld ( 1 9 8 4 ) , W e i a n d G u o ( 1 9 9 7 ) a n d E o m , H e lw e g e , a n d H u a n g ( 2 0 0 0 ) p r e s e n t e v id e n c e r e j e c t in g s u c h m o d e ls . I n s u m m a r y , o u r f o c u s is o n r e d u c e d - f o r m d e f a u lt r is k m o d e ls w it h id e n t i(cid:12) a b le e c o n o m ic f a c t o r s t h a t m a y o r m a y n o t b e t r a d a b le . 3 T h e D a t a o n D e f a u l t a b l e C o u p o n B o n d s T h e d a t a f o r t h e s t u d y is m e r g e d f r o m s e v e r a l s o u r c e s . F ir s t , c o r p o r a t e c o u p o n b o n d p r ic e s , y ie ld s a n d t r e a s u r y s e c u r it y p r ic e s a r e e x t r a c t e d f r o m t h e L e h m a n B r o t h e r s F ix e d I n c o m e D a t a b a s e . T h is d a t a b a s e h a s o v e r 2 8 , 0 0 0 in s t r u m e n t s a n d c o n t a in s in f o r m a t io n o n p u b lic ly t r a d e d n o n c o n v e r t ib le d e b t , w it h p r in c ip a l in e x c e s s o f o n e m illio n d o lla r s . F o r e a c h (cid:12) x e d in c o m e s e c u r it y , t h e d a t a b a s e h a s e n t r ie s o n , a m o n g o t h e r s , ( i) t h e m o n t h - e n d (cid:13) a t p r ic e , ( ii) t h e a c c r u e d in t e r e s t , ( iii) t h e m a t u r it y d a t e , ( iv ) t h e a m o u n t o f c o u p o n a n d p r in c ip a l, a n d ( v ) t h e y ie ld t o m a t u r it y . D e b t is s u e s a r e c la s s i(cid:12) e d a s c a lla b le , p u t a b le , o r s u b o r d in a t e d ( o r h a v in g a s in k in g f u n d p r o v is io n ) . E a c h d e b t c o n t r a c t is a s s ig n e d a n in d u s t r y c la s s i(cid:12) c a t io n a n d a c r e d it r a t in g ( F it c h , M o o d y ’ s a n d S t a n d a r d a n d P o o r ’ s ) . I n t h is s t u d y , w e e m p lo y t h e S t a n d a r d a n d P o o r ’ s c r e d it r a t in g . T h e d a t a b a s e c o v e r s t h e p e r io d f r o m J a n u a r y 1 9 7 3 t o M a r c h 1 9 9 8 . S e v e r a l e x c lu s io n a r y (cid:12) lt e r s a r e im p o s e d t o c o n s t r u c t t h e s a m p le o f d e f a u lt a b le c o u p o n b o n d s . F ir s t , t r a d e r b id q u o t e s a r e u s e d in o u r a n a ly s is ( a s k q u o t e s a r e n o t r e c o r d e d ) . B e c a u s e t h e s e c o n d a r y m a r k e t f o r c o r p o r a t e b o n d s is r e la t iv e ly illiq u id , t r a d e r s a r e o f t e n u n w illin g t o s u p p ly q u o t e s in t h e p r e s e n c e o f in s u (cid:14) c ie n t t r a d in g . F o r t h e s e illiq u id b o n d s , m a t r ix q u o t e s a r e g e n e r a t e d 1 1
( a c c o r d in g t o s o m e in t e r n a l m o d e l b y L e h m a n B r o t h e r s ) a n d r e c o r d e d in t h e d a t a b a s e . S in c e o u r p r im a r y g o a l is t o t e s t t h e p e r f o r m a n c e o f c r e d it r is k m o d e ls a g a in s t m a r k e t p r ic e s , m a t r ix q u o t e s a r e a v o id e d . S e c o n d , b o n d s w it h e m b e d d e d o p t io n s a r e d is c a r d e d . W e a ls o e lim in a t e d s u c h b o n d s a s p a s s t h r o u g h a n d a s s e t - b a c k e d s e c u r it ie s . F o r c o n s is t e n c y , o n ly r e g u la r b o n d s a r e c o n s id e r e d . T o m it ig a t e m a r k e t m ic r o s t r u c t u r e b ia s e s , d e b t s w it h t im e - t o - m a t u r it y le s s t h a n 1 y e a r a r e e x c lu d e d . N e x t , w e o n ly in c lu d e b o n d s t h a t p a y s e m i- a n n u a l c o u p o n s . I n f a c t , o n ly a f e w d e f a u lt a b le d is c o u n t b o n d s a n d q u a r t e r ly / a n n u a l c o u p o n p a y in g b o n d s e x is t in t h e d a t a b a s e . F in a lly , t o f a c ilit a t e m o d e l im p le m e n t a t io n , w e r e t a in e d (cid:12) r m s w it h a t le a s t f o u r b o n d is s u e s o u t s t a n d in g e a c h m o n t h a n d d a t a a v a ila b ilit y o f o v e r t w o y e a r s . 1 8 3 (cid:12) r m s s a t is f y a ll t h e a b o v e r e q u ir e m e n t s . T h e r e s u lt in g (cid:12) r m u n iv e r s e is m a t c h e d w it h e q u it y p r ic e a n d a c c o u n t in g d a t a f r o m C R S P a n d C O M P U S T A T , r e s p e c t iv e ly . D u e t o t h e u n a v a ila b ilit y o f e q u it y p r ic e , 8 0 (cid:12) r m s w e r e d r o p p e d . O f t h e r e m a in in g 1 0 3 (cid:12) r m s , 1 0 (cid:12) r m s h a v e d a t a m is s in g o n le v e r a g e a n d b o o k - t o - m a r k e t . W e a r e t h e r e f o r e le f t w it h a (cid:12) n a l s a m p le o f 9 3 (cid:12) r m s . T h is s a m p le is b r o a d ly d iv e r s i(cid:12) e d w it h 2 4 (cid:12) n a n c ia ls , 4 8 in d u s t r ia ls a n d 2 1 u t ilit y (cid:12) r m s . P r io r t o M a r c h 1 9 8 9 , a s f e w n o n - c a lla b le b o n d s w e r e is s u e d , t h e d a t a is s p a r s e w it h o n ly 2 0 (cid:12) r m s p e r m o n t h . I n t h e in t e r e s t o f a w id e r c r o s s - s e c t io n , w e d e c id e d t o lim it a t t e n t io n t o t h e s a m p le b e t w e e n M a r c h 1 9 8 9 a n d M a r c h 1 9 9 8 . O u r s a m p le in c lu d e s s u c h w e ll- k n o w n c o m p a n ie s a s B a n k o f A m e r ic a , F o r d , I B M , P h ilip M o r r is , a n d W a l- M a r t . T h is s a m p le h a s 4 6 , 2 6 2 c o u p o n b o n d o b s e r v a t io n s . T h e t h r e e - m o n t h t r e a s u r y b ill r a t e is t h e p r o x y f o r t h e s h o r t in t e r e s t r a t e t h r o u g h o u t ( s o u r c e : F e d e r a l R e s e r v e B o a r d ) . W e e m p lo y t r a d e r q u o t e s o n t r e a s u r y S T R I P S t o b u ild t h e t e r m s t r u c t u r e o f d e f a u lt - f r e e b o n d s . T h e r e a r e 2 0 , 1 7 3 t r e a s u r y S T R I P S q u o t e s o v e r o u r s a m p le p e r io d ( a b o u t 1 3 5 t r e a s u r y p r ic e s p e r m o n t h ) . W e u s e t r e a s u r y S T R I P S t o e s t im a t e t h e t w o - f a c t o r m o d e l o f in t e r e s t r a t e s . T o e m p ir ic a lly t e s t c r e d it r is k m o d e ls , w e c o n s t r u c t e d s e v e r a l p r o x ie s f o r t h e d is t r e s s f a c t o r , X t ( ) . T h e s e v a r ia b le s a r e c o n s t r u c t e d a s d e s c r ib e d b e lo w : n L e v e r a g e , L e v ( t ) , is d e (cid:12) n e d a s lo n g - t e r m b o o k v a lu e o f d e b t ( C O M P U S T A T q u a r t e r ly it e m n 5 1 ) d iv id e d b y t h e (cid:12) r m v a lu e . T h e (cid:12) r m v a lu e is t h e s u m o f lo n g - t e r m d e b t a n d t h e m a r k e t c a p it a liz a t io n o f c o m m o n e q u it y , M ; B o o k - t o - M a r k e t , B ( t ) / M ( t ) , is c o m p u t e d a s t h e b o o k v a lu e o f e q u it y ( C O M P U S T A T q u a r - n n t e r ly it e m 5 9 ) d iv id e d b y t h e m a r k e t v a lu e o f e q u it y ; P r o (cid:12) t a b i l i t y , P r o (cid:12) t ( t ) , is c a lc u la t e d a s o p e r a t in g in c o m e ( C O M P U S T A T q u a r t e r ly it e m 2 1 ) n d iv id e d b y n e t s a le s ( C O M P U S T A T q u a r t e r ly it e m 2 ) ; 1 2
S p r e a d t (cid:0) ( 1 ) , is t h e a v e r a g e y ie ld o n t h e (cid:12) r m ’ s d e b t m in u s t h e t h r e e - m o n t h t r e a s u r y b ill r a t e , n a s o f m o n t h t - 1 ; S c a l e d S t o c k P r i c e s t , ( ) , is t h e lo g o f t h e c u r r e n t s t o c k p r ic e n o r m a liz e d b y t h e m o n e y m a r k e t n S t ( ) s t (cid:17) a c c o u n t . T h a t is , ( ) lo g . (cid:2) (cid:1)(cid:1)(cid:1) r r r t [1 + ( 0 ) ] [1 + ( 1 ) ] [1 + ( ) ] (cid:16) (cid:17) E v e n t h o u g h lo n g - t e r m d e b t a n d b o o k v a lu e s a r e r e c o r d e d a t t h e q u a r t e r ly f r e q u e n c y , t h e s e r ie s f o r le v e r a g e a n d b o o k - t o - m a r k e t r a t io a r e m o n t h ly . F o r e a c h (cid:12) r m , t h e d e b t v a lu e a n d t h e b o o k v a lu e a r e u p d a t e d o n a q u a r t e r ly b a s is ( w h ile t h e m a r k e t v a lu e o f e q u it y is u p d a t e d m o n t h ly ) . T o c ir c u m v e n t a n y lo o k - a h e a d b ia s e s , w e u s e d e b t a n d b o o k v a lu e s f r o m t h e p r e v io u s q u a r t e r t o c o m p u t e le v e r a g e a n d b o o k - t o - m a r k e t f a c t o r s f o r t h e n e x t t h r e e m o n t h s . A c u b ic s p lin e is u s e d t o c o n v e r t t h e q u a r t e r ly p r o (cid:12) t a b ilit y m e a s u r e in t o a m o n t h ly p r o (cid:12) t a b ilit y m e a s u r e . C o r p o r a t e b o n d s a r e c la s s i(cid:12) e d in t o t h r e e c r e d it r a t in g c a t e g o r ie s . S p e c i(cid:12) c a ll y , b o n d s w it h ( n u m e r ic a l) c r e d it r a t in g u p t o 5 a r e d e s ig n a t e d a s A A - r a t e d ; b e t w e e n 6 a n d 8 a r e d e s ig n a t e d a s A - r a t e d ; a n d c r e d it r a t in g 9 a n d h ig h e r a r e d e s ig n a t e d a s B B B - r a t e d ( o r b e lo w ) . M o r e o v e r , b o n d s w it h m a t u r it y le s s t h a n 5 y e a r s a r e c a lle d s h o r t - t e r m b o n d s ; b e t w e e n 5 a n d 1 0 y e a r s a r e c la s s i(cid:12) e d a s m e d i u m - t e r m b o n d s ; a n d m a t u r it y lo n g e r t h a n 1 0 y e a r s f o r m t h e lo n g - t e r m b o n d c a t e g o r y . T a b le 1 s h o w s t h a t 1 5 % ( 5 3 % ) o f t h e b o n d is s u e s h a v e a c r e d it r a t in g o f A A ( A ) , w h ile 3 2 % h a v e a c r e d it r a t in g o f B B B o r b e lo w ( s e e D u (cid:11) e e ( 1 9 9 9 ) a n d E lt o n , G r u b e r , A g r a w a l, a n d M a n n ( 2 0 0 0 1 ) ) . T a b le 1 d is p la y s s u m m a r y s t a t is t ic s o n b o n d a t t r ib u t e s a n d (cid:12) r m - s p e c i(cid:12) c d is t r e s s f a c t o r s ( i. e . , le v e r a g e , B / M a n d p r o (cid:12) t a b ilit y ) f o r 2 5 (cid:12) r m s . W e r e p o r t t h e ( a v e r a g e ) n u m b e r o f b o n d s o u t s t a n d in g , t h e y ie ld , t h e c r e d it r a t in g , a n d t h e m a t u r it y . T h e m a t u r it y s t r u c t u r e o f b o n d s is m ix e d w it h s o m e (cid:12) r m s p r e f e r r in g s h o r t - t e r m d e b t t o it s lo n g - t e r m c o u n t e r p a r t . A c c o r d in g t o T a b le 1 , (cid:12) r m s in t h e (cid:12) n a n c ia l in d u s t r y t e n d t o h a v e t h e lo w e s t m a t u r it y , w h ile in d u s t r ia ls t e n d t o (cid:12) n a n c e lo n g - t e r m . L ik e w is e , m o r e c r e d it - w o r t h y (cid:12) r m s is s u e lo n g e r - t e r m d e b t . T h e a v e r a g e c r e d it s p r e a d b e t w e e n A A a n d B B B - r a t e d b o n d s is 5 8 b a s is p o in t s . T u r n in g t o (cid:12) r m - s p e c i(cid:12) c d is t r e s s f a c t o r s , n o t ic e t h a t le v e r a g e , B / M a n d p r o (cid:12) t a b ilit y v a r y s u b s t a n t ia lly in t h e (cid:12) r m c r o s s - s e c t io n . O n a v e r a g e , le s s c r e d it w o r t h y (cid:12) r m s a r e a s s o c ia t e d w it h h ig h e r le v e r a g e a n d a m o r e p r o n o u n c e d b o o k - t o - m a r k e t r a t io ( a n d v ic e - v e r s a ) . H o w e v e r , t h e y ie ld p a t t e r n is le s s t h a n c le a r a c r o s s in d u s t r ie s : in d u s t r ia ls h a v e t h e h ig h e s t a v e r a g e y ie ld b u t r e la t iv e ly lo w e r le v e r a g e a n d B / M . T h e r e la t io n s h ip b e t w e e n a v e r a g e p r o (cid:12) t a b ilit y a n d a v e r a g e y ie ld a p p e a r s a m b ig u o u s . T h a t is , a h ig h e r p r o (cid:12) t a b ilit y n e e d n o t t r a n s la t e in t o lo w e r c r e d it y ie ld s . F o u r e m p ir ic a l y a r d s t ic k s a r e a d o p t e d t o e v a lu a t e c r e d it r is k m o d e ls . A t t h e o u t s e t , w e d e t e r m in e w h e t h e r (cid:12) r m - s p e c i(cid:12) c d is t r e s s f a c t o r s h a v e e x p la n a t o r y a b ilit y in t h e c r o s s - s e c t io n o f y ie ld s . I n a s e c o n d e x e r c is e , w e a n a ly z e w h e t h e r t h e e s t im a t e d s t r u c t u r a l p a r a m e t e r s a r e r e a s o n a b le a n d in v e s t ig a t e t h e in - s a m p le (cid:12) t t in g e r r o r s o f c r e d it r is k m o d e ls . N e x t , w e c o n t r a s t t h e ir o u t - o f - s a m p le 1 3
p r ic in g a c c u r a c y . F in a lly , w e t a k e r is k m a n a g e m e n t p e r s p e c t iv e s a n d e x a m in e t h e h e d g in g e (cid:11) e c t iv e n e s s o f c r e d it r is k m o d e ls . E a c h y a r d s t ic k c a p t u r e s d is t in c t a s p e c t s o f m o d e l p e r f o r m a n c e . 4 F i r m - S p e c i (cid:12) c D i s t r e s s F a c t o r s a n d t h e C r o s s - s e c t i o n o f Y i e l d s A lt h o u g h t h e c h o ic e o f (cid:12) r m - s p e c i(cid:12) c d is t r e s s f a c t o r s a p p e a r s r e a s o n a b le o n e c o n o m ic g r o u n d s , w e e x a m in e w h e t h e r c r o s s - s e c t io n a l v a r ia t io n s in t h e s e v a r ia b le s e x p la in d i(cid:11) e r e n c e s in t h e c o r p o r a t e y ie ld s . T h e b a s ic t e s t in g e q u a t io n is : Y t t t (cid:28) t X t (cid:15) t ; ‘ ; (cid:1) (cid:1) (cid:1) ; L ; ( ) = (cid:5) ( ) + (cid:5) ( ) ( ) + (cid:5) ( ) + ( ) = 1 ( 2 6 ) ‘ ‘ ‘ ‘ 0 1 2 ‘ (cid:1) (cid:1) (cid:1) w h e r e r a n g e s o v e r a ll b o n d is s u e s o f a ll t h e n a m e s a n d t = 1 , , T . I n t h e r e g r e s s io n s p e c i(cid:12) c a t io n (cid:28) ‘ Y X ( 2 6 ) , d e n o t e s t e r m - t o - m a t u r it y o f b o n d , is t h e c o r r e s p o n d in g y ie ld a n d is t h e d is t r e s s ‘ ‘ ‘ t f a c t o r ( le v e r a g e , b o o k - t o - m a r k e t o r p r o (cid:12) t a b ilit y ) . T h e O L S r e g r e s s io n is p e r f o r m e d e a c h m o n t h , , T 1 (cid:17) t a n d t h e r e s u lt in g c o e (cid:14) c ie n t s a r e p o o le d in t h e t im e - s e r ie s ( i. e . , t h e r e p o r t e d (cid:5) (cid:5) ( ) ) . 2 2 t = 1 T P T h e t e s t a b le h y p o t h e s is is t h a t h ig h e r le v e ls o f le v e r a g e a n d b o o k - t o - m a r k e t ( a n d lo w e r le v e ls o f p r o (cid:12) t a b ilit y ) le a d t o h ig h e r c o r p o r a t e y ie ld s . T h a t is , m o r e d is t r e s s e d (cid:12) r m s h a v e h ig h e r c r e d it s p r e a d s . S h o w n in s q u a r e b r a c k e t s , t h e r e p o r t e d t - s t a t is t ic is t h e m e a n c o e (cid:14) c ie n t d iv id e d b y t h e s t a n d a r d e r r o r o f t h e m e a n e s t im a t e . T h e (cid:5) e s t im a t e , r e p o r t e d in T a b le 2 , is a t t h e c o r e o f t h e d e f a u lt a b le d is c o u n t r a t e s p e c i(cid:12) c a t io n 2 ( 1 1 ) . P a n e l A o f T a b le 2 d e m o n s t r a t e s t h a t t h e im p a c t o f le v e r a g e o n y ie ld s is c o n s is t e n t ly p o s it iv e a n d s t r o n g ly s ig n i(cid:12) c a n t . T h e c o e (cid:14) c ie n t (cid:5) v a r ie s b e t w e e n 0 . 2 3 a n d 1 . 9 8 w h e n b o n d s a r e g r o u p e d 2 b y c r e d it r a t in g ; a n d b e t w e e n 0 . 7 2 a n d 1 . 3 8 w h e n b o n d s a r e g r o u p e d b y m a t u r it y . F o r t h e s e t o f r e g r e s s io n s , t h e m in im u m t - s t a t is t ic is 5 . 4 . C o n s is t e n t w it h in t u it io n , t h e e (cid:11) e c t o f le v e r a g e o n y ie ld s is m o s t p r o n o u n c e d a m o n g lo w e r g r a d e b o n d s a n d a m o n g lo n g - t e r m b o n d s . F o r in s t a n c e , a 1 % c h a n g e in t h e le v e r a g e r a t io in c r e a s e s t h e y ie ld o f B B B - r a t e d b o n d s b y 1 9 . 8 b a s is p o in t s . T h e im p a c t o f le v e r a g e o n y ie ld s is t im e - s t a b le , a s r e (cid:13) e c t e d in t h e p r o p o r t io n o f t - s t a t is t ic s a b o v e 2 ( i. e . , a s s e e n b y 1 ) . M o r e o v e r , t h e g o o d n e s s - o f - (cid:12) t o f t h e m o d e l is r e a s o n a b le : f o r B B B - r a t e d t > 2 2 R (cid:28) b o n d s , t h e m a x im u m ( a d j u s t e d ) is 8 7 % in t h e b iv a r ia t e r e g r e s s io n w it h a n d L e v a s t h e ‘ ‘ e x p la n a t o r y v a r ia b le s ( a n d 1 0 . 2 % in t h e u n iv a r ia t e r e g r e s s io n w it h L e v ) . ‘ T u r n in g t o t h e y ie ld b e h a v io r w it h r e s p e c t t o b o o k - t o - m a r k e t , w e o b s e r v e a s im ila r (cid:12) n d in g : t h e B / M a t t r ib u t e is p o s it iv e ly c o r r e la t e d w it h a v e r a g e y ie ld s ( s e e P a n e l B ) . C o n s id e r B B B - r a t e d b o n d s . T h e c o e (cid:14) c ie n t (cid:5) is 0 . 2 4 w it h a t - s t a t is t ic o f 5 . 5 . N o w c o n s id e r lo n g - t e r m b o n d s , w h e r e 2 t h e e s t im a t e d (cid:5) is 0 . 5 5 ( t - s t a t is t ic o f 7 . 7 ) . L ik e le v e r a g e , t h e c o e (cid:14) c ie n t (cid:5) is m o n o t o n ic a lly 2 2 in c r e a s in g w it h b o n d m a t u r it y . T h e m a g n it u d e o f 1 s u g g e s t s t h a t t h e e (cid:11) e c t o f b o o k - t o - m a r k e t t > 2 1 4
r e m a in s r o b u s t o v e r t im e . H o w e v e r , c o m p a r in g P a n e ls A a n d B , o n e d i(cid:11) e r e n c e is a p p a r e n t . O n c e t h e m a t u r it y o f t h e b o n d is c o n t r o lle d , t h e b o o k - t o - m a r k e t f a c t o r d is p la y s s m a lle r s lo p e c o e (cid:14) c ie n t s . E c o n o m ic a lly , a 1 % c h a n g e in t h e b o o k - t o - m a r k e t f a c t o r im p lie s a 5 . 5 b a s is p o in t s c h a n g e in t h e y ie ld o f lo n g - t e r m b o n d s ( t h e le v e r a g e c o u n t e r p a r t is 1 3 . 8 b a s is p o in t s ) . O v e r a ll, t h e b o o k - t o m a r k e t f a c t o r is a n im p o r t a n t v a r ia b le in t h e c r o s s - s e c t io n o f c o r p o r a t e y ie ld s . A s e x p e c t e d , P a n e l C in d ic a t e s t h a t t h e a v e r a g e (cid:5) is n e g a t iv e . T h e c o e (cid:14) c ie n t (cid:5) r a n g e s 2 2 b e t w e e n - 1 . 7 0 a n d - 0 . 1 6 . A s in t h e c a s e o f le v e r a g e , t h e p r o (cid:12) t a b ilit y m e a s u r e h a s a b ig g e r e (cid:11) e c t o n (cid:0) lo n g - t e r m b o n d s a n d lo w - r a t e d b o n d s . B a s e d o n t h e 1 s t a t is t ic , t h e e v id e n c e is le s s t h a n s t r o n g t < 2 f o r p r o (cid:12) t a b ilit y . T h is is a ls o c o n (cid:12) r m e d b y t h e r e la t iv e ly lo w m a g n it u d e s o f t h e t - s t a t is t ic a n d t h e 2 g o o d n e s s - o f - (cid:12) t R m e a s u r e s . O u r e v id e n c e in d ic a t e s t h a t p r o (cid:12) t a b ilit y m a y b e o f s e c o n d - o r d e r im p o r t a n c e r e la t iv e t o le v e r a g e a n d B / M r a t io . T h e c o e (cid:14) c ie n t (cid:5) c a n b e in t e r p r e t e d a s t h e s lo p e o f t h e c o r p o r a t e y ie ld c u r v e . I r r e s p e c t iv e 1 o f t h e d is t r e s s f a c t o r , t h e y ie ld c u r v e is s t e e p e r f o r lo w e r - r a t e d b o n d s a n d f o r s h o r t - t e r m b o n d s . (cid:5) , t h e u n c o n d it io n a l in s t a n t a n e o u s y ie ld , is p o s it iv e a n d s t a t is t ic a lly s ig n i(cid:12) c a n t . O u r e m p ir ic a l 0 (cid:12) n d in g s o n t h e r e le v a n c e o f d is t r e s s f a c t o r s a r e r o b u s t a c r o s s s u b - p e r io d s ( n o t r e p o r t e d ) . T h is is Y t (cid:0) r t t r u e e v e n w h e n ( ) ( ) is e m p lo y e d a s t h e d e p e n d e n t v a r ia b le in ( 2 6 ) . ‘ I n s u m , t h e r e g r e s s io n r e s u lt s a r e b r o a d ly c o n s is t e n t w it h o u r m o d e lin g a p p r o a c h t h a t t h e d e f a u lt a b le d is c o u n t r a t e is a f u n c t io n o f (cid:12) r m - s p e c i(cid:12) c d is t r e s s f a c t o r s . E v e n a f t e r a c c o u n t in g f o r c r e d it r a t in g , t h e m a r g in a l im p a c t o f (cid:12) r m - s p e c i(cid:12) c d is t r e s s f a c t o r s o n y ie ld s is g e n e r a lly s ig n i(cid:12) c a n t . O v e r a ll, (cid:12) r m - s p e c i(cid:12) c v a r ia b le s a r e in f o r m a t iv e a b o u t c r o s s - s e c t io n a l v a r ia t io n s in t h e c r e d it s p r e a d s . H a v in g j u s t i(cid:12) e d o u r c h o ic e e m p ir ic a lly , w e n o w p r o c e e d t o a m o r e f o r m a l a n a ly s is o f c r e d it r is k m o d e ls a n d t o in v e s t ig a t in g t h e d e t e r m in a n t s o f d e f a u lt r is k . 5 E s t i m a t i o n o f C r e d i t R i s k M o d e l s G u id e d b y t h e o r e t ic a l a n d p r a c t ic a l c o n s id e r a t io n s , s ix c r e d it r is k m o d e ls a r e e s t im a t e d . T h e n ; (cid:1) (cid:1) (cid:1) ; N s p e c i(cid:12) c a t io n o f t h e d e f a u lt a b le d is c o u n t r a t e is a s o u t lin e d b e lo w ( = 1 ) : 1 . L e v e r a g e R a t i o M o d e l R t r t t ( ) = (cid:3) + (cid:3) ( ) + (cid:3) L e v ( ) n ;n r ;n x ;n n 0 B t n ( ) 2 . B / M R a t i o M o d e l R t r t ( ) = (cid:3) + (cid:3) ( ) + (cid:3) n ;n r ;n x ;n 0 M t n ( ) 3 . P r o (cid:12) t a b i l i t y M o d e l R t r t t ( ) = (cid:3) + (cid:3) ( ) + (cid:3) P r o (cid:12) t ( ) n ;n r ;n x ;n n 0 4 . L a g g e d Y i e l d S p r e a d M o d e l R t r t t (cid:0) ( ) = (cid:3) + (cid:3) ( ) + (cid:3) S p r e a d ( 1 ) n ;n r ;n x ;n 0 n 5 . S t o c k P r i c e M o d e l R t r t s t ( ) = (cid:3) + (cid:3) ( ) + (cid:3) ( ) n ;n r ;n x ;n n 0 6 . I n t e r e s t R a t e M o d e l R t r t ( ) = (cid:3) + (cid:3) ( ) n ;n r ;n 0 1 5
B t ( ) n t t w h e r e , r e c a ll, L e v ( ) is t h e le v e r a g e r a t io ; is t h e b o o k - t o - m a r k e t r a t io ; P r o (cid:12) t ( ) r e p r e s e n t s n n M t ( ) n t (cid:0) s t t h e (cid:12) r m p r o (cid:12) t a b ilit y ; S p r e a d ( 1 ) is t h e la g g e d c r e d it s p r e a d ; a n d ( ) is t h e s c a le d s t o c k p r ic e . n n I n e a c h c a s e , t h e d e f a u lt a b le d is c o u n t ( c o u p o n ) b o n d p r ic e c a n b e d e t e r m in e d f r o m e q u a t io n ( 1 6 ) ( ( 9 ) ) . O u r d is c u s s io n is d iv id e d in t o t h r e e p a r t s : ( i) e s t im a t in g t h e in t e r e s t r a t e p a r a m e t e r s , ( ii) e s t im a t in g t h e (cid:12) r m - s p e c i(cid:12) c d e f a u lt p a r a m e t e r s , a n d ( iii) c o n t r a s t in g t h e in - s a m p le v a lu a t io n e r r o r s a c r o s s m o d e ls . 5 . 1 I n t e r e s t R a t e P a r a m e t e r s T h e p a r a m e t e r s o f t h e in t e r e s t r a t e p r o c e s s a r e c o m m o n t o a ll (cid:12) r m s . F o r t h is r e a s o n , w e o n ly e m p lo y t r e a s u r y s e c u r it ie s t o e s t im a t e t h e in t e r e s t r a t e p a r a m e t e r s ( s e e a ls o D u (cid:11) e e ( 1 9 9 9 ) ) . I n p a r t ic u la r , w e e x p lo it a p a n e l o f t r e a s u r y S T R I P S p r ic e s . T h e r e s u lt in g in t e r e s t r a t e p a r a m e t e r s a r e (cid:12) x e d in t h e c r o s s - s e c t io n o f (cid:12) r m s . T h e e x a c t im p le m e n t a t io n p r o c e d u r e is a s f o llo w s . S t e p A . (cid:28) ‘ ; ; (cid:1) (cid:1) (cid:1) ; L C o lle c t L t r e a s u r y S T R I P S p r ic e s e a c h m o n t h . L e t ( = 1 2 ) in d e x t h e t e r m - ‘ B t ; (cid:28) t o - m a t u r it y o f t h e t r e a s u r y S T R I P S . D e n o t e t h e m a r k e t p r ic e b y ( ) a n d t h e m o d e l p r ic e ‘ B t ; (cid:28) b y ( ) . L e t t h e c o m p le t e v e c t o r o f s t r u c t u r a l p a r a m e t e r ( u n d e r t h e e q u iv a le n t m a r t in g a le ‘ (cid:17) f z ; (cid:20) ; (cid:27) ; (cid:20) ; (cid:22) ; (cid:27) ; (cid:26) g ‘ m e a s u r e ) b e d e (cid:12) n e d a s (cid:8) . F o r e a c h S T R I P S , d e (cid:12) n e t h e v a lu a t io n r r r z z z r ;z e r r o r : B t ; (cid:28) (cid:0) B t ; (cid:28) ( ) ( ) ‘ ‘ (cid:15) (cid:17) ; [ (cid:8) ] ( 2 7 ) ‘ r B t ; (cid:28) ( ) ‘ w h ic h is t h e p e r c e n t a g e d e v ia t io n o f t h e m o d e l d e t e r m in e d p r ic e f r o m t h e o b s e r v e d m a r k e t p r ic e . S t e p B . S o lv e f o r t h e p a r a m e t e r v e c t o r (cid:8) t h a t m in im iz e s t h e r o o t m e a n - s q u a r e d p e r c e n t a g e r p r ic in g e r r o r ( o n e f o r e a c h m o n t h t ) : L 1 2 v t (cid:17) j (cid:15) j ; t ; (cid:1) (cid:1) (cid:1) ; T : R M S E ( ) m in = 1 ( 2 8 ) r ‘ u r (cid:8) L u X‘ = 1 t T h is m in im iz a t io n p r o c e d u r e w ill r e s u lt in a n e s t im a t e o f t h e r is k - n e u t r a liz e d p a r a m e t e r s f o r t h e t w o - f a c t o r in t e r e s t r a t e m o d e l. T h e im p lie d p a r a m e t e r a p p r o a c h is n o w s t a n d a r d in t h e lit e r a t u r e ( s e e B r o w n a n d D y b v ig ( 1 9 8 6 ) a n d D a i a n d S in g le t o n ( 2 0 0 0 ) a n d r e f e r e n c e s t h e r e in ) . T h e v a r ia t io n a c r o s s t im e in t h e r is k - n e u t r a l p a r a m e t e r s is c o n s is t e n t w it h t h e t r a d it io n a l a s s e t p r ic in g r e s u lt s o n t h e t im e - v a r ia t io n o f t h e r is k p r e m ia in t h e (cid:12) n a n c ia l m a r k e t s . N o t e t h a t t h e o n e - f a c t o r m o d e l o f in t e r e s t r a t e s is n e s t e d w it h in t h e t w o - f a c t o r m o d e l o f (cid:20) (cid:27) in t e r e s t r a t e o u t lin e d in ( 1 2 ) - ( 1 3 ) . S e t t in g = = 0 a n d (cid:12) x in g z t o b e a c o n s t a n t , w e o b t a in z z d r t (cid:20) z (cid:0) r t d t (cid:27) d ! t t h e o n e - f a c t o r ( V a s ic e k ( 1 9 7 7 ) ) m o d e l o f in t e r e s t r a t e s : ( ) = [ ( ) ] + ( ) . I m p o s in g r r r a p p r o p r ia t e r e s t r ic t io n s o n t h e p a r a m e t e r v e c t o r in ( 2 7 ) - ( 2 8 ) , w e c a n s im ila r ly in f e r t h e p a r a m e t e r s 1 6
o f t h e o n e - f a c t o r m o d e l. I n P a n e l A a n d P a n e l B o f T a b le 3 , w e p r e s e n t t h e p a r a m e t e r e s t im a t e s a n d c o m p a r e t h e in - s a m p le a n d o u t - o f - s a m p le v a lu a t io n e r r o r s o f t h e o n e - f a c t o r a n d t h e t w o - f a c t o r m o d e l o f in t e r e s t r a t e s , r e s p e c t iv e ly . T h e e m p ir ic a l e s t im a t io n s a r e e n lig h t e n in g f r o m s e v e r a l p e r s p e c t iv e s . B e c a u s e t h e e s t im a t io n r e s u lt s a r e s im ila r a c r o s s s u b s a m p le s , c o n c e n t r a t e o n t h e f u ll s a m p le (cid:12) n d in g s . F ir s t , a d d in g a s t o c h a s t ic m e a n f a c t o r t o t h e o n e - f a c t o r m o d e l o f in t e r e s t r a t e s im p r o v e s it s p e r f o r m a n c e c o n s id e r a b ly . T h is p o in t c a n b e d e m o n s t r a t e d in s e v e r a l w a y s . I n s p e c t io n o f t h e t a b le s h o w s t h a t t h e in - s a m p le R M S E o f t h e o n e - f a c t o r m o d e l is 1 . 6 4 % , w h ile t h e c o r r e s p o n d in g R M S E f o r t h e t w o - f a c t o r m o d e l is 0 . 5 1 % . T h e r e d u c t io n in t h e in - s a m p le a b s o lu t e y ie ld e r r o r s , d e n o t e d B Y E , is a ls o s u b s t a n t ia l: t h e t w o - f a c t o r m o d e l p r o v id e s a n a v e r a g e a b s o lu t e y ie ld e r r o r o f 6 . 7 5 b a s is p o in t s v e r s u s 1 4 . 9 1 b a s is p o in t s ( h e r e b y b p ) f o r t h e o n e - f a c t o r m o d e l. V ie w e d f r o m a v a lu a t io n p e r s p e c t iv e , t h e 8 . 1 6 b p im p r o v e m e n t is e c o n o m ic a lly im p o r t a n t ( T a b le I V in D a i a n d S in g le t o n ( 2 0 0 0 ) p r o v id e s a c o m p a r is o n ) . N o t s u r p r is in g ly , t h e s u p e r io r it y o f t h e t w o - f a c t o r m o d e l o f in t e r e s t r a t e s is a ls o e v id e n t o n t h e b a s is o f o u t - o f - s a m p le g o o d n e s s - o f - (cid:12) t m e a s u r e s . T o c o m p u t e t h e o u t - o f - s a m p le v a lu a t io n e r r o r s , t (cid:0) w e in it ia lly e s t im a t e (cid:8) f r o m t r e a s u r y S T R I P S o b s e r v e d a s o f m o n t h 1 . W e t h e n u s e t h e s e r r s t r u c t u r a l p a r a m e t e r s ( a n d ) t o c a lc u la t e t h e t h e o r e t ic a l p r ic e o f S T R I P S in t h e s u b s e q u e n t m o n t h t . C o m p a r in g t h e m o d e l p r ic e t o t h e m a r k e t p r ic e a n d c o n s o lid a t in g t h e v a lu a t io n e r r o r s a c r o s s a ll S T R I P S , w e r e p o r t t h e a v e r a g e o u t - o f - s a m p le R M S E a n d B Y E ( in t h e ir r e s p e c t iv e c o lu m n s ) in c u r ly b r a c k e t s . W e c a n o b s e r v e t h a t a t w o - f a c t o r ( o n e - f a c t o r ) m o d e l o f in t e r e s t r a t e h a s a n o u t - o f - s a m p le R M S E o f 2 . 8 2 % ( 3 . 3 5 % ) a n d a n o u t - o f - s a m p le B Y E o f 2 0 . 8 2 ( 2 4 . 7 0 ) b a s is p o in t s . I n a n u t s h e ll, b a s e d o n t h e in - s a m p le a n d o u t - o f - s a m p le v a lu a t io n y a r d s t ic k s , t h e t w o - f a c t o r m o d e l o f in t e r e s t r a t e s is le s s m is - s p e c i(cid:12) e d . T h e r e f o r e , t h e t w o - f a c t o r in t e r e s t r a t e m o d e l w ill b e a d o p t e d t o e s t im a t e t h e c r e d it r is k m o d e ls t h r o u g h o u t . A s s e e n , t h e s t r u c t u r a l p a r a m e t e r s o f t h e t w o - f a c t o r ( a n d o n e - f a c t o r ) m o d e l o f in t e r e s t r a t e s a r e r e a s o n a b le . O v e r t h e f u ll s a m p le , t h e a v e r a g e lo n g - r u n m e a n , z , is 8 . 5 % w it h a d r if t c o e (cid:14) c ie n t o f 1 5 . 8 % a n d a d i(cid:11) u s io n c o e (cid:14) c ie n t o f 0 . 0 1 9 . T h e s p e e d o f a d j u s t m e n t o f r a n d z t o t h e ir m e a n s is 0 . 3 0 1 a n d 0 . 0 3 2 , r e s p e c t iv e ly . W e e s t im a t e a n e g a t iv e c o r r e la t io n b e t w e e n in t e r e s t r a t e m o v e m e n t s a n d c h a n g e s in t h e lo n g - r u n m e a n . A lb e it e s t im a t e d u n d e r t h e p h y s ic a l p r o b a b ilit y m e a s u r e , C o llin - D u f r e s n e a n d S o ln ik ( 1 9 9 9 , T a b le 1 ) r e p o r t s im ila r e s t im a t e s f o r t h e t w o - f a c t o r m o d e l ( s e e a ls o , a m o n g o t h e r s , C h e n a n d S c o t t ( 1 9 9 3 ) a n d D a i a n d S in g le t o n ( 2 0 0 0 ) ) . T h e e s t im a t e d p a r a m e t e r s s e e m t im e - s t a b le . T h is c a n b e s e e n b y b o t h t h e s m a ll s t a n d a r d e r r o r s o f t h e p a r a m e t e r e s t im a t e ( s h o w n in p a r e n t h e s is ) a n d t h e r e la t iv e ly s m a ll v a r ia t io n in t h e p a r a m e t e r e s t im a t e s f r o m t h e 8 9 : 0 3 - 9 3 : 1 2 s u b s a m p le t o t h e 9 4 : 0 1 - 9 8 : 0 3 s u b s a m p le . M o r e o v e r , t h e c o e (cid:14) c ie n t o f v a r ia t io n , c o m p u t e d a s t h e s t a n d a r d d e v ia t io n o f t h e e s t im a t e d iv id e d b y m e a n 1 7
e s t im a t e , a r e a ll le s s t h a n 0 . 5 . O t h e r p a r a m e t e r s o f t h e c r e d it r is k m o d e ls , a s w e d e s c r ib e n e x t , a r e c o n d it io n a l o n t h e e s t im a t e d in t e r e s t r a t e p a r a m e t e r s . 5 . 2 F i r m - S p e c i (cid:12) c D e f a u l t P a r a m e t e r s T h e r e m a in in g t a s k is t o e s t im a t e t h e p a r a m e t e r s o f t h e d e f a u lt a b le d is c o u n t r a t e a n d t h e p a r a m e t e r s o f t h e d is t r e s s p r o c e s s . T a k e t h e le v e r a g e r a t io m o d e l a s a n e x a m p le a n d d e (cid:12) n e t h e (cid:12) r m - (cid:17) f ; ; ; (cid:20) ; (cid:22) ; (cid:27) ; (cid:26) n ; (cid:1) (cid:1) (cid:1) ; N g s p e c i(cid:12) c p a r a m e t e r s b y t h e v e c t o r : (cid:8) (cid:3) (cid:3) (cid:3) : = 1 . x ;n ;n r ;n x ;n x ;n x ;n x ;n r ;x ;n 0 A s b e f o r e , w e m in im iz e t h e r o o t m e a n - s q u a r e d p e r c e n t a g e p r ic in g e r r o r : K 1 2 v t (cid:17) j (cid:15) j ; t ; (cid:1) (cid:1) (cid:1) ; T k ; (cid:1) (cid:1) (cid:1) ; K ; R M S E ( ) m in = 1 a n d = 1 ( 2 9 ) x ;n x ;k ;n u (cid:8) x ;n K u Xk = 1 t (cid:0) P t ;(cid:28) P t ;(cid:28) ( ) ( ) k k P t ; (cid:28) (cid:15) (cid:17) . H e r e , ( ) is t h e m a r k e t p r ic e o f t h e d e f a u lt a b le c o u p o n w h e r e le t t in g [ (cid:8) ] x ;k x ;n P t ;(cid:28) ( ) k P t ; (cid:28) b o n d w h ile ( ) is t h e t h e o r e t ic a l p r ic e im p lic it in t h e le v e r a g e r a t io m o d e l. W h ile t h e r e is a n a b u n d a n c e o f t r e a s u r y S T R I P S , t h e n u m b e r o f d e f a u lt a b le c o u p o n b o n d s is o f t e n in s u (cid:14) c ie n t t o e s t im a t e t h e 7 s t r u c t u r a l p a r a m e t e r s c o n t a in e d in (cid:8) . T o a c h ie v e a x ;n c o m p r o m is e , w e m o d if y o u r a p p r o a c h in a n e le m e n t a r y w a y . W e p o o l c o r p o r a t e b o n d s e a c h q u a r t e r a n d t h e n m in im iz e ( 2 9 ) , w h ic h h a s t h e im p lic a t io n o f a s s e r t in g t h e c o n s t a n c y o f r is k p r e m ia f o r a q u a r t e r . L e n g t h e n in g t h e p e r io d o f e s t im a t io n is s u b j e c t t o g r e a t e r m is - s p e c i(cid:12) c a t io n . T h is e s t im a t io n p r o c e d u r e is r e p e a t e d f o r e a c h (cid:12) r m a n d f o r e a c h o f t h e s ix c r e d it r is k m o d e ls . I n s u m m a r y , t h e e s t im a t io n o f e a c h c r e d it r is k m o d e l is c a r r ie d in t w o p h a s e s : ( 1 ) T h e p a r a m e t e r s o f t h e t w o - f a c t o r in t e r e s t r a t e m o d e l a r e e s t im a t e d f r o m t r e a s u r y S T R I P S . ( 2 ) C o n d it io n a l o n t h e e s t im a t e d t r e a s u r y p a r a m e t e r s , t h e p a r a m e t e r s o f t h e d e f a u lt a b le d is c o u n t r a t e a n d t h e d is t r e s s f a c t o r a r e e s t im a t e d f r o m in d iv id u a l c o r p o r a t e b o n d s . I n o t h e r w o r d s , t h e p a r a m e t e r s o f t h e d e f a u lt a b le d is c o u n t r a t e a r e u p d a t e d o n ly o n c e e v e r y t h r e e m o n t h s , w h ile t h e s t r u c t u r a l p a r a m e t e r s d r iv in g t r e a s u r y a r e u p d a t e d e a c h m o n t h . I n r e p o r t in g t h e p a r a m e t e r e s t im a t e s o f c r e d it r is k m o d e ls , t w o d e c is io n s a r e m a d e t o c o n s e r v e o n s p a c e . F ir s t , w e o n ly p r e s e n t t h e p a r a m e t e r e s t im a t e s f o r t h e le v e r a g e r a t io m o d e l a n d t h e B / M r a t io m o d e l. S e c o n d , r a t h e r t h a n r e p o r t p a r a m e t e r e s t im a t e s f o r e a c h (cid:12) r m , w e a g g r e g a t e p a r a m e t e r e s t im a t e s b y t h e ir c r e d it r a t in g , m a t u r it y a n d in d u s t r ia l c la s s i(cid:12) c a t io n , r e s p e c t iv e ly . I n d o in g t h e a f o r e m e n t io n e d , t h e p a r a m e t e r s a r e (cid:12) r s t a v e r a g e d in t h e t im e - s e r ie s f o r e a c h (cid:12) r m a n d t h e n a c r o s s (cid:12) r m s . N a t u r a lly , t h e la r g e n u m b e r o f (cid:12) r m s in o u r s a m p le m a k e s it im p r a c t ic a l t o d is p la y p a r a m e t e r e s t im a t e s a t t h e in d iv id u a l (cid:12) r m le v e l. T a b le 4 p r e s e n t s t h e a v e r a g e v a lu e a n d t h e c o e (cid:14) c ie n t o f v a r ia t io n f o r e a c h p a r a m e t e r in (cid:8) . x S t a r t w it h t h e le v e r a g e r a t io m o d e l in P a n e l A . S t r e n g t h e n in g o u r p r io r s f r o m t h e c r o s s - s e c t io n a l 1 8
r e g r e s s io n s , t h e c o e (cid:14) c ie n t (cid:3) is p o s it iv e . T h e a v e r a g e (cid:3) a c r o s s a ll t h e (cid:12) r m s a n d t im e p e r io d s x x is 0 . 0 0 5 1 ( a lt h o u g h n o t r e p o r t e d , it is s t a t is t ic a lly s ig n i(cid:12) c a n t f o r m o s t in d iv id u a l (cid:12) r m s ) . C o n s e q u e n t ly , in t h e p r e s e n c e o f in t e r e s t r a t e r is k , t h e m a r g in a l e (cid:11) e c t o f le v e r a g e is t o e n h a n c e t h e d e f a u lt a b le d is c o u n t r a t e . C o r p o r a t e y ie ld s a n d c r e d it s p r e a d s a c c o r d in g ly r is e w h e n (cid:12) r m le v e r a g e in c r e a s e s d y n a m ic a lly o v e r t im e . O f p a r t ic u la r in t e r e s t is t h e f a c t t h a t (cid:3) is h ig h e s t a m o n g x A A - r a t e d b o n d s a n d t h e lo w e s t a m o n g u t ilit y (cid:12) r m s . A lo n g t h e m a t u r it y s p e c t r u m , t h e m a g n it u d e o f t h e e s t im a t e in d ic a t e s t h a t le v e r a g e r e la t e d d is t r e s s is m o r e r e le v a n t f o r lo n g - d a t e d c o r p o r a t e b o n d s t h a n f o r s h o r t - d a t e d c o r p o r a t e b o n d s . W it h r e g a r d t o t h e r o le o f in t e r e s t r a t e r is k in t h e d e f a u lt a b le d is c o u n t r a t e , t h e s ig n o f (cid:3) r is p o s it iv e a n d le s s t h a n u n it y . S p e c i(cid:12) c a lly , t h e lo w e r t h e c r e d it r a t in g o f t h e b o n d , t h e m o r e p o s it iv e is t h e p a r a m e t e r e s t im a t e : (cid:3) e q u a ls 0 . 7 6 7 f o r A A - r a t e d b o n d s , 0 . 8 2 3 f o r A - r a t e d b o n d s r a n d 0 . 8 7 4 f o r B B B - r a t e d b o n d s . A s in D u (cid:11) e e ( 1 9 9 9 ) , t h e p a r a m e t e r e s t im a t e s s u b s t a n t ia t e t h e c o m m o n in t u it io n t h a t a n u p w a r d s h if t in t h e s h o r t r a t e r a is e s t h e d e f a u lt a b le d is c o u n t r a t e . O u r e v id e n c e r e v e a ls t h a t t h e e (cid:11) e c t o f in t e r e s t r a t e is g e n e r a lly t h e s t r o n g e s t a m o n g s h o r t - t e r m b o n d s . C o m p a r in g t h e c o e (cid:14) c ie n t o f v a r ia t io n f o r (cid:3) v e r s u s (cid:3) , w e n o t e t h a t t h e C V f o r in t e r e s t r a t e r x ( d is t r e s s ) s e n s it iv it y is m o s t ly le s s ( m o r e ) t h a n 0 . 5 . T h e r e s p o n s e o f d e f a u lt a b le d is c o u n t r a t e t o a c h a n g in g in t e r e s t r a t e e n v ir o n m e n t is m o r e u n if o r m r e la t iv e t o c h a n g in g (cid:12) r m - s p e c i(cid:12) c d is t r e s s r is k . D u r in g o u r s a m p le p e r io d , t h e a v e r a g e e s t im a t e o f t h e in s t a n t a n e o u s c r e d it y ie ld , (cid:3) , r a n g e s 0 b e t w e e n 1 . 8 % f o r A A - r a t e d b o n d s a n d 2 . 3 % f o r B B B - r a t e d b o n d s . C o n s e q u e n t ly , a s p r e v io u s ly a s s e r t e d , (cid:3) is m o n o t o n ic a lly in c r e a s in g w h e n t h e c r e d it r a t in g w o r s e n s . C o n s is t e n t w it h t h e 0 o b s e r v e d y ie ld s t r u c t u r e o f c o r p o r a t e b o n d s , t h e in s t a n t a n e o u s c r e d it y ie ld is h ig h e r f o r lo n g - t e r m b o n d s r e la t iv e t o s h o r t - t e r m c o u n t e r p a r t s . O n b a la n c e , t h e e s t im a t e d p a r a m e t e r s (cid:3) , (cid:3) a n d (cid:3) r x 0 a r e e a c h p la u s ib le . A s t h e o r y r e q u ir e s , t h e d e f a u lt a b le d is c o u n t r a t e is p o s it iv e in g e n e r a l ( g iv e n t h e p o s it iv it y o f r ( t ) a n d L e v ( t ) ) . N o w s h if t f o c u s t o t h e f o r c in g p r o c e s s f o r a (cid:12) r m ’ s le v e r a g e . T h e d a t a s u p p o r t s t h e id e a t h a t (cid:22) le v e r a g e is a m e a n - r e v e r t in g s t o c h a s t ic p r o c e s s : t h e a v e r a g e is 4 6 . 4 % f o r A A - r a t e d b o n d s a n d x 4 7 . 9 % f o r B B B - r a t e d b o n d s ( a n d s t a t is t ic a lly s ig n i(cid:12) c a n t w it h C V f a r b e lo w 0 . 5 ) . R e g a r d le s s o f t h e c r e d it r a t in g a n d b o n d m a t u r it y , t h e s p e e d o f a d j u s t m e n t is r e m a r k a b ly s t a b le . T h e a v e r a g e (cid:20) is in t h e n e ig h b o r h o o d o f 0 . 2 6 a n d im p lie s a h a lf - lif e o f 2 . 6 5 y e a r s . T h e v o la t ilit y p a r a m e t e r , x (cid:27) (cid:26) , is a b o u t 1 2 % a n d is in t h e r a n g e o f - 0 . 1 1 5 t o - 0 . 1 0 . T h e s m a ll R M S E a n d A P E ( le s s x r ;x t h a n 2 % ) s u g g e s t s t h a t t h e a d o p t e d o n e - f a c t o r s p e c i(cid:12) c a t io n m a y b e a r e a s o n a b le c a n d id a t e f o r c h a r a c t e r iz in g t h e r is k - n e u t r a l d y n a m ic s o f t h e le v e r a g e r a t io . W h e n d is t r e s s is p r o x ie d b y b o o k - t o - m a r k e t , t h e s t r u c t u r a l p a r a m e t e r s a r e s im ila r ( a n d r e p o r t e d in P a n e l B o f T a b le 4 ) . W e a g a in r e a c h t h e c o n c lu s io n t h a t t h e m a g n it u d e o f (cid:3) is s m a ll x r e la t iv e t o t h e m a g n it u d e o f (cid:3) . A llo w in g f o r p o s s ib le s c a lin g d i(cid:11) e r e n c e s in t h e n o m in a l in t e r e s t r 1 9
r a t e a n d t h e (cid:12) r m - s p e c i(cid:12) c d is t r e s s f a c t o r s , o n e in t e r p r e t a t io n o f t h is (cid:12) n d in g is t h a t t h e in t e r e s t r a t e r is k c o m p o n e n t m a y b e o f h ig h e r - o r d e r r e le v a n c e . W e a ls o d iv id e d t h e f u ll s a m p le in t o t w o s u b s a m p le s a n d f o u n d a v e r a g e p a r a m e t e r s c lo s e t o t h o s e p r e s e n t e d in T a b le 4 . T h e s u b s a m p le r e s u lt s a n d p a r a m e t e r e s t im a t e s f r o m o t h e r c r e d it r is k m o d e ls a r e a v a ila b le u p o n r e q u e s t . 5 . 3 C o m p a r i s o n o f I n - S a m p l e V a l u a t i o n E r r o r s B e c a u s e c r e d it r is k m o d e ls a r e o f t e n e m p lo y e d in m a r k in g - t o - m a r k e t o t h e r illiq u i d s e c u r it ie s , t h e in - s a m p le p e r f o r m a n c e is o f in t e r e s t . D o e s a t h r e e - f a c t o r c r e d it r is k m o d e l w it h id e n t i(cid:12) a b le (cid:12) r m s p e c i(cid:12) c d is t r e s s f a c t o r s im p r o v e u p o n a t w o - f a c t o r c o u n t e r p a r t w it h (cid:12) r m - s p e c i(cid:12) c d is t r e s s c o n s id e r a t io n s a b s e n t ? T a b le 5 p r o v id e s a s n a p - s h o t o f in - s a m p le v a lu a t io n e r r o r s a c r o s s t h e t w o d im e n s io n s o f c r e d it r a t in g a n d m a t u r it y . F o r e a c h c r e d it r is k m o d e l, w e p r o v id e t h e a v e r a g e ( m in im iz e d ) R M S E a n d t h e c o r r e s p o n d in g a v e r a g e B Y E v a lu a t io n e r r o r s . T h e r e s u lt s f r o m t h is lin e o f in v e s t ig a t io n a r e s u m m a r iz e d b e lo w : 1 . O v e r a ll, t h e in c lu s io n o f (cid:12) r m - s p e c i(cid:12) c d is t r e s s f a c t o r s p r o v id e o n ly a m a r g in a l im p r o v e m e n t o v e r t h e in t e r e s t r a t e c r e d it r is k m o d e l. C o n t r a r y t o o u r e x p e c t a t io n s , b a s e d o n t h e m in im iz e d R M S E , t h e m a x im u m im p r o v e m e n t is o f t h e o r d e r o f 0 . 1 7 % ( in t h e in t e r s e c t io n o f A A - r a t e d a n d lo n g - t e r m b o n d s ) . T h e m a x im u m d i(cid:11) e r e n c e in a b s o lu t e y ie ld e r r o r s , B Y E , a m o u n t s t o 2 . 3 1 b a s is p o in t s ( a c r o s s a ll m o d e ls ) . T h e s y s t e m a t ic d is t r e s s f a c t o r is a n im p o r t a n t s o u r c e o f c r e d it r is k . 2 . A m o n g t h e s e t o f c r e d it r is k m o d e ls w it h a (cid:12) r m - s p e c i(cid:12) c d is t r e s s f a c t o r , t h e s t o c k p r ic e m o d e l is t h e le a s t m is - s p e c i(cid:12) e d in - s a m p le . W h e n a v e r a g e d o v e r t h e e n t ir e b o n d s a m p le , t h e R M S E a n d B Y E o f t h is m o d e l is 1 . 2 6 % a n d 2 0 . 5 2 b a s is p o in t s ( c o m p a r e d t h e m a x im u m e r r o r s o f 1 . 3 5 % a n d 2 1 . 6 2 ) , r e s p e c t iv e ly . 3 . C o n t r o llin g f o r m a t u r it y , t h e a v e r a g e B Y E w o r s e n s w it h t h e c r e d it r a t in g o f t h e b o n d ( e s p e c ia lly a m o n g m e d iu m - t e r m b o n d s t h a t a r e B B B - r a t e d ) . T h e y ie ld b a s is p o in t e r r o r s a r e d e c lin in g w it h m a t u r it y : a ll m o d e ls f a c e a p a r t ic u la r h u r d le (cid:12) t t in g s h o r t - t e r m b o n d s . 4 . E v e n t h o u g h le v e r a g e a n d B / M f a c t o r s a r e o f t e n id e n t i(cid:12) e d w it h d is t r e s s , t h e p e r f o r m a n c e o f t h e m o d e ls d o n o t d e p a r t s u b s t a n t iv e ly . I n s e v e r a l c r e d it r a t in g - m a t u r it y g r o u p in g s , t h e 8 p a r a m e t e r in t e r e s t r a t e m o d e l h a s lo w e r v a lu a t io n e r r o r s r e la t iv e t o t h e m o r e e la b o r a t e ly s p e c i(cid:12) e d 1 3 p a r a m e t e r m o d e ls . M o r e c o m p le x m o d e ls n e e d n o t n e c e s s a r ily p e r f o r m b e t t e r in - s a m p le . T h e r e m a y b e s e v e r a l r e a s o n s f o r t h e d o c u m e n t e d p h e n o m e n o n t h a t t h e d is c r e p a n c y b e t w e e n t h e v a lu a t io n e r r o r s f r o m v a r io u s c r e d it r is k m o d e ls is s m a ll in m a g n it u d e . F o r o n e , t h e p r e v a ilin g 2 0
(cid:12) x e d - in c o m e d a t a b a s e s a r e t ilt e d t o w a r d s in v e s t m e n t g r a d e b o n d s . F o r t h is c la s s o f b o n d s , it is e n t ir e ly p o s s ib le t h a t s y s t e m a t ic d is t r e s s f a c t o r s c a p t u r e t h e (cid:12) r s t - o r d e r e (cid:11) e c t o f d e f a u lt . F ir m s p e c i(cid:12) c d e f a u lt c o m p o n e n t m a y b e m o r e r e le v a n t f o r p r ic in g s p e c u la t iv e ( h ig h y ie ld ) b o n d s . S e c o n d , t h e a d o p t e d p r a c t ic e o f u p d a t in g t h e p a r a m e t e r s e t p o t e n t ia lly in t e r a c t s w it h t h e illiq u id it y o f t h e c o r p o r a t e (cid:12) x e d in c o m e m a r k e t . A t a n a b s t r a c t le v e l, t h e f r e q u e n t u p d a t in g o f t h e p a r a m e t e r s c a n h a v e t h e e (cid:11) e c t o f r e d u c in g t h e v a lu a t io n e r r o r s o f a ll p r ic in g m o d e ls . S o m e o f t h e s e c o n c e r n s w ill b e a d d r e s s e d in t h e e m p ir ic a l e x e r c is e s t o f o llo w . I t s u (cid:14) c e s t o s a y t h a t t h e n a t u r e o f t h e in - s a m p le (cid:12) n d in g s h a v e p u s h e d u s t o u n d e r s t a n d t h e o u t - o f - s a m p le p r ic in g a n d d y n a m ic h e d g in g p e r f o r m a n c e o f c r e d it r is k m o d e ls in e v e n g r e a t e r d e t a il. 6 P r i c i n g A c c u r a c y o f C r e d i t R i s k M o d e l s O u t - o f - s a m p le p r ic in g p e r f o r m a n c e n e e d n o t im p r o v e w h e n a d d it io n a l p a r a m e t e r s a r e a d d e d t o a v a lu a t io n m o d e l. T h is is b e c a u s e e x t r a p a r a m e t e r s h a v e id e n t i(cid:12) c a t io n p r o b le m s a n d m a y p e n a liz e o u t - o f - s a m p le a c c u r a c y . I n t h e e m p ir ic a l a n a ly s is t o f o llo w , w e (cid:12) r s t p r e s e n t a n in t e g r a t e d p ic t u r e o f m o d e l p e r f o r m a n c e w h e n d e f a u lt a b le b o n d s a r e c la s s i(cid:12) e d a c c o r d in g t o c r e d it r a t in g , m a t u r it y , in d u s t r y a (cid:14) lia t io n , a n d in d iv id u a l (cid:12) r m s . S e c o n d , a s e t o f r o b u s t n e s s c h e c k s a r e p e r f o r m e d . L a s t ly , w e p r e s e n t a s p e c i(cid:12) c a t io n a n a ly s is o f m o d e l m is p r ic in g . O u r f u n d a m e n t a l e m p h a s is c o n t in u e s t o b e o n e x p la in in g v a r ia t io n s in d e f a u lt a b le b o n d y ie ld s u s in g (cid:12) r m - s p e c i(cid:12) c a n d s y s t e m a t ic d e f a u lt f a c t o r s . 6 . 1 C o n s o l i d a t e d P i c t u r e o f M o d e l E r r o r s t T h e im p le m e n t a t io n o f t h e o u t - o f - s a m p le t e s t is s t r a ig h t f o r w a r d . T o c o m p u t e t h e t im e v a lu a t io n e r r o r s , w e (cid:12) r s t e s t im a t e t h e p a r a m e t e r v e c t o r s , (cid:8) a n d (cid:8) , u s in g t h e c r o s s - s e c t io n o f S T R I P S r x ;n ( p r e v io u s m o n t h ) a n d d e f a u lt a b le c o r p o r a t e b o n d s ( in t h e p r e v io u s q u a r t e r ) . U s in g p a r a m e t e r s s o e s t im a t e d , w e c a lc u la t e t h e m o d e l d e t e r m in e d b o n d p r ic e w it h c o n t e m p o r a n e o u s in t e r e s t r a t e a n d (cid:12) r m - s p e c i(cid:12) c d is t r e s s f a c t o r a s in p u t s . T h e lo n g - r u n in t e r e s t r a t e is s e t a t it s la g g e d v a lu e , a s it is e s t im a t e d w it h in t h e p a r a m e t e r v e c t o r (cid:8) . N e x t w e c o m p u t e t h e a b s o lu t e p e r c e n t a g e r p r ic in g e r r o r , A P E , a s t h e a b s o lu t e v a lu a t io n e r r o r d iv id e d b y t h e m a r k e t b o n d p r ic e . A s im ila r c a lc u la t io n is m a d e f o r t h e a b s o lu t e y ie ld b a s is p o in t e r r o r , B Y E , d e (cid:12) n e d a s t h e a b s o lu t e d e v ia t io n o f t h e m o d e l y ie ld f r o m t h e m a r k e t o b s e r v e d y ie ld . T h is p r o c e d u r e is r e p e a t e d f o r e v e r y b o n d a n d f o r e a c h (cid:12) r m . A llo w in g f o r p o s s ib le d e p a r t u r e s in t h e n u m b e r o f s t r u c t u r a l p a r a m e t e r s , a s im ila r p r o c e d u r e is f o llo w e d f o r o t h e r c r e d it r is k m o d e ls . T a b le 6 p u t s t h e o u t - o f - s a m p le p r ic in g r e s u lt s in t o p e r s p e c t iv e . F ir s t , m o s t t h r e e - f a c t o r c r e d it r is k m o d e ls d o b e t t e r t h a n t h e t w o - f a c t o r in t e r e s t r a t e o n ly m o d e l. F o r in s t a n c e , m e a s u r e d b y 2 1
b o t h t h e A P E a n d B Y E , t h e le v e r a g e r a t io m o d e l a n d b o o k - t o - m a r k e t r a t io m o d e l c o n s is t e n t ly p e r f o r m b e t t e r t h a n t h e in t e r e s t r a t e m o d e l. B a s e d o n t h e e n t ir e b o n d s a m p le , t h e im p r o v e m e n t is n o t la r g e in m a g n it u d e , h o w e v e r . M o r e p r e c is e ly , t h e d i(cid:11) e r e n c e b e t w e e n t h e B / M r a t io m o d e l a n d in t e r e s t r a t e m o d e l is 0 . 0 3 % b a s e d o n t h e A P E , a n d 0 . 6 7 b a s is p o in t s b a s e d o n t h e B Y E . T h o u g h , o n t h e s u r f a c e , t h e p r ic in g im p r o v e m e n t a p p e a r s t o b e o b s e r v a t io n a lly s m a ll, g iv e n t h e la r g e n o t io n a l p r in c ip a l, it m a y s t ill b e e c o n o m ic a lly r e le v a n t . I n s p ir it o f o u r (cid:12) n d in g s , G u p t a a n d S u b r a h m a n y a m ( 2 0 0 0 ) a ls o (cid:12) n d lo w m a g n it u d e s o f o u t - o f - s a m p le p r ic in g e r r o r s in t h e c a p a n d (cid:13) o o r m a r k e t s . T h e lo w p r ic in g e r r o r s m a y b e a f e a t u r e c o m m o n t o m o s t (cid:12) x e d in c o m e m a r k e t s ( a ls o s e e D a i a n d S in g le t o n ( 2 0 0 0 ) a n d D u (cid:11) e e ( 1 9 9 9 ) ) . I n o u r c a s e , t h e m a x im u m A P E a n d B Y E in t h e \ A ll- A ll" c a t e g o r y is 1 . 8 8 % a n d 3 3 . 4 1 b p , r e s p e c t iv e ly . S e c o n d , c o m p a r in g t h e e n t r ie s in T a b le 5 a n d T a b le 6 , it is n o t s u r p r is in g t h a t t h e o u t - o f s a m p le v a lu a t io n e r r o r s a r e s t r ic t ly h ig h e r t h a n t h e c o r r e s p o n d in g in - s a m p le v a lu a t io n e r r o r s . T h e d e t e r io r a t io n in m o d e l v a lu a t io n s a m o u n t t o a b o u t 6 b a s is p o in t s w h e n a v e r a g e d o v e r a ll b o n d s , a n d a b o u t 9 b a s is p o in t s a m o n g B B B - r a t e d b o n d s . B e c a u s e t h e a s k p r ic e is n o t r e p o r t e d in t h e L e h m a n d a t a b a s e , w e a r e u n a b le t o c o m p a r e t h e v a lu a t io n e r r o r s r e la t iv e t o t h e b id - a s k s p r e a d s . I n a la t e r e x e r c is e , w e p r o v id e a s p e c i(cid:12) c a t io n a n a ly s is o f p r ic in g e r r o r s ( e s p e c ia lly a f t e r c o n t r o llin g f o r c r e d it r a t in g a n d m a t u r it y e (cid:11) e c t s ) . U n lik e o u r (cid:12) n d in g s f r o m in - s a m p le p r ic in g e r r o r s , t h ir d , t h e s t o c k p r ic e m o d e l h a s t h e w o r s t p e r f o r m a n c e o u t - o f - s a m p le ( in a ll c r e d it r a t in g a n d m a t u r it y g r o u p in g s ) . T h e B Y E ( A P E ) o f t h is m o d e l r a n g e s b e t w e e n 2 6 . 3 6 b p t o 3 8 . 6 9 b p ( 0 . 9 1 % t o 3 . 3 4 % ) c o m p a r e d t o 2 4 . 2 1 b p t o 4 2 . 6 2 b p ( 0 . 6 2 % t o 2 . 6 4 % ) f o r t h e in t e r e s t r a t e m o d e l. T h is p o t e n t ia l in c o n s is t e n c y m a y b e r e la t e d t o t h e s u b s t a n t ia l v o la t ilit y o f e q u it ie s r e la t iv e t o c o r p o r a t e y ie ld s . T h e r e f o r e , t h e u s e o f s c a le d s t o c k p r ic e p r o d u c e s u n w a n t e d v o la t ilit y in t h e d e f a u lt a b le d is c o u n t r a t e . H o w e v e r , in t h e h e d g in g s e c t io n , w e s h o w t h a t t h e s t o c k p r ic e is e (cid:11) e c t iv e in e x p la in in g p r ic e c h a n g e s . F in a lly , e v e n t h o u g h t h e o u t - o f - s a m p le p e r f o r m a n c e o f t h e le v e r a g e , B / M , p r o (cid:12) t a b ilit y , a n d t h e la g g e d s p r e a d , m o d e ls is c lo s e , t h e r e la t iv e r a n k in g b e t w e e n c r e d it r is k m o d e ls is a lt e r e d in - s a m p le v e r s u s o u t - o f - s a m p le . F o u r t h , t h e m o d e l m is p r ic in g d is p la y s s y s t e m a t ic b ia s e s a c r o s s c r e d it r a t in g a n d m a t u r it y . T h e p r in c ip a l e v id e n c e is o u t lin e d b e lo w : (cid:15) T h e a b s o lu t e y ie ld b a s is p o in t e r r o r s a r e g e n e r a lly h ig h e r f o r lo w e r r a t e d b o n d s . I n t h e c a s e o f t h e le v e r a g e r a t io m o d e l, t h e d i(cid:11) e r e n c e b e t w e e n t h e p r ic in g e r r o r s f o r A A - r a t e d v e r s u s B B B - r a t e d b o n d s is a b o u t 8 b p . T h e y a r e a ls o h ig h e r f o r s h o r t e r m a t u r it y b o n d s , w it h a d i(cid:11) e r e n c e o f a b o u t 7 b p ; (cid:15) C o n s id e r m o d e l p e r f o r m a n c e a c r o s s t h e m a t u r it y d im e n s io n . B a s e d o n t h e A P E y a r d s t ic k , t h e B / M r a t io m o d e l h a s t h e b e s t p e r f o r m a n c e . W h e n t h e B Y E v a lu a t io n e r r o r s a r e t h e 2 2
b a s is , t h e B / M is a ls o t h e b e s t m o d e l f o r s h o r t - t e r m a n d m e d iu m - t e r m b o n d s . I n c o n t r a s t , t h e p r o (cid:12) t a b ilit y m o d e l p r o d u c e s t h e lo w e s t B Y E (cid:12) t t in g e r r o r s a m o n g lo n g - t e r m b o n d s ; (cid:15) N o w c o n s id e r t h e r a t in g d im e n s io n . B o t h t h e B / M r a t io m o d e l a n d t h e le v e r a g e r a t io m o d e l a r e s u p e r io r m o d e ls c o m p a r e d t o t h e in t e r e s t r a t e m o d e l f o r A A - r a t e d a n d A - r a t e d b o n d s . T h e r e v e r s e c o n c lu s io n h o ld s t r u e f o r B B B - r a t e b o n d s ; (cid:15) R e t u r n in g t o lo n g - t e r m b o n d s , t h e d i(cid:11) e r e n c e in B Y E b e t w e e n t h e le v e r a g e r a t io m o d e l ( a n d a ls o B / M , la g g e d s p r e a d a n d p r o (cid:12) t a b ilit y m o d e ls ) a n d t h e in t e r e s t r a t e o n ly m o d e l is m o r e t h a n 2 b p . I n p a r t ic u la r , t h e s m a ll o v e r a ll d i(cid:11) e r e n c e b e t w e e n t h e c r e d it m o d e ls w it h a n d w it h o u t (cid:12) r m - s p e c i(cid:12) c v a r ia b le s is m a in ly d u e t o t h e ir c lo s e p e r f o r m a n c e in t h e s h o r t a n d m e d iu m t e r m c a t e g o r y ; (cid:15) R a r e ly is t h e la g g e d s p r e a d m o d e l s u p e r io r t o t h e le v e r a g e r a t io m o d e l a n d t h e B / M r a t io m o d e l. O v e r a ll, le v e r a g e a n d b o o k - t o - m a r k e t f a c t o r s e n h a n c e t h e w o r k in g o f t h e c r e d it r is k m o d e l. T h e a v e r a g e m is p r ic in g p a t t e r n s f o r e a c h c r e d it r is k m o d e l w a r r a n t a f e w c o m m e n t s . F ig u r e 1 a n d F ig u r e 2 d is p la y s t h e m e a n y ie ld b a s is p o in t e r r o r b y c r e d it r a t in g a n d m a t u r it y . F r o m F ig u r e 1 a n d \ A ll B o n d s , " w e c a n m a k e t h e o b s e r v a t io n t h a t ( w it h t h e e x c e p t io n o f t h e s t o c k p r ic e m o d e l) a ll c r e d it r is k m o d e ls o v e r - p r ic e t h e m a r k e t o b s e r v e d y ie ld . M o r e o v e r , t h e in t e r e s t r a t e m o d e l h a s t h e w o r s t a v e r a g e o v e r - p r ic in g o f 6 . 9 3 b p . O n t h e o t h e r h a n d , A - r a t e d a n d B B B - r a t e d b o n d s a r e u n d e r p r ic e d b y t h e s t o c k p r ic e m o d e l. T h e m is p r ic in g w o r s e n s a s t h e r a t in g d e c r e a s e s . A d i(cid:11) e r e n t m is p r ic in g p a t t e r n e m e r g e a c r o s s m a t u r it y : a ll m o d e ls o v e r - p r ic e s h o r t - t e r m b o n d s a n d u n d e r - p r ic e m e d iu m - t e r m b o n d s . T h e c r e d it r is k m o d e ls b e h a v e d i(cid:11) e r e n t ly in t h e lo n g - t e r m b o n d c a t e g o r y w it h le v e r a g e a n d B / M m o d e ls o v e r - p r ic in g , w h ile t h e in t e r e s t r a t e m o d e l u n d e r - p r ic e s . M o v in g o n t o P a n e l A o f T a b le 7 , w e (cid:12) r s t r e p o r t t h e o u t - o f - s a m p le p r ic in g a n d y ie ld p e r f o r m a n c e , b y in d u s t r y . W h ile t h e c o n c lu s io n s r e g a r d in g t h e r e la t iv e r a n k in g o f a lt e r n a t iv e c r e d it r is k m o d e ls e s s e n t ia lly h o ld , o n e a d d it io n a l in s ig h t e m e r g e s . T h a t is , m o s t m o d e ls (cid:12) t t h e u t ilit y b o n d s t h e b e s t , f o llo w e d b y in d u s t r ia l a n d t h e n (cid:12) n a n c ia l (cid:12) r m s . H o w e v e r , t h is c o n c lu s io n m u s t b e in t e r p r e t e d w it h c a u t io n , a s c o n s id e r a t io n s o f b o n d m a t u r it y a n d c a s h (cid:13) o w s t a b ilit y v a r y a c r o s s in d u s t r ie s . F o r e x a m p le , s t a b le o p e r a t in g c a s h (cid:13) o w s m a y t r a n s la t e in t o a lo w e r d e f a u lt p r o b a b ilit y f o r u t ilit ie s . P a n e l B o f T a b le 7 d e m o n s t r a t e s t h a t t h e d i(cid:11) e r e n c e s b e t w e e n c r e d it r is k m o d e ls s u r f a c e m o r e c le a r ly w h e n t h e p r ic in g e r r o r is c o n s id e r e d a t t h e in d iv id u a l (cid:12) r m le v e l. S p e c i(cid:12) c a lly , w e s h o w t h a t (cid:12) r m - s p e c i(cid:12) c d is t r e s s f a c t o r s h a v e e x p la n a t o r y a b ilit y b e y o n d s y s t e m a t ic d is t r e s s f a c t o r s . A m o n g o u r s a m p le o f 9 3 (cid:12) r m s , w e o b s e r v e t h a t t h e le v e r a g e r a t io m o d e l o u t p e r f o r m s t h e in t e r e s t r a t e 2 3
m o d e l f o r 7 1 % o f t h e (cid:12) r m s ; t h e c o r r e s p o n d in g (cid:12) g u r e s f o r b o o k - t o - m a r k e t r a t io m o d e l a n d t h e la g g e d s p r e a d m o d e l is 6 6 % a n d 6 9 % , r e s p e c t iv e ly . P a ir w is e c o m p a r is o n o f t h e le v e r a g e r a t io m o d e l a n d t h e b o o k - t o - m a r k e t r a t io m o d e ls s h o w s t h a t t h e f o r m e r o u t p e r f o r m s t h e la t t e r f o r 5 1 % o f t h e (cid:12) r m s . H e n c e , t h e t w o m o d e ls h a v e a c o m p a r a b le r e la t iv e p e r f o r m a n c e . B y f a r , o n e s u r p r is in g r e s u lt is t h e s m a ll im p r o v e m e n t m a d e b y t h e la g g e d s p r e a d m o d e l o v e r t h e in t e r e s t r a t e m o d e l. T h is m o d e l u s e s t h e la g g e d a v e r a g e s p r e a d in e x p la in in g t h e c u r r e n t s p r e a d . T h e e m p ir ic a l o u t c o m e is t h e r e f o r e p u z z lin g f r o m a n a u t o r e g r e s s iv e v ie w p o in t . I n s u m m a r y , (cid:12) r m s p e c i(cid:12) c c h a r a c t e r is t ic s , e s p e c ia lly le v e r a g e r a t io a n d b o o k - t o - m a r k e t e q u it y r a t io h e lp lo w e r t h e o u t - o f - s a m p le p r ic in g e r r o r s . C o n t r a r y t o w h a t o n e m ig h t e x p e c t , e v e n t h e w o r s t p e r f o r m in g m o d e ls ( t h e s t o c k p r ic e a n d p r o (cid:12) t a b ilit y m o d e ls ) h a v e a r e la t iv e s m a ll o u t - o f - s a m p le p r ic in g e r r o r . T h e c la s s o f c r e d it r is k m o d e ls c o n s id e r e d h a v e p e r f o r m a n c e a d e q u a t e f o r m a r k in g r is k e x p o s u r e s . 6 . 2 R o b u s t n e s s H o w r o b u s t a r e o u r e m p ir ic a l (cid:12) n d in g s t o p e r t u r b a t io n s in t e s t d e s ig n ? T o a d d r e s s t h is q u e s t io n , s e v e r a l e m p ir ic a l e x e r c is e s a r e c o n d u c t e d . I n t h e (cid:12) r s t e x p e r im e n t , w e d iv id e d t h e f u ll s a m p le p e r io d in t o t w o s u b s a m p le s . T h e y ie ld b a s is p o in t e r r o r s f o r t h e 1 9 9 4 - 1 9 9 8 s u b s a m p le ( s h o w n in T a b le 8 ) c o n (cid:12) r m s t h a t m o d e lin g in t e r e s t r a t e r is k is o f f u n d a m e n t a l im p o r t a n c e . A lb e it t h e B Y E a r e lo w e r o v e r t h is s u b s a m p le , t h e m a x im u m im p r o v e m e n t b y a n y c r e d it r is k m o d e l o v e r t h e in t e r e s t r a t e m o d e l in t h e \ A ll" c a t e g o r y is a b o u t 0 . 5 b a s is p o in t s . I t is r e a s s u r in g t h a t B Y E a g a in d is p la y s a U - s h a p e d m a t u r it y p a t t e r n , w it h B Y E o f t h e s h o r t - t e r m b o n d s t y p ic a lly m o r e p r o n o u n c e d t h a n t h e lo n g - t e r m b o n d s . B e c a u s e t h e m e d ia n c a n b e a m o r e r o b u s t s t a t is t ic , w e c o m p u t e t h e m e d ia n B Y E a c r o s s (cid:12) r m s e a c h m o n t h ; t h e r e p o r t e d m e d ia n B Y E is t h e a v e r a g e m e d ia n in t h e t im e - s e r ie s . C o m p a r in g t h e c o r r e s p o n d in g e n t r ie s in T a b le 8 a n d T a b le 6 , t h e m e d iu m B Y E is g e n e r a lly s m a lle r t h a n t h e m e a n . F o r in s t a n c e , t h e m e d ia n B Y E f o r t h e in t e r e s t r a t e ( le v e r a g e r a t io ) m o d e l is 2 3 . 9 8 b p ( 2 3 . 4 2 b p ) v e r s u s 2 8 . 0 7 b p ( 2 7 . 6 0 b p ) f o r t h e m e a n . O u r d is c u s s io n s a b o u t c r e d it r is k m o d e ls a r e n o t d e p e n d e n t o n t h e c h o ic e o f t h e m e a s u r e o f c e n t r a l t e n d e n c y . I n p a r t ic u la r , t h e d o c u m e n t e d r e s u lt s a r e n o t o v e r ly in (cid:13) u e n c e d b y e it h e r a g g r e g a t io n o r t h e e x is t e n c e o f o u t lie r s . O n e r e m a in in g c o n c e r n is t h e c o n s e q u e n c e o f f r e q u e n t ly u p d a t in g s t r u c t u r a l p a r a m e t e r s o n m o d e l p e r f o r m a n c e . T o in v e s t ig a t e t h is q u e s t io n , w e t o o k a r a t h e r e x t r e m e a p p r o a c h . I n t h is e m p ir ic a l e x e r c is e , w e t im e - a v e r a g e d t h e s t r u c t u r a l p a r a m e t e r v a lu e s a n d k e p t t h e m c o n s t a n t o v e r t h e e n t ir e s a m p le o f 8 9 : 0 3 t h r o u g h 9 8 : 0 3 . R e c a ll t h a t t h e c o n s t a n c y o f t h e r is k - n e u t r a l p a r a m e t e r s is t a n t a m o u n t t o t h e c o n s t a n c y o f t h e r is k p r e m ia , w h ic h is c o u n t e r f a c t u a l. T h e n , u p d a t in g v a lu e s f o r t h e n o m in a l in t e r e s t r a t e , t h e (cid:12) r m - s p e c i(cid:12) c d is t r e s s f a c t o r a n d t h e s t o c h a s t ic lo n g - r u n m e a n 2 4
in t e r e s t r a t e , w e r e c o m p u t e d t h e p r ic in g e r r o r s f o r a ll t h e b o n d s . I n t h e la s t p a n e l o f T a b le 8 , w e r e p o r t t h e r e s u lt in g y ie ld b a s is p o in t e r r o r s . W e c a n a g a in m a k e t h e p o in t t h a t t h e y ie ld b a s is p o in t e r r o r s a r e m o n o t o n ic a lly in c r e a s in g w h e n c r e d it - w o r t h in e s s o f t h e b o n d s w o r s e n s . I t is e v id e n t t h a t (cid:12) r m - s p e c i(cid:12) c v a r ia b le s h a v e in c r e m e n t a l p o w e r in e x p la in in g t h e y ie ld s t r u c t u r e o f c o r p o r a t e b o n d s : t h e le v e r a g e r a t io m o d e l a n d t h e B / M r a t io m o d e l n o w o u t - p e r f o r m s t h e in t e r e s t r a t e m o d e l b y a w id e r m a r g in . I n la r g e p a r t , o u r c e n t r a l (cid:12) n d in g s a r e u n a (cid:11) e c t e d b y h o w t h e s t r u c t u r a l p a r a m e t e r s a r e u p d a t e d . T h e e m p ir ic a l r e s u lt s a r e m o s t ly r o b u s t . 6 . 3 S p e c i (cid:12) c a t i o n A n a l y s i s o f P r i c i n g E r r o r s T w o k e y is s u e s a r e a d d r e s s e d in t h is s u b s e c t io n . F ir s t , a r e m o d e l b ia s e s - p o o le d b y c r e d it r a t in g o r m a t u r it y - lin k e d t o d y n a m ic v a r ia t io n s in c e r t a in s y s t e m a t ic d is t r e s s f a c t o r s ? S e c o n d , a r e c r o s s - s e c t io n a l v a r ia t io n s in m o d e l e r r o r s r e la t e d t o e x c lu d e d (cid:12) r m - s p e c i(cid:12) c f a c t o r s ? E a c h is s u e is e la b o r a t e d in t u r n . T o in v e s t ig a t e w h e t h e r t h e u n e x p la in e d p o r t io n o f t h e p r ic in g e r r o r is d u e t o s y s t e m a t ic d is t r e s s f a c t o r s o t h e r t h a n t h e s h o r t in t e r e s t r a t e , t h e f o llo w in g O L S r e g r e s s io n is e s t im a t e d : t (cid:17) (cid:17) t (cid:17) t (cid:17) t (cid:15) t ; A P E ( ) = + G I P ( ) + T E R M ( ) + D E F ( ) + ( ) ( 3 0 ) g p d 0 G I P ( t ) f o r e a c h c r e d it r is k m o d e l. I n e q u a t io n ( 3 0 ) , is t h e g r o w t h r a t e o f in d u s t r ia l p r o d u c t io n t T E R M ( t ) ( d u r in g m o n t h ) ; , t h e t e r m p r e m iu m , is t h e y ie ld d i(cid:11) e r e n t ia l b e t w e e n t h e 3 0 - y e a r D E F ( t ) t r e a s u r y b o n d a n d t h e t h r e e - m o n t h t r e a s u r y b ill r a t e ; a n d , t h e d e f a u lt p r e m iu m , is t h e y ie ld d i(cid:11) e r e n t ia l b e t w e e n B B B - r a t e d b o n d s a n d A A A - r a t e d b o n d s . T a b le 9 r e p o r t s t h e r e s u lt s o f t h e t im e - s e r ie s r e g r e s s io n , w h e r e t h e d e p e n d e n t v a r ia b le , A P E , is t h e e q u a lly - w e ig h t e d a b s o lu t e p e r c e n t a g e p r ic in g e r r o r o v e r a ll b o n d s . T h e t - s t a t is t ic s a r e c o m p u t e d u s in g r o b u s t s t a n d a r d e r r o r s . S e v e r a l p o in t s c a n b e m a d e b a s e d o n t h is t a b le . C o n s id e r t h e c o m b in e d e x p la n a t o r y a b ilit y o f 2 R t h e s y s t e m a t ic f a c t o r s . W e c a n (cid:12) r s t m a k e t h e o b s e r v a t io n t h a t t h e g o o d n e s s - o f - (cid:12) t - s t a t is t ic s a r e h ig h e r f o r A A - r a t e d b o n d s a n d m e d iu m - t e r m b o n d s . S e c o n d , a m o n g c r e d it r is k m o d e ls , t h e s t o c k p r ic e m o d e l a n d t h e p r o (cid:12) t a b ilit y m o d e l a p p e a r m o s t m is - s p e c i(cid:12) e d w h e n r a n k e d b y t h e ir a v e r a g e 2 R , a c r o s s a ll c a t e g o r ie s . T h e r e f o r e , if o n e w e r e t o c o n s id e r e x p a n d in g o n t h e s y s t e m a t ic f a c t o r s , t h e n t h e s t o c k p r ic e m o d e l a n d t h e p r o (cid:12) t a b ilit y m o d e l w o u ld b e n e (cid:12) t t h e m o s t , e s p e c ia lly in t h e m e d iu m - t e r m c a t e g o r y . R e g a r d le s s o f t h e c r e d it r is k a n d t h e m a t u r it y o f t h e b o n d , m o s t o f t h e c r e d it r is k m o d e ls e x p e r ie n c e a d e t e r io r a t io n in t h e p r ic in g q u a lit y in p e r io d s o f h ig h in d u s t r ia l g r o w t h . T h is is (cid:17) in f e r r e d f r o m n o t in g t h a t t h e c o e (cid:14) c ie n t is g e n e r a lly p o s it iv e . I t is a ls o a p p a r e n t t h a t G I P g im p a c t s t h e p r ic in g q u a lit y o f A A - r a t e d a n d s h o r t - t e r m b o n d s t h e m o s t . C o n t in u in g , p e r io d s o f 2 5
h ig h t e r m p r e m iu m s ig n i(cid:12) c a n t ly im p a c t s t h e p r ic in g o f s h o r t - t e r m b o n d s . O n e p o s s ib le e x p la n a t io n is t h a t t h e a d o p t e d t w o - f a c t o r in t e r e s t r a t e m o d e l p e r f o r m s a d e q u a t e ly f o r m e d iu m a n d lo n g - t e r m b o n d s . O u r r e s u lt s b r o a d ly s u g g e s t t h a t if t h e n u m b e r o f s y s t e m a t ic f a c t o r s w a s t o b e in c r e a s e d , t h e n t h e d e f a u lt p r e m iu m w o u ld b e a g o o d c h o ic e . A s s e e n , t h e d e f a u lt p r e m iu m is p e r h a p s t h e m o s t s ig n i(cid:12) c a n t s y s t e m a t ic d is t r e s s f a c t o r . A r is e in t h e d e f a u lt p r e m iu m lo w e r s t h e a b s o lu t e p r ic in g e r r o r s o f A A - r a t e d b o n d s . A m o n g A A - r a t e d b o n d s , w e s e e t h a t w it h t h e e x c e p t io n o f le v e r a g e a n d B / M r a t io m o d e ls , m o s t m o d e l e r r o r s a r e s ig n i(cid:12) c a n t ly r e la t e d t o d e f a u lt p r e m iu m . A m o n g B B B - r a t e d b o n d s , e a c h c r e d it r is k m o d e lin g e r r o r is s ig n i(cid:12) c a n t ly r e la t e d t o D E F ( t ) . S o , in c lu d in g d e f a u lt p r e m iu m in t h e d e f a u lt a b le d is c o u n t r a t e s p e c i(cid:12) c a t io n c a n h e lp m it ig a t e t h e p r ic in g e r r o r s o f a c r e d it r is k m o d e l. I f t h e d e f a u lt p r e m iu m w e r e t o b e in t r o d u c e d a s a s y s t e m a t ic d is t r e s s f a c t o r , o n e w o u ld n e e d t o c o n s t r u c t a z e r o - c o u p o n c r e d it s p r e a d c u r v e t h a t is r ic h e n o u g h t o p e r m it e s t im a t io n o f t h e r is k - n e u t r a l d e f a u lt p r e m iu m p r o c e s s . C u r r e n t ly , t h e d a t a f o r s u c h a n e x e r c is e is n o t r e a d ily a v a ila b le . F o r t h e s e r e a s o n s , w e h a v e m a in t a in e d f o c u s o n t h e in t e r e s t r a t e f a c t o r t h r o u g h o u t . L e t u s n o w r e c o n c ile t h e s e c o n d is s u e in s o m e d e t a il. N o t ic e t h a t , u n lik e w it h a d d it io n a l s y s t e m a t ic f a c t o r s , t w o (cid:12) r m - s p e c i(cid:12) c f a c t o r s c a n b e a c c o m m o d a t e d w it h in o u r e m p ir ic a l f r a m e w o r k ( p r o v id e d t h e r e a r e s u (cid:14) c ie n t n u m b e r o f b o n d s ) . H o w c a n a c r e d it r is k m o d e l w it h t w o (cid:12) r m s p e c i(cid:12) c f a c t o r s b e e x p e c t e d t o f a r e r e la t iv e t o t h e s e t o f o n e - f a c t o r m o d e ls a lr e a d y e s t im a t e d ? I n t h is r e g a r d , w e e x p lo r e t w o c r o s s - s e c t io n a l r e g r e s s io n s p e c i(cid:12) c a t io n s . F ir s t , w e r e g r e s s e d t h e a b s o lu t e p e r c e n t a g e p r ic in g e r r o r s o f t h e le v e r a g e r a t io m o d e l o n t h e B / M r a t io ( a ll t - s t a t is t ic s ‘ ; (cid:1) (cid:1) (cid:1) ; L a r e s ig n i(cid:12) c a n t , a n d = 1 ) , a n d o b t a in e d : ‘ B 2 A l l - B o n d s j : : (cid:15) R A P E = 0 2 5 5 + 0 0 2 1 + , = 5 . 6 % ‘ ‘ L e v ‘ M ‘ B 2 B B B - R a t e d j : : (cid:15) R A P E = 0 2 8 3 + 0 0 1 9 + = 9 . 9 % , ‘ ‘ L e v ‘ M w h e r e it is u n d e r s t o o d t h a t t h e r e p o r t e d c o e (cid:14) c ie n t s a r e a v e r a g e s a c r o s s a ll t h e m o n t h ly r e g r e s s io n s . S e c o n d , w e r e g r e s s e d t h e p r ic in g e r r o r s o f t h e B / M r a t io m o d e l o n t h e le v e r a g e r a t io , a n d o b t a in e d : 2 A l l - B o n d s j : : (cid:15) R A P E = 0 1 9 0 + 0 1 6 3 L e v + , = 4 . 8 % , ‘ ‘ ‘ B / M 2 B B B - R a t e d j : : (cid:15) R A P E = 0 0 9 4 + 0 4 1 2 L e v + , = 1 3 . 0 % . ‘ ‘ ‘ B / M I n e a c h o f t h e r e g r e s s io n s , w e in v e s t ig a t e w h e t h e r t h e p r ic in g e r r o r s a r e s ig n i(cid:12) c a n t ly c o r r e la t e d w it h a n o m it t e d (cid:12) r m - s p e c i(cid:12) c v a r ia b le . T h e r e s u lt s r e in f o r c e o u r e a r lie r a s s e r t io n t h a t it m a y b e p o s s ib le t o lo w e r t h e p r ic in g e r r o r s o f B B B - r a t e d b o n d s . F o r e x a m p le , t h e le v e r a g e f a c t o r is p o s it iv e ly a n d s ig n i(cid:12) c a n t ly r e la t e d t o t h e p r ic in g e r r o r s o f t h e B / M r a t io m o d e l. T h e a v e r a g e g o o d n e s s - o f - (cid:12) t s t a t is t ic s f r o m t h is r e g r e s s io n is 2 6
a b o u t 1 3 . 0 % . W h ile B / M is a ls o p o s it iv e ly r e la t e d t o t h e p r ic in g e r r o r s o f t h e le v e r a g e r a t io m o d e l, t h e c o e (cid:14) c ie n t a r e m u c h s m a lle r . A lt h o u g h n o t d o n e h e r e , a c r e d it r is k m o d e l w it h t w o (cid:12) r m - s p e c i(cid:12) c v a r ia b le s m a y p r o v e b e n e (cid:12) c ia l f r o m a v a lu a t io n p e r s p e c t iv e . G iv e n d a t a a n d s p a c e c o n s id e r a t io n s , w e le a v e a f o r m a l a n a ly s is o f t h e f o u r - f a c t o r c r e d it r is k m o d e ls t o a f o llo w - u p e m p ir ic a l p r o j e c t . 7 H e d g i n g P e r f o r m a n c e o f C r e d i t R i s k M o d e l s W e n o w p r o c e e d t o e x a m in e t h e d y n a m ic h e d g in g p e r f o r m a n c e o f c r e d it r is k m o d e ls . I n t h e h e d g in g s t r a t e g y , a s m a n y in s t r u m e n t s a s s o u r c e s o f r is k s a r e e m p lo y e d t o c r e a t e a d e lt a - n e u t r a l h e d g e in a ll d im e n s io n s . F o u r c r e d it r is k m o d e ls a r e c o n s id e r e d : ( i) t h e le v e r a g e r a t io m o d e l, ( ii) t h e b o o k - t o - m a r k e t r a t io m o d e l, ( iii) t h e s t o c k p r ic e m o d e l, a n d ( iv ) t h e in t e r e s t r a t e m o d e l. T h e (cid:12) r s t t w o c r e d it r is k m o d e ls p e r f o r m r e la t iv e ly b e t t e r b a s e d o n o u t - o f - s a m p le p r ic in g , w h ile t h e o t h e r t w o m o d e ls r e ly o n t r a d e d s e c u r it ie s a s f a c t o r s . (cid:28) T o (cid:12) x m a in id e a s , s u p p o s e t h e t a r g e t h e d g e is a s h o r t p o s it io n in a c o r p o r a t e b o n d w it h P t ; (cid:28) p e r io d s t o e x p ir a t io n a n d m a r k e t p r ic e ( ) . T a k e t h e s t o c k p r ic e m o d e l f o r a b e n c h m a r k illu s t r a t io n . I n t h is m o d e l, t h e r e a r e t h r e e s o u r c e s o f u n c e r t a in t y : t h e s h o r t in t e r e s t r a t e , t h e lo n g - t e r m m e a n o f t h e s h o r t in t e r e s t r a t e a n d t h e is s u in g (cid:12) r m ’ s s t o c k . T h e r e f o r e , t h e d e lt a - n e u t r a l (cid:28) h e d g e e m p lo y s t h r e e in s t r u m e n t s : ( i) a z e r o - c o u p o n t r e a s u r y w it h m a t u r it y , ( ii) a n o t h e r z e r o - 1 (cid:28) c o u p o n t r e a s u r y w it h m a t u r it y , a n d ( iii) t h e is s u in g (cid:12) r m ’ s s t o c k . I t is u n d e r s t o o d t h a t t h e 2 p o s it io n in g in b o n d s n e u t r a liz e s in t e r e s t r a t e r is k a n d t h e p o s it io n in g in e q u it y n e u t r a liz e s e q u it y r is k . B y a s t a n d a r d a r g u m e n t , t h e c u r r e n t v a lu e o f t h e r e p lic a t in g p o r t f o lio is g iv e n b y : w t w t B t ; (cid:28) w t B t ; (cid:28) w t S t ; ( ) + ( ) ( ) + ( ) ( ) + ( ) ( ) ( 3 1 ) 0 1 1 2 2 3 w t w t w t w h e r e ( ) , ( ) a n d ( ) , a r e t h e w e ig h t s o n t h e t w o t r e a s u r y s e c u r it ie s a n d t h e is s u in g (cid:12) r m ’ s 1 2 3 s t o c k , r e s p e c t iv e ly . L e t t h e c u r r e n t s t o c k p r ic e b e d e n o t e d b y S ( t ) , a n d t h e y e t u n d e t e r m in e d w t d S t r t S t d t (cid:27) S t d ! t r e s id u a l c a s h p o s it io n b e ( ) . U s in g t h e f a c t t h a t ( ) = ( ) ( ) + ( ) ( ) a n d I t o ’ s x x 0 le m m a , w e a r r iv e a t t h e p o s it io n in g b e lo w ( s e e t h e A p p e n d ix f o r d e t a ils ) : 2 2 @ B t ;(cid:28) @ B t ;(cid:28) ( ) ( ) (cid:0) (cid:1) (cid:1) z r @ r @ z w t ; ( ) = ( 3 2 ) 1 2 1 2 1 @ B t ;(cid:28) @ B t ;(cid:28) @ B t ;(cid:28) @ B t ;(cid:28) ( ) ( ) ( ) ( ) (cid:0) @ r @ z @ z @ r 1 1 @ B t ;(cid:28) @ B t ;(cid:28) ( ) ( ) (cid:0) (cid:1) (cid:1) z r @ r @ z w t ; ; ( ) = a n d ( 3 3 ) 2 1 2 1 2 @ B t ;(cid:28) @ B t ;(cid:28) @ B t ;(cid:28) @ B t ;(cid:28) ( ) ( ) ( ) ( ) (cid:0) @ r @ z @ z @ r (cid:1) x w t : ( ) = ( 3 4 ) 3 S t ( ) 2 7
T f w d a w p m b t w v p f h b M o c m u @ B t ;(cid:28) @ B t ;(cid:28) ( ) ( ) h e p a r t ia l d e r iv a t iv e s , a n d , a r e in a n a ly t ic a l c lo s e d - f o r m u s in g t h e b o n d v a lu a t io n @ r @ z t o r m u la ( 1 4 ) . T o m a k e t h e p o s it io n s e lf - (cid:12) n a n c in g a t e a c h d a t e , w e im p o s e t h e r e s t r ic t io n t h a t : t P t ; (cid:28) (cid:0) w t B t ; (cid:28) (cid:0) w t B t ; (cid:28) (cid:0) w t S t ( ) = ( ) ( ) ( ) ( ) ( ) ( ) ( ) . T h e lo c a l r is k e x p o s u r e s f o r t h e 0 1 1 2 2 3 e f a u lt a b le c o u p o n b o n d a r e t h e a g g r e g a t e d f a c e a n d c o u p o n e x p o s u r e s a s m a d e e x a c t b e lo w : (cid:28) (cid:3) (cid:3) t ; (cid:28) (cid:17) F t ; (cid:28) c t u t ; u d u ; (cid:1) ( ) (cid:1) ( ) + ( + ) (cid:1) ( ) ( 3 5 ) r r r Z 0 (cid:28) (cid:3) (cid:3) t ; (cid:28) (cid:17) F t ; (cid:28) c t u t ; u d u ; (cid:1) ( ) (cid:1) ( ) + ( + ) (cid:1) ( ) ( 3 6 ) z z z Z 0 n d (cid:12) n a lly , (cid:28) (cid:3) (cid:3) t ; (cid:28) (cid:17) F t ; (cid:28) c t u t ; u d u ; (cid:1) ( ) (cid:1) ( ) + ( + ) (cid:1) ( ) ( 3 7 ) x x x Z 0 (cid:3) (cid:3) (cid:3) t ; (cid:28) t ; (cid:28) t ; (cid:28) h e r e (cid:1) ( ) , (cid:1) ( ) a n d (cid:1) ( ) a r e a s d is p la y e d in ( 2 1 ) - ( 2 3 ) . I n s u m , in v o k in g t h e M a r k o v r z x r o p e r t y o f t h e m o d e l, t h e p o s it io n in g in t h e r e p lic a t in g p o r t f o lio ( 3 1 ) is a f u n c t io n o f c u r r e n t a r k e t p r ic e s . T o s t u d y t h e h e d g in g e (cid:11) e c t iv e n e s s o f t h is m o d e l, w e m a in t a in a s h o r t p o s it io n in t h e d e f a u lt a b le o n d a n d c o n s t r u c t t h e a f o r e m e n t io n e d h e d g e p o r t f o lio . W e liq u id a t e t h e c o m b in e d p o s it io n a t t t im e + (cid:1) a n d c o m p u t e t h e p e r c e n t a g e h e d g in g e r r o r a s : 1 f w t r t t w t B t t ; (cid:28) (cid:0) t w B t t ; (cid:28) (cid:0) t H t t ( ) e x p [ ( ) (cid:1) ] + ( ) ( + (cid:1) (cid:1) ) + ( + (cid:1) (cid:1) ) ( + (cid:1) ) = 0 1 1 2 2 P t ; (cid:28) ( ) w t S t t (cid:0) P t t ; (cid:28) (cid:0) t ; + ( ) ( + (cid:1) ) ( + (cid:1) (cid:1) ) ( 3 8 ) 3 o h e r e t h e t e r m in t h e c u r ly b r a c k e t s r e p r e s e n t s t h e d is c r e p a n c y b e t w e e n t h e r e p lic a t in g p o r t f o lio a lu e a n d t h e v a lu e o f t h e s h o r t e d t a r g e t h e d g e . T h e r e f o r e , u n d e r t h e s t a t e d c o n v e n t io n , t h e c a s h o s it io n a p p r e c ia t e s a t t h e r is k - f r e e in t e r e s t r a t e . R e b a la n c in g t h e r e p lic a t in g p o r t f o lio , w e im p le m e n t t h is h e d g in g s t r a t e g y e a c h p e r io d a n d o r e a c h o f t h e 9 3 in d iv id u a l (cid:12) r m s . A s a c o n s e q u e n c e , w e h a v e a c r o s s - s e c t io n o f p e r c e n t a g e t e d g in g e r r o r s { o n e f o r e a c h p e r io d . T h e r e p o r t e d a b s o lu t e p e r c e n t a g e h e d g in g e r r o r is g iv e n K 1 t t (cid:17) j H t t j y : A H E ( + (cid:1) ) ( + (cid:1) ) , a n d t h e m e a n p e r c e n t a g e h e d g in g e r r o r is g iv e n b y : k = 1 K K 1 P t t (cid:17) H t t H E ( + (cid:1) ) ( + (cid:1) ) . B e c a u s e t h e p a r a m e t e r s o f t h e c r e d it r is k m o d e ls a r e u p d a t e d k = 1 K P n c e e a c h q u a r t e r , t h e r e b a la n c in g f r e q u e n c y is s e t t o t h r e e m o n t h s a n d t h e h e d g in g e r r o r s a r e t = a lc u la t e d o v e r t h e f o llo w in g o n e m o n t h ( i. e . , (cid:1) = 1 1 2 ) . I m p le m e n t in g a d y n a m ic h e d g in g s t r a t e g y w it h t h e le v e r a g e r a t io m o d e l ( a n d t h e b o o k - t o a r k e t r a t io m o d e l) p o s e s a d ile m m a , a s le v e r a g e ( a n d b o o k v a lu e ) is n o n - t r a d e d . H o w e v e r , w e s e a s in s t r u m e n t s in c o n s t r u c t in g t h e h e d g e t r e a s u r ie s a n d in d iv id u a l e q u it y p r ic e t h a t h e d g e s t h e 2 8
m a r k e t c o m p o n e n t o f le v e r a g e a n d b o o k - t o - m a r k e t . I n p a r t ic u la r , t h e lo n g - t e r m d e b t c o m p o n e n t t o f le v e r a g e is le f t u n h e d g e d o v e r t h e n e x t o n e - m o n t h p e r io d . T h a t is , w r it e le v e r a g e a s L e v ( ) = D t = q ( ) q , w h e r e is t h e n u m b e r o f s h a r e s o u t s t a n d in g a n d D ( t ) is t h e d e b t v a lu e f r o m t h e S t D t = q ( ) + ( ) p r e v io u s p e r io d ( t o a v o id lo o k - a h e a d b ia s e s ) . P r o c e e d in g in t h e s a m e f a s h io n a s b e f o r e , w e c a n D t = q ( ) 2 w t w t w t (cid:0) (cid:2) d e r iv e ( ) a n d ( ) a s s h o w n in ( 3 2 ) - ( 3 3 ) a n d ( ) = (cid:1) . R e p la c e D ( t ) w it h x 1 2 3 S t D t = q ( ( ) + ( ) ) B ( t ) t o g e t t h e r e s p e c t iv e p o s it io n in g f o r t h e b o o k - t o - m a r k e t r a t io m o d e l. T h e p e r c e n t a g e h e d g in g e r r o r s f o r t h e r e s p e c t iv e c r e d it r is k m o d e l c a n n o w b e c o m p u t e d b y a p p e a lin g t o e q u a t io n ( 3 8 ) . P a n e l A o f T a b le 1 0 p r e s e n t s t h e h e d g in g e r r o r r e s u lt s . A n u m b e r o f p o in t s a r e w o r t h h ig h lig h t in g . F ir s t , u s in g e q u it y o f a (cid:12) r m a s a h e d g in g in s t r u m e n t r e d u c e s t h e m o d e l h e d g in g e r r o r s . S p e c i(cid:12) c a lly , m e a s u r e d b y b o t h t h e a b s o lu t e ( p e r c e n t a g e ) h e d g in g e r r o r s a n d t h e m e a n h e d g in g e r r o r s , t h e p e r f o r m a n c e o f t h e s t o c k p r ic e m o d e l is r e la t iv e ly b e t t e r t h a n it s c o m p e t it o r s . T o b e c o n c r e t e , t h e a v e r a g e a b s o lu t e ( m e a n ) h e d g in g e r r o r f o r t h e s t o c k p r ic e m o d e l is 1 . 4 3 % ( 0 . 5 7 % ) . T h e c o r r e s p o n d in g h e d g in g e r r o r s f o r t h e in t e r e s t r a t e m o d e l is 1 . 5 4 % ( 0 . 6 5 % ) . T h a t t h e u s e o f e q u it y p o s it io n in g im p r o v e s h e d g in g e (cid:11) e c t iv e n e s s is a ls o o b s e r v e d in s u b s a m p le s . T h e s a m e c o n c lu s io n h o ld s w h e n w e r e p la c e a v e r a g e h e d g in g e r r o r w it h m e d ia n e r r o r s ( n o t r e p o r t e d ) . T h e (cid:12) n d in g t h a t s t o c k p r ic e m o d e l p r o v id e s s u p e r io r h e d g in g p e r f o r m a n c e is in c o n t r a s t t o t h e r e s u lt t h a t s t o c k p r ic e m o d e l h a s t h e w o r s t o u t - o f - s a m p le p r ic in g p e r f o r m a n c e . T h e r e f o r e , w e a r e p o t e n t ia lly c o n f r o n t e d w it h a s it u a t io n w h e r e a c r e d it r is k m o d e l w it h t h e b e s t d y n a m ic h e d g in g p e r f o r m a n c e m a y in f a c t p r o v id e t h e w o r s t o u t - o f - s a m p le p e r f o r m a n c e . S im u la t io n s c o n (cid:12) r m t h a t t h is is p o s s ib le w h e n t h e in t r o d u c t io n o f t h e n o n - t r a d e d e x p la n a t o r y v a r ia b le h in d e r s t h e w o r k in g o f t h e t r a d e d v a r ia b le s . S e c o n d , le a v in g t h e d e b t c o m p o n e n t a n d t h e b o o k c o m p o n e n t o f le v e r a g e a n d b o o k - t o - m a r k e t u n a c c o u n t e d in t h e d e lt a - n e u t r a l s t r a t e g y c a n im p a ir t h e h e d g in g a b ilit y o f t h e c r e d it r is k m o d e ls . D e s p it e t h e u s e o f e q u it y a s a n in s t r u m e n t in t h e s e m o d e ls , t h e a b s o lu t e h e d g in g e r r o r s f r o m t h e le v e r a g e r a t io m o d e l a r e s o m e t im e s e v e n h ig h e r t h a n t h a t o f t h e in t e r e s t r a t e m o d e l ( in 8 o u t 1 6 c a t e g o r ie s w it h A H E a n d 4 o u t o f 1 6 w it h M H E ) . W h e n a v e r a g e d o v e r a ll b o n d s , t h e a b s o lu t e h e d g in g e r r o r o f t h e le v e r a g e r a t io ( b o o k - t o - m a r k e t ) m o d e l is 1 . 5 3 % ( 1 . 6 1 % ) c o m p a r e d t o 1 . 5 4 % f o r t h e in t e r e s t r a t e m o d e l. O n e le s s o n in h e r e n t in t h is h e d g in g e x e r c is e is t h a t c r e d it r is k m o d e ls r e ly in g o n p u r e ly t r a d e d f a c t o r t e n d t o p r o d u c e b e t t e r h e d g in g e (cid:11) e c t iv e n e s s . M o r e o v e r , in f o r m a t io n a b o u t d y n a m ic c h a n g e s in t h e d e b t / b o o k v a lu e o f t h e (cid:12) r m is e c o n o m ic a lly r e le v a n t f o r t h e p r o (cid:12) t / lo s s a c c o u n t s o f t h e h e d g e r . T h ir d , f o r a ll c r e d it r is k m o d e ls , t h e d e lt a - n e u t r a l a b s o lu t e h e d g in g e r r o r s a r e c o n s is t e n t ly lo w e r f o r s h o r t - t e r m b o n d s . T h a t is , h ig h e r t h e m a t u r it y o f t h e b o n d , t h e m o r e v a r ia b le is t h e h e d g e , e v e n a f t e r c o n t r o llin g f o r c r e d it r a t in g . T h e h e d g in g e r r o r s a r e f o u n d t o r is e w it h m a t u r it y , b u t n o t a s f a s t a s m a t u r it y . T h e la t t e r r e s u lt is c o n s is t e n t w it h t h e o u t - o f - s a m p le p r ic in g e x e r c is e s 2 9
w h e r e t h e a b s o lu t e y ie ld e r r o r s w e r e r e la t iv e ly h ig h e r f o r s h o r t - t e r m b o n d s . S h if t in g f o c u s t o t h e m e a n h e d g in g e r r o r s , a ll t h e c r e d it r is k m o d e ls o v e r - h e d g e t h e t a r g e t s h o r t p o s it io n , r e g a r d le s s o f t h e b o n d m a t u r it y . W e c a n a ls o m a k e t h e o b s e r v a t io n t h a t t h e M H E a r e h u m p - s h a p e d , w it h m e d iu m - t e r m b o n d s o v e r - s h o o t in g it s t a r g e t p o s it io n t h e m o s t . F o u r t h , t h e h e d g in g p e r f o r m a n c e is g e n e r a lly b e t t e r f o r h ig h q u a lit y b o r r o w e r s t h a n f o r le s s c r e d it w o r t h y b o r r o w e r s . I n t h is c a s e , t h e r e la t iv e r a n k in g f r o m t h e a b s o lu t e h e d g in g e r r o r s a n d m e a n h e d g in g e r r o r s a r e in a g r e e m e n t . E a c h f o r s h o r t - t e r m a n d m e d iu m - t e r m b o n d s , t h e h e d g in g e r r o r s a r e U - s h a p e d in c r e d it r a t in g f o r e a c h m o d e l: g o in g d o w n (cid:12) r s t f r o m A A - r a t e d t o A - r a t e d , a n d t h e n r is in g t o B B B - r a t e d b o n d s . A s s t r e s s e d e a r lie r , t h e a v e r a g e m a t u r it y o f t h e A A b o n d s in o u r s a m p le is m u c h lo n g e r t h a n t h a t o f t h e A b o n d s . T h is a s p e c t o f t h e b o n d d a t a m a y e x p la in t h e b e t t e r h e d g in g p e r f o r m a n c e f o r t h e A - r a t e d b o n d s r e la t iv e t o t h e A A - r a t e d b o n d s . A lt h o u g h n o t s h o w n in a t a b le , f o r a ll m o d e ls , t h e h e d g in g p e r f o r m a n c e is t h e b e s t f o r u t ilit y (cid:12) r m s , w h ile t h e r e is n o c le a r p a t t e r n in t h e h e d g in g p e r f o r m a n c e b e t w e e n (cid:12) n a n c ia l a n d in d u s t r ia l (cid:12) r m s . T h is r e s u lt a ls o m ir r o r s t h e o u t - o f - s a m p le p r ic in g p e r f o r m a n c e in t h e p r e v io u s s e c t io n . W h e n w e s p lit t h e s a m p le p e r io d in t o t w o p a r t s , t h e h e d g in g e r r o r s a r e v ir t u a lly s im ila r . O u r c o n c lu s io n s a b o u t h e d g in g e (cid:11) e c t iv e n e s s a r e n o t a f a c t o r o f t h e s a m p le . F in a lly , d e lt a - n e u t r a l h e d g in g e r r o r s f o r 2 5 in d iv id u a l (cid:12) r m s a r e p r e s e n t e d in P a n e l B o f T a b le 1 0 . O n e n o t e w o r t h y a s p e c t o f t h is t a b le is t h a t a ll f o u r m o d e ls p e r f o r m c o n s is t e n t ly a c r o s s (cid:12) r m s . A ll t h e f o u r m o d e ls g e n e r a t e r e la t iv e ly lo w h e d g in g e r r o r s f o r t h e s a m e g r o u p o f (cid:12) r m s , w h ile p r o d u c in g h ig h h e d g in g e r r o r s f o r o t h e r g r o u p s o f (cid:12) r m s . O v e r a ll, t h e in s t r u m e n t s d o a f a ir ly g o o d j o b n e u t r a liz in g (cid:12) r m le v e l d e f a u lt r is k . 8 C o n c l u d i n g R e m a r k s a n d F u t u r e W o r k I n s p ir e d b y t h e s t r a n d o f r e s e a r c h in D u (cid:11) e e ( 1 9 9 9 ) , D u (cid:14) e a n d S in g le t o n ( 1 9 9 7 ) , J a r r o w a n d T u r n b u ll ( 1 9 9 5 ) , L o n g s t a (cid:11) a n d S c h w a r t z ( 1 9 9 5 ) , a n d M a d a n a n d U n a l ( 1 9 9 8 ) , t h is p a p e r h a s d e v e lo p e d a t h e o r e t ic a l a n d e m p ir ic a l f r a m e w o r k f o r a n a ly z in g v a r ia t io n s in d e f a u lt a b le b o n d y ie ld s . T h e v a lu a t io n m e t h o d o lo g y r e lie s o n a s u r p r is e s t o p p in g t im e a p p r o a c h a n d im p o s e s t h e a s s u m p t io n t h a t r e c o v e r y is a f r a c t io n o f t h e p r e - d e f a u lt v a lu e o f t h e d e f a u lt a b le d e b t ( D u (cid:14) e a n d S in g le t o n ( 1 9 9 9 ) ) . S p e c i(cid:12) c a lly , t h e t h e o r e t ic a l a p p r o a c h o (cid:11) e r s t h e v e r s a t ilit y t h a t t h e d e f a u lt a b le d is c o u n t r a t e c a n b e f o r m u la t e d a s a f u n c t io n o f (cid:12) r m - s p e c i(cid:12) c d is t r e s s f a c t o r s a s w e ll a s s y s t e m a t ic d is t r e s s f a c t o r s . O u r e m p ir ic a l in v e s t ig a t io n p o s e d t w o q u e s t io n s o f b r o a d e c o n o m ic in t e r e s t : 1 . W h e n (cid:12) t t e d t o d e f a u lt a b le c o u p o n b o n d s , w h ic h c r e d it r is k m o d e l p r o v id e s s u p e r io r o u t - o f s a m p le p r ic in g a n d h e d g in g p e r f o r m a n c e ? 3 0
2 . W h a t t y p e o f f a c t o r s a r e im p o r t a n t f o r e x p la in in g c r e d it r is k ? D o s y s t e m a t ic ( o r (cid:12) r m s p e c i(cid:12) c ) d is t r e s s f a c t o r s d r iv e t h e p r ic e o f d e f a u lt ? O u r a n a ly s is o f c r e d it r is k m o d e ls is b a s e d o n m e a s u r a b le y a r d s t ic k s . F ir s t , t h e e c o n o m e t r ic f r a m e w o r k v e r i(cid:12) e s t h a t s u c h (cid:12) r m - s p e c i(cid:12) c f a c t o r s a s le v e r a g e a n d b o o k - t o - m a r k e t a c c o u n t f o r c r o s s s e c t io n a l v a r ia t io n s in c o r p o r a t e y ie ld s . S e c o n d , b a s e d o n t h e in - s a m p le a n d o u t - o f - s a m p le p r ic in g e r r o r m e t r ic s , it is s h o w n t h a t in t e r e s t r a t e c o n s id e r a t io n s a r e o f (cid:12) r s t - o r d e r p r o m in e n c e in e x p la in in g t h e p r ic in g a n d h e d g in g o f c o r p o r a t e b o n d s . A t t h e s a m e t im e , t h e e m p ir ic a l e x a m in a t io n le a d s t o t h e c o n c lu s io n t h a t (cid:12) r m - s p e c i(cid:12) c d is t r e s s f a c t o r s a r e n o t r e d u n d a n t . P a r t ic u la r ly p a r a m e t e r iz e d c r e d it r is k m o d e ls t h a t in c o r p o r a t e le v e r a g e a n d b o o k - t o - m a r k e t r e d u c e y ie ld e r r o r s f o r t h e m a j o r it y o f t h e (cid:12) r m s . T h is p o r t io n o f t h e in q u ir y s u g g e s t t h a t m o d e l p e r f o r m a n c e is b e t t e r f o r ( i) h ig h g r a d e b o n d s t h a n f o r lo w g r a d e b o n d s ( o f c o m p a r a b le m a t u r it y ) , a n d f o r ( ii) lo n g - m a t u r it y b o n d s t h a n f o r s h o r t - m a t u r it y b o n d s ( o f c o m p a r a b le c r e d it r a t in g ) . V a lu a t io n e r r o r s a r e c o r r e la t e d w it h d e f a u lt p r e m iu m ; if o n e w e r e t o c o n s id e r b r o a d e n in g t h e s e t o f s y s t e m a t ic d e f a u lt f a c t o r s , t h e d e f a u lt p r e m iu m w o u ld b e a g o o d m o d e lin g c h o ic e . F in a lly , t h e u s e o f in d iv id u a l e q u it y a s a n in s t r u m e n t is d e s ir a b le f r o m d y n a m ic h e d g in g s t a n d p o in t s . C r e d it r is k m o d e ls n e u t r a liz in g e q u it y r is k m it ig a t e d e lt a - n e u t r a l h e d g in g e r r o r s . S in c e c a lla b ilit y , p u t a b ilit y a n d c o n v e r t ib ilit y a r e c o m m o n ly a s s o c ia t e d f e a t u r e s o f c o r p o r a t e c o u p o n b o n d s , m o r e m o d e lin g e (cid:11) o r t is n e e d e d t o c h a r a c t e r iz e o p t io n a lit y . T h e c la s s o f c r e d it r is k m o d e ls p r o p o s e d h e r e c a n b e u s e d t o u n d e r s t a n d w h y (cid:12) r m s e m p lo y c e r t a in t y p e o f d e r iv a t iv e s . M o r e o v e r , c lo s e d - f o r m m o d e lin g a n d e m p ir ic a l t e s t in g o f c r e d it d e r iv a t iv e s c o n t in u e s t o b e a n is s u e r e le v a n t t o p r a c t it io n e r s . U s in g o u r a s s u m p t io n s a b o u t t h e d e f a u lt a b le d is c o u n t r a t e , t h e e x is t in g s e t - u p c a n b e a d a p t e d t o p r ic e o p t io n c o n t r a c t s t h a t a r e d e f a u lt c o n t in g e n t . W h e t h e r it is a n e m b e d d e d o p t io n o r a c o m p le x c r e d it d e r iv a t iv e , it s in t r in s ic v a lu e c a n b e m a d e t o d e p e n d o n b o t h s y s t e m a t ic d is t r e s s f a c t o r s a n d is s u e r - s p e c i(cid:12) c d is t r e s s f a c t o r s . I n p r in c ip le , t h e e m p ir ic a l w o r k c a n b e e x t e n d e d t o t h e c la s s o f n o n - a (cid:14) n e d e f a u lt r is k m o d e ls . T h e a v a ila b ilit y o f s p e c u la t iv e g r a d e b o n d s c a n f u r t h e r e n h a n c e o u r c o m p r e h e n s io n o f d e f a u lt . E v e n t h o u g h in t u it iv e ly a p p e a lin g , o n e c o u ld r e la x t h e p r e v a ilin g a s s u m p t io n a b o u t r e c o v e r y in d e f a u lt . M u c h r e m a in s t o b e r e c o n c ile d a b o u t d e f a u lt r is k . 3 1
A p p e n d i x : P r o o f o f R e s u l t s P r o o f o f t h e D e f a u l t a b l e C o u p o n B o n d P r i c e i n E q u a t i o n ( 7 ) y t (cid:21) I n D u (cid:14) e a n d S in g le t o n ( 1 9 9 9 ) , t h e r e c o v e r y in d e f a u lt is : ( ) = ( a d a p t e d p r o p o r t io n o f t h e p r e - d e f a u lt b o n d v a lu e . I n t h is c a s e , t (cid:28) u + Q P t ; (cid:28) E (cid:0) r s h s d s d C u F (cid:0) ( ) = e x p [ ( ) + ( ) ] ( ) + + e x p t (cid:26) (cid:18) (cid:19) (cid:18) Z Z Z t t u t (cid:28) + ; (cid:28) (cid:0) r s h s d s d u : h u (cid:21) u P u ) e x p [ ( ) + ( ) ] + ( ) ( ) ( (cid:18) (cid:19) (cid:27) Z Z t t N o w d e (cid:12) n e t h e d is c o u n t e d g a in s p r o c e s s : t u G t (cid:17) (cid:0) d C u r s h s d s ( ) ( ) + e x p ( ( ) + ( ) ) (cid:18) (cid:19) Z Z 0 0 t u h u (cid:21) u P u ; (cid:28) (cid:0) r s h s d ( ) ( ) ( ) e x p ( ( ) + ( ) ) (cid:18) Z Z 0 0 t (cid:0) r s h s d s P t ; (cid:28) ; + e x p ( ( ) + ( ) ) ( ) (cid:18) (cid:19) Z 0 w h ic h is a m a r t in g a le ( b e c a u s e it is t h e c o n d it io n a l e x p e c t a t io n o f a t e r m I t f o llo w s b y a n a p p lic a t io n o f I t o ’ s le m m a t h a t : Q E d P t ; (cid:28) f c t h t (cid:21) t P t ; (cid:28) (cid:0) r t h t P 0 = [ ( ) ] + ( ) + ( ) ( ) ( ) [ ( ) + ( ) ] ( t o r t h a t Q E d P t ; (cid:28) f c t r t (cid:0) (cid:21) t h t P t ; (cid:28) 0 = [ ( ) ] + ( ) + [ ( ) + ( 1 ( ) ) ( ) ] ( ) t T h is im p lie s b y I t o ’ s L e m m a o n c e a g a in t h a t t u L t (cid:17) c u (cid:0) r s h s (cid:0) (cid:21) s d s d ( ) ( ) e x p [ ( ) + ( ) ( 1 ( ) ] (cid:18) (cid:19) Z Z 0 0 t (cid:0) r s h s (cid:0) (cid:21) s d s P t ; (cid:28) e x p [ ( ) + ( ) ( 1 ( ) ] ( ) (cid:18) (cid:19) Z 0 Q L t L T is a m a r t in g a le . E q u a t in g ( ) t o t h e c o n d it io n a l e x p e c t a t io n o f ( ) 2 P ; t (cid:28) F c o n d it io n ( 0 + ) = , w e o b t a in t h e d e s ir e d r e s u lt . 3 2 t P ) t + t s (cid:19) in a t ; (cid:28) g d u + a n d t ( (cid:28) r [ d u l r g ) t : u ; (cid:28) ) s ( ) a n d d t ; s in g , + o w h m t h h ic h s d ( ) ] v a r ia e t e r m is a n s (cid:19) ( 3 9 ) b le ) . ( 4 0 ) ( 4 1 ) in a l
P t w e I C ( r o o f o f D e l t a - N e u t r a l H e d g e S u p p o s e w e u s e a c o m b in a t io n o h e d g e t h e d e f a u lt a b le c o u p o n b V t w t ( ) = ( ) + 0 B t ; (cid:28) h e r e ( ) is t h e d e f a u lt - f r e e ‘ q u it y p r ic e . T h e n , Q d V t (cid:0) E d V t ( ) [ ( ) ] = t t is a ls o t r u e t h a t : @ P t ; (cid:28) 1 ( ) d d P t (cid:0) E d P t ( ( ) [ ( ) ] = @ S S o m p a r in g ( 4 3 ) a n d ( 4 4 ) a n d s o lv 2 3 4 ) d is p la y e d in t h e t e x t . s i n E q u a t i o n s ( 3 2 ) - ( 3 4 ) o f t w o d e f a u lt - f r e e d is c o u n t b o n d s a n d t h o n d . C o n s id e r a r e p lic a t in g p o r t f o lio o f t h e w t B t ; (cid:28) w t B t ; (cid:28) w t S t ( ) ( ) + ( ) ( ) + ( ) ( ) 1 1 2 2 3 (cid:28) d is c o u n t b o n d p r ic e w it h m a t u r it y a n d ‘ w t d S (cid:0) E d S ( ) ( [ ] ) 3 @ B t ; (cid:28) @ B ( ) 1 d r (cid:0) E d r w t w t + ( [ ] ) ( ) + ( ) 1 2 (cid:26) @ r @ B t ; (cid:28) @ B ( ) 1 d z (cid:0) E d z w t w t + ( [ ] ) ( ) + ( ) 1 2 (cid:26) @ z @ P t ; (cid:28) ( ) S (cid:0) E d S d r (cid:0) E d r d z (cid:0) E [ ] ) + ( [ ] ) + ( @ r in g a s e t o f s im u lt a n e o u s e q u a t io n s p r o v e t 3 3 e e q u t y p e ; S t ( ) t ; (cid:28) ( 2 @ r t ; (cid:28) ( 2 @ z d z [ ] ) h e p o it y o : r e p r ) (cid:27) ) : (cid:27) @ P ( @ s it io f t e s e t ; (cid:28) z n in h n ) g e t : s ( (cid:12) ( ( ( 3 r m 4 2 ) t h e 4 3 ) 4 4 ) 2 ) -
B F B t B f C o C o C a D o D o D 1 D t D s D F E b E e R e f e r e n c e s r iy s , E . , a n d F . d e V a r e n n e , 1 9 9 7 , \ V a lu in g r is k y (cid:12) x e d r a t e d e b t : a n e x t e n s io n , " J o u r n a l o f i n a n c i a l a n d Q u a n t i t a t i v e A n a ly s i s , V o l. 3 2 , N o . 2 , 2 3 9 - 2 4 8 . r o w n , S . , a n d P . D y b v ig , 1 9 8 6 , \ T h e e m p ir ic a l im p lic a t io n s o f t h e C o x , I n g e r s o ll, R o s s t h e o r y o f h e t e r m s t r u c t u r e o f in t e r e s t r a t e s , " J o u r n a l o f F i n a n c e 4 1 , 6 1 6 - 6 2 8 . u h le r , W . , M . U h r ig - H o m b u r g , U . W a lt e r , a n d T . W e b b e r , 1 9 9 9 , \ A n e m p ir ic a l c o m p a r is o n o f o r w a r d a n d s p o t r a t e m o d e ls f o r v a lu in g in t e r e s t r a t e o p t io n s , " J o u r n a l o f F i n a n c e 5 4 , 2 6 9 - 3 0 5 . h e n , R . , a n d L . S c o t t , 1 9 9 3 , \ M a x im u m lik e lih o o d e s t im a t io n f o r a m u lt if a c t o r e q u ilib r iu m m o d e l f t h e t e r m s t r u c t u r e o f in t e r e s t r a t e s , " J o u r n a l o f F i x e d I n c o m e 3 , 1 4 - 3 1 . o llin - D u f r e s n e , P . , a n d R . G o ld s t e in , 1 9 9 9 , \ D o c r e d it s p r e a d s r e (cid:13) e c t s t a t io n a r y le v e r a g e ? R e c n c ilin g s t r u c t u r a l a n d r e d u c e d - f o r m f r a m e w o r k s , " J o u r n a l o f F i n a n c e ( f o r t h c o m in g ) . o llin - D u f r e s n e , P . , a n d B . S o ln ik , 1 9 9 9 , \ O n t h e t e r m s t r u c t u r e o f d e f a u lt p r e m ia in t h e s w a p n d L I B O R m a r k e t s , " J o u r n a l o f F i n a n c e ( f o r t h c o m in g ) . a i, Q . , a n d K . S in g le t o n , 2 0 0 0 , \ S p e c i(cid:12) c a t io n a n a ly s is o f a (cid:14) n e t e r m s t r u c t u r e m o d e ls , " J o u r n a l f F i n a n c e 5 5 ( 5 ) , 1 9 4 3 - 1 9 7 8 . u (cid:11) e e , G . , 1 9 9 8 , \ T h e r e la t io n b e t w e e n t r e a s u r y y ie ld s a n d c o r p o r a t e b o n d y ie ld s p r e a d s , " J o u r n a l f F i n a n c e 5 3 , 2 2 2 5 - 2 2 4 2 . u (cid:11) e e , G . , 1 9 9 9 , \ E s t im a t in g t h e p r ic e o f d e f a u lt r is k , " R e v i e w o f F i n a n c i a l S t u d i e s 1 2 , N o . 1 , 9 7 - 2 2 6 . u (cid:14) e , D . , 1 9 9 6 , D y n a m ic A s s e t P r ic in g T h e o r y , 2 n d e d it io n , P r in c e t o n U n iv e r s it y P r e s s , P r in c e o n , N e w J e r s e y . u (cid:14) e , D . , a n d K . S in g le t o n , 1 9 9 7 , \ A n e c o n o m e t r ic m o d e l o f t h e t e r m s t r u c t u r e o f in t e r e s t r a t e w a p y ie ld s , " J o u r n a l o f F i n a n c e 5 2 , N o . 4 , 1 2 8 7 - 1 3 2 2 . u (cid:14) e , D . , a n d K . S in g le t o n , 1 9 9 9 , \ M o d e lin g t e r m s t r u c t u r e s o f d e f a u lt r is k y b o n d s , " R e v i e w o f i n a n c i a l S t u d i e s 1 2 , N o . 4 , 6 8 7 - 7 2 0 . lt o n , E . , M . G r u b e r , D . A g r a w a l, a n d C . M a n n , 2 0 0 1 , \ E x p la in in g t h e r a t e s p r e a d o n c o r p o r a t e o n d s , " J o u r n a l o f F i n a n c e 5 6 , N o . 1 , 2 4 7 - 2 7 7 . o m , Y . , J . H e lw e g e , a n d J . H u a n g , 2 0 0 0 , \ S t r u c t u r a l m o d e ls o f c o r p o r a t e b o n d p r ic in g : A n m p ir ic a l a n a ly s is , " m i m e o , P e n n s y lv a n ia S t a t e U n iv e r s it y . 3 4
F 4 G o U J r J r J s L R L F L g L d M 1 M c M o S R S J a m a , E . a n d K . F r e n c h , 1 9 9 2 , \ T h e c r o s s - s e c t io n o f e x p e c t e d s t o c k r e t u r n s , " J o u r n a l o f F i n a n c e 7 , N o . 2 , 4 2 7 - 4 6 5 . u p t a , A . , a n d M . S u b r a h m a n y a m , 2 0 0 0 , \ A n e s t im a t io n o f t h e s t a t ic a n d d y n a m ic p e r f o r m a n c e f in t e r e s t r a t e o p t io n p r ic in g m o d e ls in t h e d o lla r c a p - (cid:13) o o r m a r k e t s , " m i m e o , C a s e W e s t e r n n iv e r s it y a n d N e w Y o r k U n iv e r s it y . a r r o w , R . , D . L a n d o , a n d S . T u r n b u ll, 1 9 9 7 , \ A M a r k o v m o d e l f o r t h e t e r m s t r u c t u r e o f c r e d it is k s p r e a d s , " R e v i e w o f F i n a n c i a l S t u d i e s 1 0 , 4 8 1 - 5 2 3 . a r r o w , R . , a n d S . T u r n b u ll, 1 9 9 5 , \ P r ic in g d e r iv a t iv e s o n (cid:12) n a n c ia l s e c u r it ie s s u b j e c t t o c r e d it is k , " J o u r n a l o f F i n a n c e 5 0 , 5 3 - 8 5 . o n e s , E . , S . M a s o n , a n d E . R o s e n f e ld , 1 9 8 4 , \ C o n t in g e n t c la im s a n a ly s is o f c o r p o r a t e c a p it a l t r u c t u r e s : A n e m p ir ic a l in v e s t ig a t io n , " J o u r n a l o f F i n a n c e , 6 1 1 - 6 2 5 . a n d o , D a v id , 1 9 9 8 , \ O n C o x P r o c e s s e s a n d C r e d it R is k y S e c u r it ie s , " R e v i e w o f D e r i v a t i v e s e s e a r c h 2 , 9 9 - 1 2 0 . it t e r m a n , R . a n d J . S c h e in k m a n , 1 9 9 1 , \ C o m m o n f a c t o r s a (cid:11) e c t in g b o n d r e t u r n s , " J o u r n a l o f i x e d I n c o m e 1 , 5 4 - 6 1 . o n g s t a (cid:11) , F . , a n d E . S c h w a r t z , 1 9 9 2 , \ I n t e r e s t r a t e v o la t ilit y a n d t h e t e r m s t r u c t u r e : A t w o - f a c t o r e n e r a l e q u ilib r iu m m o d e l, " J o u r n a l o f F i n a n c e 4 7 , 1 2 5 9 - 1 2 8 2 . o n g s t a (cid:11) , F . , a n d E . S c h w a r t z , 1 9 9 5 , \ A s im p le a p p r o a c h t o v a lu in g r is k y (cid:12) x e d a n d (cid:13) o a t in g r a t e e b t , " J o u r n a l o f F i n a n c e 5 0 , 7 8 9 - 8 1 9 . a d a n , D . , a n d H . U n a l, 1 9 9 8 , \ P r ic in g t h e r is k s o f d e f a u lt , " R e v i e w o f D e r i v a t i v e s R e s e a r c h 2 , 2 1 - 1 6 0 . a d a n , D . , a n d H . U n a l, 2 0 0 0 , \ P r ic in g r is k y d e b t : a t w o - f a c t o r h a z a r d - r a t e m o d e l w it h c o m p le x a p it a l s t r u c t u r e , " J o u r n a l o f F i n a n c i a l a n d Q u a n t i t a t i v e A n a ly s i s 3 5 , N o . 1 , 4 3 - 6 5 . e r t o n , R . , 1 9 7 4 , \ O n t h e p r ic in g o f c o r p o r a t e d e b t : t h e r is k s t r u c t u r e o f in t e r e s t r a t e s , " J o u r n a l f F i n a n c e 2 9 , 4 4 9 - 4 7 0 . c h w a r t z , T . , 1 9 9 8 , \ E s t im a t in g t h e T e r m S t r u c t u r e s o f C o r p o r a t e D e b t , " R e v i e w o f D e r i v a t i v e s e s e a r c h 2 , 1 9 3 - 2 3 0 . h u m w a y , T y le r , 2 0 0 1 , \ F o r e c a s t in g b a n k r u p t c y m o r e a c c u r a t e ly : A s im p le h a z a r d r a t e m o d e l, " o u r n a l o f B u s i n e s s 7 4 , N o . 1 , 1 0 1 - 1 2 4 . 3 5
T F V n W S it m a n , i n a n c e , a s ic e k , a n c i a l E e i, D . , c h w a r t z S V O c a a . , a o l. 4 . , 1 o n o n d n d n 3 9 m D M d R . W , N o . 1 , 7 7 , \ A n i c s 5 , 1 . G u o , e r t o n m e 1 7 1 s s e ls , 1 9 8 8 , - 1 9 . e q u ilib r iu m 7 - 1 8 8 . 9 9 7 , \ P r ic in o d e ls , " J o u r \ c g n T h e h a r a c r is k y a l o f d t F e e d i t r e x e r m in iz a t io b t : A e d I n 3 6 a n t s n o f n e m c o m e o f t h e p ir , S e c a p it a l s t r u c t u t e r m s t r u c t u r ic a l c o m p a r is o p t e m b e r , 8 { 2 8 r e e , " n . o c T f h o ic h e t h e e , " J o u L o J o u r r n a l n g s t a n o (cid:11) a f l F a n o f i d
T a b l e 1 : Y i e l d , C r e d I n t h is t a b le , w e r e p o r t t h e f o llo w in g b o n d a a v e r a g e y ie ld ( in % ) , ( iii) t h e a v e r a g e n u m e s o u r c e is L e h m a n B r o t h e r s F ix e d I n c o m e D B / t e r m d e b t a n d m a r k e t v a lu e o f e q u it y ) ; is o p e r a t in g in c o m e d iv id e d b y n e t s a le s . ( C R S P ) . B o n d s w it h c r e d it r a t in g b e t w e e n B B B - r a t e d h ig h e r t h a n 9 a s ( o r b e lo w ) . n o n - p u t a b le . W e c la s s if y (cid:12) r m - s p e c i(cid:12) c d a t b r e v it y , o n ly t h e in f o r m a t io n o n 2 5 (cid:12) r m s M a r c h 1 9 8 9 t h r o u g h M a r c h 1 9 9 8 . N a m e N o . o f B o n d s A m r C o r p 1 1 .1 B o e in g C o m p a n y 7 .8 B a n k o f A m e r ic a 1 7 .7 B e ll S o u t h C o r p 7 .5 B e a r S t e a r n s C o . 1 0 .7 B a n k e r s T r u s t N Y 1 2 .2 C it ic o r p 1 1 .5 C o a s t a l C o r p 8 .7 C h a s e M a n h a t t e n 9 .2 C s x C o r p 6 .8 D e lt a A ir lin e s 7 .9 C o n s o lid a t e d E d is o n 8 .4 E d is o n I n t e r n a it o n a l 1 0 .1 E n r o n C o r p 8 .0 F o r d M o t o r C o m p a n y 7 .5 F ir s t C h ic a g o N B D 9 .0 F ir s t U n io n C o r p 9 .9 I B M 6 .2 I n t e r n a t io n a l P a p e r 7 .2 M e r r ill L y n c h & C o . 1 7 .6 P h ilip M o r r is C o . 1 8 .4 U s x C o r p 8 .7 N ia g a r a M o h a w k H ld . 7 .3 T im e W a r n e r , I n c 7 .7 W a l- M a r t S t o r e s I n c 1 1 .2 A ll 4 6 2 6 2 U t ilit y 1 0 3 3 6 F in a n c ia l 1 5 4 5 2 I n d u s t r ia l 2 0 4 7 4 A A 7 0 8 1 A 2 4 4 1 0 B B B a n d b e lo w 1 4 7 7 1 i t R a t i n g a n d F i r m - S p e c i (cid:12) c D i s t r e s s F a c t o r s t t r ib u t e s : ( i) t h e a v e r a g e n u m b e r o f b o n d s o u t s t a n d in g p e r m o n t h , ( ii) t h e r ic a l c r e d it r a t in g ( a s s ig n e d b y S & P ) , a n d ( iv ) t h e a v e r a g e m a t u r it y . T h e L e v e r a g e a t a b a s e . is lo n g - t e r m d e b t d iv id e d b y (cid:12) r m v a lu e ( s u m o f lo n g - M P r o (cid:12) t is b o o k v a lu e o f e q u it y d iv id e d b y t h e m a r k e t v a lu e o f e q u it y ; A ll a c c o u n t in g ( e q u it y p r ic e ) in f o r m a t io n is t a k e n f r o m C O M P U S T A T A A - r a t e d A - r a t e d 1 a n d 5 a r e d e s ig n a t e d a s ; b e t w e e n 6 t o 8 a s ; a n d E a c h o f t h e 9 3 (cid:12) r m s in o u r s a m p le a r e c o u p o n - p a y in g , n o n - c a lla b le a n d a b y t h e ir in d u s t r y a (cid:14) lia t io n , i.e ., (cid:12) n a n c ia l, in d u s t r ia l a n d u t ilit y . F o r is s h o w n . T h e n u m b e r o f o b s e r v a t io n s is 4 6 ,2 6 2 . T h e s a m p le p e r io d is B o n d A t t r i b u t e s D i s t r e s s F a c t o r s Y i e l d C r e d i t M a t u r i t y L e v e r a g e B / M P r o (cid:12) t R a t i n g 8 .2 4 1 1 .1 1 5 .4 0 .5 3 0 0 .7 3 9 0 .1 3 2 7 .3 9 3 .9 2 9 .7 0 .1 1 9 0 .5 0 4 0 .0 9 0 7 .1 8 7 .4 6 .9 0 .3 3 7 0 .5 3 1 0 .2 6 8 6 .8 4 2 .0 2 2 .4 0 .1 6 7 0 .3 5 5 0 .4 4 4 6 .9 7 7 .1 6 .0 0 .6 1 3 0 .7 1 0 0 .5 6 8 7 .0 0 6 .4 6 .6 0 .5 4 1 0 .6 7 9 0 .0 8 7 7 .6 6 6 .8 5 .7 0 .5 3 5 0 .8 3 1 0 .1 9 9 7 .4 3 1 1 .7 1 1 .2 0 .4 8 3 0 .6 8 4 0 .1 0 5 8 .3 2 8 .4 6 .1 0 .5 8 8 1 .5 7 0 0 .1 6 1 7 .2 6 9 .7 1 1 .6 0 .3 0 2 0 .5 0 3 0 .1 8 5 8 .6 5 1 1 .5 1 5 .5 0 .4 4 0 0 .6 2 7 0 .0 7 9 6 .6 4 5 .8 5 .6 0 .3 5 6 0 .7 4 8 0 .2 7 8 6 .5 9 6 .1 4 .3 0 .4 4 8 0 .7 4 2 0 .3 5 2 7 .3 7 9 .8 8 .3 0 .3 3 7 0 .3 9 6 0 .1 1 0 7 .2 9 6 .5 1 9 .9 0 .7 7 1 0 .7 3 7 0 .1 9 2 7 .8 8 7 .9 7 .2 0 .4 3 3 1 .0 5 2 0 .2 1 5 6 .9 1 8 .1 7 .1 0 .3 0 9 0 .6 3 9 0 .2 3 0 7 .0 4 7 .0 2 0 .1 0 .1 8 2 0 .3 8 8 0 .1 5 7 7 .1 4 8 .0 9 .2 0 .3 3 1 0 .7 2 8 0 .1 3 6 6 .7 3 5 .9 5 .1 0 .6 2 1 0 .7 1 0 0 .5 1 4 6 .9 1 7 .0 5 .4 0 .1 9 0 0 .2 0 6 0 .2 2 8 7 .9 0 1 1 .6 1 4 .7 0 .4 4 6 0 .7 5 7 0 .1 1 2 8 .0 0 1 1 .6 7 .5 0 .6 4 1 1 .4 2 2 0 .2 3 3 7 .6 7 1 1 .0 1 3 .7 0 .3 8 6 0 .2 4 1 0 .1 6 6 6 .7 1 4 .0 8 .5 0 .1 3 2 0 .2 2 6 0 .0 6 0 7 .2 6 7 .7 9 .4 3 0 .4 2 7 0 .6 3 5 0 .2 2 9 7 .0 8 8 .4 8 .1 4 0 .4 4 3 0 .7 5 5 0 .3 1 4 7 .2 2 6 .9 6 .8 0 0 .5 4 6 0 .7 3 8 0 .2 6 4 7 .3 9 8 .2 1 2 .1 2 0 .3 4 1 0 .4 9 6 0 .1 6 3 7 .0 7 4 .3 1 1 .3 2 0 .3 3 4 0 .5 3 4 0 .1 6 6 7 .0 8 7 .1 8 .5 8 0 .4 3 2 0 .6 2 2 0 .2 6 5 7 .6 5 1 0 .6 9 .8 9 0 .4 7 7 0 .7 1 0 0 .2 0 9 3 7
T h e (cid:17) D d b e t w y ie ld O L S % ) . is t h r E e . T e e g r e s s io n t (cid:15) t F ( ) + ( e n t h e 3 0 d i(cid:11) e r e n c e T h e t - s t a h e r e g r e s o u t - o f - s a A A A B B B S h o r t M e d i u L o n g T a b l e 9 : A n a l y s i s o f C r e d i t R a t i n g a n d M a t u r i t y R e l a t e d B i a s e s 0 t (cid:17) (cid:17) t (cid:17) t r e s u lt s a r e b a s e d o n t h e f o llo w in g t im e - s e r ie s s p e c i(cid:12) c a t io n : A P E ( ) = + G I P ( ) + T E R M ( ) + g p G I P T E R M ) . is t h e g r o w t h r a t e o f in d u s t r ia l p r o d u c t io n , , t h e t e r m p r e m iu m , is t h e y ie ld d i(cid:11) e r e n c e D E F y e a r t r e a s u r y b o n d r a t e a n d t h e 3 - m o n t h t r e a s u r y b ill r a t e , a n d , t h e d e f a u lt p r e m iu m , is t h e b e t w e e n B B B - r a t e d c o r p o r a t e b o n d s a n d A A A - r a t e d c o r p o r a t e b o n d s . T h e m e t h o d o f e s t im a t io n is 2 2 R R t is t ic is c o m p u t e d u s in g a h e t e r o s k e d a s ic a lly c o n s is t e n t e s t im a t o r . is t h e a d j u s t e d - s t a t is t ic ( in s io n s a r e d o n e b y r a t in g a n d m a t u r it y . T h e s a m p le p e r io d is M a r c h 1 9 8 9 t h r o u g h M a r c h 1 9 9 8 . A P E m p le a b s o lu t e p e r c e n t a g e p r ic in g e r r o r s a g g r e g a t e d a c r o s s t h e r e s p e c t iv e c a t e g o r y . 2 0 0 (cid:17) (cid:17) (cid:17) (cid:17) (cid:17) (cid:17) (cid:17) (cid:17) R t ( ) t ( ) t ( ) t ( ) g g p p d d L e v e r a g e 0 .0 1 6 3 .7 4 0 .0 0 4 2 .1 5 0 .0 0 1 1 .2 5 - 0 .0 0 7 - 1 .3 3 1 0 .5 B / M 0 .0 1 7 3 .9 2 0 .0 0 4 2 .1 1 0 .0 0 1 0 .9 1 - 0 .0 0 7 - 1 .4 3 9 .9 P r o (cid:12) t a b ilit y 0 .0 3 8 5 .5 5 0 .0 0 2 0 .6 3 0 .0 0 1 1 .3 8 - 0 .0 2 6 - 3 .3 4 1 4 .5 L a g g e d S p r e a d 0 .0 2 1 4 .9 5 0 .0 0 4 2 .2 4 0 .0 0 0 0 .5 6 - 0 .0 1 1 - 2 .3 4 1 5 .4 S t o c k P r ic e 0 .0 2 4 5 .3 6 - 0 .0 0 0 - 0 .1 6 0 .0 0 1 1 .6 2 - 0 .0 1 2 - 2 .2 7 6 .1 I n t e r e s t R a t e 0 .0 2 0 4 .7 9 0 .0 0 4 2 .1 8 0 .0 0 0 0 .4 6 - 0 .0 1 0 - 2 .0 2 1 2 .8 L e v e r a g e 0 .0 0 2 0 .6 5 0 .0 0 3 1 .8 2 0 .0 0 1 1 .4 2 0 .0 1 0 2 .5 4 6 .3 B / M 0 .0 0 2 1 .6 5 0 .0 0 3 1 .4 3 0 .0 0 1 2 .5 3 0 .0 1 0 2 .5 3 6 .1 P r o (cid:12) t a b ilit y 0 .0 2 7 5 .7 5 0 .0 0 3 1 .4 2 0 .0 0 4 5 .5 9 - 0 .0 0 6 - 1 .1 2 2 7 .6 L a g g e d S p r e a d 0 .0 0 3 0 .8 2 0 .0 0 3 1 .6 9 0 .0 0 1 1 .3 5 0 .0 1 0 2 .4 7 5 .7 S t o c k P r ic e 0 .0 0 2 0 .4 7 - 0 .0 0 2 - 1 .0 2 0 .0 0 1 1 .4 1 0 .0 1 9 3 .6 4 1 7 .5 I n t e r e s t R a t e 0 .0 0 2 0 .5 1 0 .0 0 3 1 .9 8 0 .0 0 1 0 .9 2 0 .0 1 2 2 .8 6 6 .6 L e v e r a g e 0 .0 0 4 1 .0 6 0 .0 0 2 1 .5 0 0 .0 0 1 1 .2 8 0 .0 1 5 3 .4 5 9 .9 B / M 0 .0 0 6 1 .8 0 0 .0 0 2 1 .2 1 0 .0 0 1 1 .4 8 0 .0 1 2 2 .8 9 7 .4 P r o (cid:12) t a b ilit y 0 .0 4 6 7 .6 2 0 .0 0 3 1 .2 9 0 .0 0 1 1 .3 6 - 0 .0 1 9 - 2 .7 1 1 3 .6 L a g g e d S p r e a d 0 .0 0 4 0 .9 9 0 .0 0 3 1 .6 4 0 .0 0 1 1 .2 3 0 .0 1 5 3 .4 3 9 .7 S t o c k P r ic e 0 .0 0 9 2 .3 5 0 .0 0 3 1 .6 6 0 .0 0 1 1 .2 2 0 .0 1 4 3 .1 0 7 .9 I n t e r e s t R a t e 0 .0 0 7 2 .0 0 0 .0 0 2 0 .9 9 0 .0 0 1 2 .4 1 0 .0 0 8 1 .8 3 6 .5 L e v e r a g e 0 .0 0 1 0 .4 4 0 .0 0 2 2 .2 9 0 .0 0 1 3 .5 7 0 .0 0 6 3 .2 4 1 9 .1 B / M 0 .0 0 1 0 .8 0 0 .0 0 1 2 .1 8 0 .0 0 1 3 .4 5 0 .0 0 5 2 .8 7 1 7 .1 P r o (cid:12) t a b ilit y 0 .0 2 0 7 .2 5 0 .0 0 2 1 .5 3 0 .0 0 2 3 .7 0 - 0 .0 0 8 - 2 .5 0 2 2 .7 L a g g e d S p r e a d 0 .0 0 1 0 .4 2 0 .0 0 2 2 .3 1 0 .0 0 1 3 .5 1 0 .0 0 6 3 .2 1 1 8 .8 S t o c k P r ic e 0 .0 0 1 0 .1 7 - 0 .0 0 2 - 1 .4 0 0 .0 0 1 1 .8 1 0 .0 1 2 3 .5 3 1 9 .2 I n t e r e s t R a t e 0 .0 0 1 0 .5 1 0 .0 0 2 2 .4 7 0 .0 0 1 3 .2 1 0 .0 0 6 3 .1 7 1 7 .5 L e v e r a g e - 0 .0 1 2 - 2 .3 0 0 .0 0 3 1 .3 4 - 0 .0 0 0 - 0 .0 7 0 .0 3 5 6 .0 1 2 6 .1 B / M - 0 .0 1 0 - 2 .0 3 0 .0 0 2 1 .0 9 - 0 .0 0 0 - 0 .1 5 0 .0 3 3 5 .7 7 2 4 .8 m P r o (cid:12) t a b ilit y - 0 .0 2 7 4 .2 4 0 .0 0 4 1 .5 9 0 .0 0 2 2 .1 5 0 .0 0 6 0 .8 3 5 .3 L a g g e d S p r e a d 0 .0 1 1 - 2 .1 9 0 .0 0 3 1 .3 7 - 0 .0 0 0 - 0 .1 8 0 .0 3 4 6 .0 6 2 6 .3 S t o c k P r ic e - 0 .0 0 8 - 1 .8 3 - 0 .0 0 1 - 0 .5 7 0 .0 0 0 0 .2 4 0 .0 3 9 7 .5 1 4 2 .7 I n t e r e s t R a t e - 0 .0 0 9 - 1 .9 5 0 .0 0 3 1 .3 1 0 .0 0 0 0 .0 8 0 .0 3 2 5 .7 0 2 3 .8 L e v e r a g e 0 .0 1 8 2 .9 5 0 .0 0 5 1 .8 9 0 .0 0 1 1 .1 1 0 .0 0 1 0 .1 0 3 .6 B / M 0 .0 1 8 2 .9 2 0 .0 0 5 1 .6 9 0 .0 0 1 1 .1 6 0 .0 0 1 0 .1 4 2 .7 P r o (cid:12) t a b ilit y 0 .0 4 6 6 .1 3 0 .0 0 3 0 .8 9 0 .0 0 1 1 .1 2 - 0 .0 0 5 - 0 .5 5 1 .0 L a g g e d S p r e a d 0 .0 2 0 3 .1 8 0 .0 0 5 1 .7 0 0 .0 0 1 0 .9 3 - 0 .0 0 0 - 0 .0 1 2 .4 S t o c k P r ic e 0 .0 2 9 4 .9 0 0 .0 0 3 1 .3 3 0 .0 0 1 1 .5 4 - 0 .0 0 6 - 0 .8 5 4 .8 I n t e r e s t R a t e 0 .0 2 0 3 .0 9 0 .0 0 6 1 .9 3 - 0 .0 0 1 - 0 .5 9 0 .0 0 6 0 .8 0 0 .8 4 6
38 Term [63.7] [6.2] [18.8] f15.6g [78.6] [5.8] [7.7] f13.4g [62.8] [2.1] [-10.9] f14.5g Long 113 7.55 0.01 1.38 23.4 82.4 7.79 0.01 0.55 19.8 46.7 8.55 0.00 -1.70 19.9 56.5 Term [48.9] [14.6] [17.6] f16.5g [46.6] [17.5] [9.0] f14.4g [48.4] [19.0] [-5.8] f3.4g Medium 172 6.68 0.09 1.21 26.6 90.8 7.07 0.09 0.27 23.1 76.2 7.35 0.10 -0.73 11.8 39.8 Term [35.3] [16.4] [17.8] f7.2g [35.8] [16.8] [10.3] f7.3g [38.4] [16.5] [-4.0] f2.7g Short 155 5.90 0.29 0.72 49.2 90.8 6.13 0.29 0.23 48.6 69.7 6.31 0.28 -0.21 43.3 32.1 Rated [50.7] [23.9] [17.4] f10.2g [56.3] [25.2] [5.5] f6.1g [59.7] [24.8] [-8.36] f10.3g BBB 138 6.44 0.08 1.98 49.6 87.2 7.28 0.08 0.24 45.1 50.5 7.73 0.07 -1.16 42.7 48.6 Rated [57.9] [16.5] [5.4] f1.5g [56.8] [16.5] [11.0] f2.7g [49.7] [16.0] [-2.3] f2.8g A 231 6.99 0.06 0.23 44.8 29.4 6.86 0.06 0.31 47.9 54.1 7.14 0.05 -0.18 45.3 23.9 Rated [61.9] [13.7] [11.0] f7.9g [23.2] [13.6] [2.5] f8.1g [60.0] [13.8] [-1.21] f7.1g AA 69 6.80 0.04 0.66 47.7 46.8 6.39 0.05 1.00 49.6 60.55 7.07 0.04 -0.16 47.0 16.5 (cid:5)0 (cid:5)1 (cid:5)2 R 2 1t>2 (cid:5)0 (cid:5)1 (cid:5)2 R 2 1t>2 (cid:5)0 (cid:5)1 (cid:5)2 R 2 1t<(cid:0) 2 Subsample L Panel A: Leverage Ratio Panel B: B/M Ratio Panel C: Pro(cid:12)tability results are virtually similar). (t((cid:5)2) <(cid:0)2). L represents the number of yields in the OLS regression. Only the results for full sample (89:03 - 98:03) are reported (the subsample The R 2 in curly brackets is from the regression with (cid:5)1 (cid:17) 0. The counting indicator 1t>2 (1t<(cid:0) 2) is the proportion of months in which t((cid:5)2) > 2 (in square brackets) is the average coe(cid:14)cient divided by the standard error of the coe(cid:14)cient. The reported R is the adjusted-R statistic (in %). 2 2 years are classi(cid:12)ed as long-term. The regressions are performed each month and the coe(cid:14)cients are pooled over the entire sample. The t-statistic less than 5 years are classi(cid:12)ed as short-term; maturities between 5 and 10 years are classi(cid:12)ed as medium-term; and maturities longer than 10 (cid:12)rm-speci(cid:12)c distress factor,X‘,is either leverage, book-to-market,or pro(cid:12)tability. (cid:28)‘ is the maturityofthe bond (inyears). Allbonds with maturity All regression results are based on the cross-sectional regression: Y‘(t) = (cid:5)0(t) + (cid:5)1(t) (cid:28)‘(t) + (cid:5)2(t) X‘(t), ‘ = 1;(cid:1)(cid:1)(cid:1);L, and t = 1;(cid:1)(cid:1)(cid:1);T. The Table 2: Cross-Sectional Analysis of Corporate Yields
39 CV 0.236 0.223 0.618 0.337 0.218 0.283 -0.110 0.204 0.276 0.717 Subsample SE 0.009 0.003 0.005 0.001 0.005 0.001 0.003 f2.92g f20.23g 0.002 0.014 0.006 f3.32g f20.93g 94:01-98:03 Mean 0.294 0.080 0.054 0.027 0.168 0.019 -0.206 0.51 6.79 0.087 0.351 0.061 1.45 10.51 CV 0.100 0.226 0.424 0.435 0.140 0.269 -0.088 0.168 0.154 0.438 Subsample SE 0.009 0.001 0.002 0.002 0.003 0.001 0.002 f2.73g f20.26g 0.002 0.008 0.005 f3.40g f28.01g 89:03-93:12 Mean 0.306 0.090 0.043 0.036 0.149 0.020 -0.206 0.50 6.72 0.108 0.403 0.091 1.80 18.78 CV 0.178 0.225 0.556 0.436 0.195 0.276 -0.098 0.211 0.222 0.574 Sample SE 0.006 0.001 0.003 0.001 0.003 0.001 0.002 f2.82g f20.82g 0.002 0.008 0.004 f3.36g f24.70g Full Mean 0.301 0.085 0.048 0.032 0.158 0.019 -0.206 0.51 6.75 0.098 0.379 0.077 1.64 14.91 (%) (bp) (%) (bp) (cid:20)r z (cid:27)r (cid:20)z (cid:22)z (cid:27)z (cid:26)r;z RMSE BYE z (cid:20)r (cid:27)r RMSE BYE of Interest Rates of Interest Rates Two Factor Model One Factor Model montht-1 treasury STRIPS. There are a total of 20,173treasury strips (about 135 STRIPS per month) brackets). To be consistent with the existing literature, the month t out-of-sample errors are computed using structural parameters estimated from Second, BYE denotes absolute yield errors (in basis points, bp). In-sample errors are (cid:12)rst shown followed by the out-of-sample errors (in curly CV). Two type of pricing errors are recorded. First, RMSE denotes the square-root of the sum of squared percentage pricing errors (in %). (cid:12) (cid:12) corresponding model bond price. We P report(cid:12)the average p(cid:12)arameter value, the standard error (denoted SE) and the coe(cid:14)cient of variation (denoted (cid:12) (cid:12) pricing errors: RMSE(t) (cid:17) min r L j=1 B(t;(cid:28)‘) , t = 1;(cid:1)(cid:1)(cid:1);T, where B(t;(cid:28)) is the market price of treasury STRIPS and B(t;(cid:28)) is the 1 L B(t;(cid:28)‘) (cid:0)B(t;(cid:28)‘) 2 totreasurySTRIPS(source: LehmanBrothers FixedIncomeDatabase). Theestimationprocedure involvesminimizingthesumofsquaredpercentage (cid:26)r;z (cid:17)Covt(!r(t);!z(t)). The one-factor interestrate modelis described by dr(t)=(cid:20)r[z(cid:0)r(t)]dt + (cid:27)rd!r(t). All modelbond prices are (cid:12)tted is described by dr(t) = (cid:20)r[z(t) (cid:0) r(t)]dt + (cid:27)rd!r(t), and dz(t) = (cid:20)z[(cid:22)z (cid:0) z(t)]dt + (cid:27)zd!z(t). z(t) is the stochastic mean interest rate and This table reports the parameter estimates of the one-factor and the two-factor model of interest rates. The two-factor interest rate model Interest Rates Table 3: Estimation of the Treasury Parameters and Performance of One-Factor Versus Two-Factor Models of
40 CV 0.207 0.053 0.956 0.043 0.024 0.045 -0.066 0.190 0.053 0.865 0.065 0.017 0.012 -0.022 Industrial Mean 0.020 0.798 0.004 0.457 0.266 0.123 -0.103 1.09 1.36 0.020 0.802 0.004 0.460 0.267 0.119 -0.100 1.09 1.36 CV 0.206 0.027 1.003 0.025 0.022 0.049 -0.151 0.199 0.022 0.542 0.107 0.017 0.080 -0.135 Utility Mean 0.018 0.815 0.002 0.495 0.266 0.121 -0.109 1.00 1.22 0.018 0.821 0.001 0.521 0.267 0.124 -0.106 1.00 1.22 CV 0.275 0.069 0.808 0.052 0.017 0.022 -0.022 0.269 0.104 0.603 0.033 0.015 0.001 -0.002 Financial Mean 0.019 0.835 0.002 0.426 0.267 0.120 -0.101 1.01 1.24 0.019 0.847 0.001 0.427 0.270 0.120 -0.100 1.03 1.27 CV 0.198 0.076 0.944 0.050 0.038 0.065 -0.140 0.180 0.060 1.013 0.076 0.025 0.051 -0.081 Long Mean 0.021 0.779 0.006 0.472 0.262 0.124 -0.109 1.34 1.67 0.021 0.789 0.005 0.498 0.265 0.121 -0.104 1.35 1.67 CV 0.222 0.045 0.715 0.048 0.011 0.015 -0.017 0.235 0.056 0.794 0.049 0.009 0.005 -0.002 Medium Mean 0.018 0.830 0.002 0.459 0.269 0.120 -0.100 0.93 1.13 0.018 0.834 0.001 0.457 0.270 0.120 -0.100 0.94 1.14 CV 0.325 0.052 0.214 0.107 0.011 0.014 -0.018 0.273 0.069 0.695 0.170 0.008 0.001 -0.001 Short Mean 0.019 0.839 0.001 0.418 0.269 0.120 -0.100 0.74 0.93 0.019 0.843 0.001 0.403 0.269 0.120 -0.100 0.73 0.91 CV 0.241 0.052 1.324 0.030 0.018 0.029 -0.031 0.227 0.081 1.120 0.053 0.025 0.015 -0.033 BBB Mean 0.023 0.829 0.003 0.479 0.268 0.121 -0.101 1.16 1.47 0.022 0.835 0.004 0.487 0.267 0.119 -0.101 1.15 1.46 CV 0.222 0.050 0.906 0.040 0.023 0.047 -0.068 0.209 0.047 0.811 0.060 0.012 0.010 -0.007 A Mean 0.019 0.810 0.003 0.445 0.266 0.122 -0.103 1.01 1.23 0.019 0.814 0.002 0.443 0.269 0.119 -0.099 1.01 1.24 CV 0.278 0.064 1.125 0.058 0.056 0.091 -0.233 0.258 0.057 0.642 0.189 0.030 0.125 -0.213 AA Mean 0.018 0.782 0.006 0.464 0.264 0.122 -0.115 1.10 1.36 0.017 0.802 0.002 0.502 0.266 0.127 -0.110 1.11 1.37 (cid:3)0 (cid:3)r (cid:3)x (cid:22)x (cid:20)x (cid:27)x (cid:26)r;x APE RMSE (cid:3)0 (cid:3)r (cid:3)x (cid:22)x (cid:20)x (cid:27)x (cid:26)r;x APE RMSE R(t)=(cid:3)0+(cid:3)rr(t)+(cid:3)xLev(t) R(t)=(cid:3)0+(cid:3)rr(t)+(cid:3)xB(t)/M(t) Panel A: Leverage Ratio Model Panel B: Book-to-Market Ratio Model in-sample absolute percentage pricing error and RMSE denotes the in-sample root mean-squared percentage pricing error. variation. To conserve on space, only the estimation results from the leverage ratio model and the book-to-market ratio model are shown. APE denotes the errors between the market bond price and the model-determined bond price (for each (cid:12)rm). Mean is the average parameter value and CV is the coe(cid:14)cient of only once each quarter (and conditional on the estimated treasury parameters). Each model is estimated by minimizing the sum of squared percentage pricing X(t)surrogates some (cid:12)rm-speci(cid:12)c distress factor. Asthe number ofstructural parameters is relatively large compared to available bonds, the estimation is done All estimation results rely on the following speci(cid:12)cation of the defaultable discount rate: R(t)=(cid:3)0+(cid:3)rr(t)+(cid:3)xX(t), where r(t) is the spot interest rate and Table 4: Parameter Estimates of Credit Risk Models
41 Interest Rate 0.83 25.97 1.54 21.80 1.94 17.83 1.44 21.98 Stock Price 0.77 24.23 1.57 22.33 1.82 17.09 1.40 21.58 Lagged Spread 0.82 26.03 1.69 23.69 1.98 18.13 1.51 22.63 BBB Pro(cid:12)tability 0.78 24.70 1.64 23.14 1.86 17.58 1.44 21.88 B/M 0.81 25.88 1.68 23.50 1.94 17.90 1.49 22.44 Leverage 0.81 25.79 1.68 23.57 1.96 18.02 1.50 22.48 Interest Rate 0.75 26.01 1.35 19.37 1.87 16.84 1.29 21.84 Stock Price 0.73 23.59 1.28 18.81 1.70 15.71 1.21 20.28 Lagged Spread 0.76 25.82 1.33 19.17 1.84 16.42 1.29 21.48 A Pro(cid:12)tability 0.72 24.72 1.31 19.18 1.72 15.80 1.22 20.97 B/M 0.74 25.64 1.31 19.14 1.79 16.31 1.27 21.33 Leverage 0.74 25.55 1.31 19.06 1.80 16.29 1.26 21.27 Interest Rate 0.73 24.15 1.29 18.62 1.89 17.02 1.31 20.09 Stock Price 0.67 23.32 1.24 18.35 1.77 16.95 1.22 19.54 Lagged Spread 0.73 23.62 1.29 18.79 1.90 17.41 1.31 19.80 AA Pro(cid:12)tability 0.88 25.56 1.41 19.97 1.93 17.31 1.39 20.80 B/M 0.74 24.18 1.37 19.34 1.92 17.49 1.35 20.31 Leverage 0.72 24.74 1.39 19.51 1.90 17.29 1.35 20.46 Interest Rate 0.77 25.60 1.40 19.88 1.91 17.31 1.34 21.62 Stock Price 0.73 23.62 1.35 19.40 1.76 16.33 1.26 20.52 Lagged Spread 0.77 25.44 1.42 20.10 1.89 17.00 1.35 21.57 All Pro(cid:12)tability 0.76 24.60 1.42 20.25 1.80 16.54 1.31 21.17 B/M 0.75 25.36 1.41 20.11 1.86 16.87 1.33 21.50 Leverage 0.75 25.26 1.42 20.05 1.86 16.85 1.33 21.46 RMSE BYE RMSE BYE RMSE BYE RMSE BYE Rating Model Short Medium Long All through 98:03. estimation procedure is explained in Tables 3 and 4). RMSE is expressed in % and BYE is in basis points. The sample period is 89:03 and "Interest Rate" is the interest rate only model. All model determined prices are based on implied structural parameters (the "Pro(cid:12)tability" is the pro(cid:12)tability model, \Lagged Spread" is the lagged yield spread model, "Stock Price" is the stock price model, Second, BYE is the absolute yield basis point error. \Leverage"is the leverage ratiomodel, "B/M"is the book-to-marketratiomodel, Two types of in-sample valuation errors are reported in this table. First, RMSE is the root-mean square percentage pricing error. Table 5: In-Sample Valuation Errors of Credit Risk Models
42 Interest Rate 0.88 42.62 1.95 31.76 2.64 27.69 1.68 31.48 Stock Price 1.13 38.63 3.03 37.27 3.34 33.20 2.31 36.80 Lagged Spread 0.88 34.23 2.26 32.52 3.02 30.12 1.78 31.74 BBB Pro(cid:12)tability 0.88 34.35 2.27 32.97 2.96 29.30 1.78 31.85 B/M 0.86 34.77 2.21 33.46 3.02 30.86 1.76 32.48 Leverage 0.86 34.20 2.24 32.76 2.98 30.26 1.76 31.87 Interest Rate 0.69 30.91 1.51 24.67 2.27 24.49 1.30 27.38 Stock Price 1.21 38.68 2.00 31.20 2.59 26.49 1.81 33.51 Lagged Spread 0.68 30.13 1.46 23.92 2.09 22.28 1.27 26.39 A Pro(cid:12)tability 0.70 30.26 1.53 24.19 2.01 21.64 1.26 26.31 B/M 0.66 29.56 1.44 23.55 2.04 21.76 1.24 25.80 Leverage 0.67 30.33 1.47 24.70 2.08 22.10 1.26 26.67 Interest Rate 0.62 26.44 1.43 24.29 2.49 24.21 1.41 24.63 Stock Price 0.91 32.58 1.65 26.90 2.74 26.36 1.67 29.05 Lagged Spread 0.63 26.60 1.34 22.81 2.43 23.56 1.38 24.05 AA Pro(cid:12)tability 0.70 27.75 1.48 23.42 2.51 23.05 1.50 24.80 B/M 0.62 26.47 1.35 23.25 2.38 23.52 1.36 24.26 Leverage 0.62 26.15 1.33 22.91 2.35 23.25 1.35 23.97 Interest Rate 0.73 31.22 1.62 26.57 2.44 25.94 1.42 28.07 Stock Price 1.13 37.31 2.19 31.92 2.78 28.20 1.88 33.41 Lagged Spread 0.72 30.57 1.64 26.08 2.30 23.83 1.41 27.46 All Pro(cid:12)tability 0.74 30.92 1.71 26.49 2.29 23.49 1.43 27.62 B/M 0.70 30.35 1.61 26.17 2.26 23.90 1.39 27.40 Leverage 0.71 30.63 1.63 26.50 2.26 23.76 1.39 27.60 RMSE BYE RMSE BYE RMSE BYE RMSE BYE Rating Model Short Medium Long All market price, averaged over all bonds. BYE is the absolute yield basis point error. The sample period is 89:03 through 98:03. 3 and 4). APE, the absolute percentage pricing error, is the absolute di(cid:11)erence between the market and the model price divided by the rateonlymodel. All model determined prices arebasedonimplied structuralparameters(theestimation procedure is explained in Tables model, \Lagged Spread"is thelaggedyield spread model, "Stock Price"isthe stockprice model, and"Interest Rate" is theinterest risk models. \Leverage" is the leverage ratio model, "B/M" is the book-to-market ratio model, "Pro(cid:12)tability" is the pro(cid:12)tability The pricing errors are consolidated by credit rating and maturity. We have reported the out-of-sample pricing performance of six default Table 6: Out-of-Sample Pricing and Yield Performance of Credit Risk Models (By Rating and Maturity)
43 Interest Rate 0.74 30.60 1.41 25.21 2.54 25.65 1.52 27.19 Stock Price 1.09 35.60 1.75 29.63 2.86 27.67 1.92 31.80 Lagged Spread 0.71 29.65 1.36 24.25 2.34 23.50 1.48 26.17 Industrial Pro(cid:12)tability 0.70 29.67 1.28 23.24 2.31 23.29 1.44 25.91 B/M 0.70 29.36 1.35 24.42 2.32 23.71 1.46 26.18 Leverage 0.71 29.60 1.37 24.60 2.33 23.60 1.48 26.28 Interest Rate 0.62 26.05 1.34 22.98 2.33 24.65 1.26 24.46 Stock Price 0.83 32.01 1.74 28.67 2.64 28.39 1.59 30.03 Lagged Spread 0.64 26.15 1.32 22.43 2.32 24.36 1.24 24.16 Utility Pro(cid:12)tability 0.71 26.41 1.39 22.70 2.31 22.23 1.29 24.09 B/M 0.62 25.60 1.27 21.91 2.22 23.91 1.20 23.67 Leverage 0.62 25.40 1.27 21.79 2.11 22.97 1.18 23.41 Interest Rate 0.72 32.75 1.70 27.27 2.20 25.57 1.28 29.03 Stock Price 1.23 40.16 2.42 33.16 2.59 29.10 1.85 35.39 Lagged Spread 0.72 32.13 1.75 26.81 2.14 24.68 1.28 28.49 Financial Pro(cid:12)tability 0.73 32.21 1.77 27.25 2.14 24.49 1.30 28.82 B/M 0.70 32.02 1.72 27.05 2.11 24.50 1.26 28.48 Leverage 0.71 32.73 1.74 27.92 2.14 24.81 1.27 29.23 RMSE BYE RMSE BYE RMSE BYE RMSE BYE Industry Model Short Medium Long All Panel A: Pricing Errors by Industry and Maturity as described in Tables 1 and 2. The sample period is 89:03through 98:03. Panel B present the out-of-sampleAPE and BYE for 25 individual(cid:12)rms. market price, averaged over allbonds. BYE is the absolute yield basis point error. Industry and maturityclassi(cid:12)cation used here are the sameones in Tables 3 and 4). APE, the absolute percentage pricing error, is the absolute di(cid:11)erence between the market and the model price divided by the Rate"is theinterest rate onlymodel. Allmodeldetermined prices are based onimpliedstructural parameters(the estimationprocedure isexplained "Pro(cid:12)tability"isthepro(cid:12)tabilitymodel,\LaggedSpread"isthelaggedyieldspreadmodel,"StockPrice"isthestockpricemodel,and"Interest out-of-sample pricing performance of six default risk models. \Leverage" is the leverage ratio model, "B/M" is the book-to-market ratio model, In Panel A and Panel B, the pricing errors are consolidated by industry and maturity and individual (cid:12)rms, respectively. We have reported the Table 7: Out-of-Sample Pricing and Yield Performance of Credit Risk Models (By Industry and Individual Firms)
44 WalmartStore 1.20 20.96 1.21 21.25 1.13 19.28 1.18 20.40 1.98 32.27 1.17 20.21 TimeWarner 2.46 39.90 2.67 36.67 2.49 42.58 1.72 37.75 3.14 42.83 2.46 42.84 Niagara M. P. 1.97 34.97 1.84 32.87 2.02 35.11 1.92 34.23 2.56 46.17 1.78 31.78 Usx 2.23 32.06 2.20 31.75 2.24 32.36 2.24 32.32 2.54 34.71 2.22 31.99 Philip Morris 0.79 20.58 0.79 20.60 0.80 20.88 0.80 21.44 1.10 27.20 0.82 21.81 Merrill Lynch 0.84 26.04 0.84 25.86 0.84 25.97 1.18 34.93 1.38 34.14 0.82 26.62 Int’l Paper 1.29 21.73 1.33 22.71 1.33 22.76 1.31 22.61 1.51 25.64 1.37 23.34 Ibm 2.12 27.99 2.20 28.09 2.11 27.96 2.13 27.55 2.23 32.50 2.11 27.83 First Union 1.11 22.56 1.11 22.53 1.11 22.67 1.10 22.65 1.27 25.73 1.14 23.42 First Chicago 1.67 31.84 2.02 37.27 1.67 31.78 2.01 36.74 2.32 38.99 1.74 32.08 Ford Motor 1.96 25.74 2.02 26.74 2.03 26.82 2.21 29.14 2.07 30.35 1.91 24.84 Enron Corp 1.19 21.19 1.20 21.22 1.21 21.59 1.27 22.89 2.01 32.27 1.20 22.39 Edison Int’l 0.86 22.50 0.82 21.65 0.86 22.64 0.86 22.68 1.18 31.34 0.68 21.99 Edison N.Y. 0.92 20.50 0.91 20.27 0.91 20.35 0.99 22.62 1.23 28.63 0.92 20.62 Delta Air. 2.35 35.75 2.37 36.14 2.34 32.29 2.36 35.91 2.14 30.26 2.42 36.81 Csx Corp 1.32 21.74 1.32 21.75 1.31 21.46 1.35 22.03 1.70 28.44 1.34 22.00 Chase Manhat 1.97 38.24 1.95 39.30 2.04 36.99 2.00 36.68 1.93 45.94 1.94 37.93 Coastal Corp 1.55 25.03 1.62 26.70 1.62 26.42 1.71 28.51 1.96 31.62 1.79 29.37 Citicorp 1.38 33.80 1.39 42.07 1.42 34.98 1.37 33.97 1.37 34.50 1.47 34.38 Bankers Trust 1.15 25.26 1.14 25.17 1.09 24.89 1.20 26.59 1.46 28.92 1.15 25.57 Bear Stern 1.10 22.82 1.08 22.45 1.03 21.27 1.00 20.23 1.93 38.70 1.09 22.71 Bell South 3.49 42.24 4.28 52.40 5.17 62.85 5.63 45.74 3.01 37.57 5.15 62.78 Bank America 1.29 24.41 1.24 23.76 1.31 24.94 1.18 22.87 1.35 25.39 1.35 25.63 Boeing Co. 2.73 26.90 2.57 26.44 2.60 26.66 2.76 27.85 4.19 35.12 2.69 27.93 Amr Corp 2.08 34.47 2.11 34.83 2.11 34.81 2.11 34.87 2.32 32.03 2.08 34.82 APE BYE APE BYE APE BYE APE BYE APE BYE APE BYE Leverage B/M Ratio Lagged Spread Pro(cid:12)tability Stock Price Interest Rate Panel B of Table 7: Pricing Errors for IndividualFirms
45 Interest Rate 34.61 27.67 28.12 29.81 28.98 26.86 24.29 25.87 45.27 45.86 45.52 45.43 Stock Price 34.61 32.27 32.32 32.98 31.48 31.36 28.96 30.63 41.39 42.51 47.36 44.40 Lagged Spread 33.58 27.35 28.09 29.37 28.16 28.72 27.53 26.05 45.64 44.75 45.07 45.51 BBB Pro(cid:12)tability 33.75 27.88 26.85 29.19 28.74 28.74 26.69 26.39 42.09 44.51 45.66 43.61 B/M 34.05 27.90 28.56 29.85 28.93 29.11 27.65 26.50 43.25 44.03 45.12 44.29 Leverage 33.65 27.85 28.02 29.56 27.90 28.79 27.36 25.95 44.27 43.74 44.48 44.77 Interest Rate 25.14 21.04 25.34 23.55 28.05 22.54 23.02 24.36 43.53 44.32 43.27 43.79 Stock Price 31.86 26.53 28.64 28.87 33.68 26.78 23.12 27.65 39.37 41.21 46.44 41.73 Lagged Spread 24.96 20.80 25.06 23.31 27.40 22.03 20.56 23.22 40.34 43.63 41.96 41.73 A Pro(cid:12)tability 24.65 20.65 24.27 22.87 27.79 21.27 19.84 23.05 38.21 41.28 41.15 39.69 B/M 24.48 20.67 24.26 22.86 26.80 21.67 20.14 22.74 39.62 41.98 41.97 40.84 Leverage 24.78 20.73 24.77 23.14 27.18 22.21 20.18 23.14 40.06 42.54 42.33 41.12 Interest Rate 25.53 25.27 28.19 26.36 23.63 21.62 21.23 21.20 34.11 40.64 44.27 38.36 Stock Price 30.54 27.80 27.73 28.58 29.11 23.51 23.44 25.49 35.18 41.38 41.07 39.55 Lagged Spread 25.26 25.24 28.01 26.21 23.93 19.73 20.58 20.56 33.66 39.16 44.13 37.57 AA Pro(cid:12)tability 27.23 24.10 26.78 26.02 25.15 21.48 20.66 21.99 35.05 39.61 43.52 38.65 B/M 25.26 24.12 27.39 25.56 24.15 20.91 20.48 21.07 33.35 40.17 44.02 38.39 Leverage 24.61 22.96 27.21 24.92 23.68 21.00 20.85 21.26 34.27 39.26 42.39 38.34 Interest Rate 27.73 23.62 26.74 25.81 27.26 23.22 23.10 23.98 41.48 43.38 43.82 42.66 Stock Price 32.44 28.32 29.80 30.06 31.13 26.59 24.13 27.29 38.29 40.68 46.03 40.89 Lagged Spread 27.26 23.36 26.52 25.48 26.63 22.78 20.80 23.48 39.74 42.55 42.91 41.38 All Pro(cid:12)tability 27.54 23.25 25.63 25.27 27.07 22.94 20.52 23.56 38.65 41.69 42.09 40.45 B/M 27.21 23.37 26.32 25.41 26.51 22.78 20.75 23.40 39.03 41.85 42.62 40.92 Leverage 27.12 23.10 26.23 25.27 26.44 22.94 20.90 23.42 39.78 42.01 42.52 41.19 Short Med. Long All Short Med. Long All Short Med. Long All Basis Point Errors Constant Parameters Rating Model BYE, 1994-1999 Subsample Median Yield BYE, absolute yield basis point error averaged over all bonds. median yield error across (cid:12)rms at each date. Next, we time-average the monthly medians to get the averaged median. As before, BYE is the mean determined prices are based on impliedstructural parameters. The medianabsolute yieldbasis error is computedintwosteps. First, wecomputethe Spread"is the laggedyield spread model,"Stock Price"is the stock price model,and "InterestRate" is the interest rate only model. Allmodel maturity. \Leverage" is the leverage ratio model, "B/M" is the book-to-market ratio model, "Pro(cid:12)tability"is the pro(cid:12)tability model, \Lagged mean yield basis point errors, when the parameters are kept constant over the entire sample. All pricing errors are consolidated by credit rating and Wereportthe(i)the(out-of-sample)yieldbasispointerrorsfromthe1994:03-1998:03sub-sample,(ii)themedianyieldbasispointerrors,and(iii)the Table 8: Robustness Issues (Median and Subsample Analysis of Yield Basis Points Errors)
47 Interest Rate 1.02 0.53 1.65 0.52 2.26 0.59 1.55 0.57 Stock Price 0.99 0.47 1.67 0.31 2.22 0.58 1.47 0.48 BBB B/M 1.10 0.61 1.91 0.56 2.41 0.66 1.63 0.61 Leverage 1.03 0.53 1.73 0.35 2.33 0.68 1.55 0.54 Interest Rate 0.98 0.55 1.51 0.55 1.64 0.40 1.42 0.40 Stock Price 0.95 0.58 1.40 0.57 1.77 0.47 1.32 0.55 A B/M 1.04 0.53 1.58 0.53 1.70 0.50 1.48 0.58 Leverage 0.98 0.51 1.49 0.52 1.67 0.53 1.40 0.56 Interest Rate 1.12 0.63 1.89 0.71 2.36 0.70 1.75 0.73 Stock Price 1.01 0.54 1.75 0.67 2.31 0.54 1.62 0.61 AA B/M 1.13 0.59 1.85 0.61 2.49 0.51 1.74 0.61 Leverage 1.10 0.61 1.79 0.60 2.40 0.45 1.68 0.60 Interest Rate 1.03 0.58 1.78 0.69 2.08 0.61 1.54 0.65 Stock Price 0.99 0.54 1.57 0.60 2.03 0.51 1.43 0.57 All B/M 1.12 0.64 1.91 0.78 2.12 0.52 1.61 0.67 Leverage 1.05 0.61 1.72 0.64 2.07 0.54 1.53 0.62 AHE MHE AHE MHE AHE MHE AHE MHE Rating Model Short Medium Long All Panel A: Delta-Neutral Hedging Errors Across Credit Rating and Maturity ratio model,(iii) the stock price model,and (iv) the interest rate only model. The sampleperiod is 03:89 through 03:98. in the replicating portfolio net of the short position in the target. Four credit risk models are considered: (i) the leverage ratio model, (ii) the B/M computed using implied structural parameters. All hedges are liquidated in the following30 days. The hedging error is de(cid:12)ned as the long position hedge, as manyinstruments as sources of risks are employedto create a delta-neutral hedge in all dimensions. The positioning in each instrument is MHE is the average mean percentage hedging error. The target hedge is a short position in the corporate coupon bond. In constructing the Thistableevaluates the dynamicperformance ofcredit riskmodels. Foreach model,AHE isthe averageabsolutepercentage hedgingerrorand Table 10: Hedging E(cid:11)ectiveness of Credit Risk Models
48 Walmart Store 0.60 0.34 0.53 0.37 0.68 0.29 0.52 0.45 Time Warner 1.10 0.15 1.04 0.25 2.06 -0.55 0.99 0.27 Niagara M. P. 1.49 0.32 1.74 0.41 1.74 -0.11 1.30 0.41 Usx 2.11 0.88 2.17 0.85 2.10 0.97 2.22 0.93 Philip Morris 1.38 0.69 1.39 0.69 1.39 0.78 1.40 0.66 Merrill Lynch 1.16 0.63 1.27 0.70 1.14 0.75 1.39 0.78 Int’l Paper 0.66 0.55 0.78 0.53 0.78 0.56 0.66 0.51 Ibm 2.05 1.30 2.13 0.42 1.92 0.85 1.46 1.15 First Union 1.92 0.64 2.06 0.56 1.75 0.83 1.99 0.57 First Chicago 1.96 0.38 2.06 0.47 0.95 0.37 0.87 0.37 Ford Motor 0.74 0.19 0.84 0.16 1.59 0.09 0.73 0.27 Enron Corp 1.85 0.71 1.88 0.72 1.63 0.69 1.53 0.63 Edison Int’l 0.96 0.12 1.09 0.24 0.82 0.10 0.86 0.29 Edison N.Y. 1.68 0.86 1.70 0.89 1.59 0.81 1.68 0.86 Delta Air. 1.92 0.80 2.05 0.81 2.06 0.79 2.00 0.78 Csx Corp 1.75 0.76 1.83 0.80 1.77 0.68 1.90 0.64 Chase Manhat 1.90 0.55 2.03 0.84 2.12 0.88 2.03 0.45 Coastal Corp 1.54 0.23 1.63 0.19 1.32 0.47 1.51 0.22 Citicorp 2.28 0.07 2.24 -0.18 1.99 0.15 2.36 0.11 Bankers Trust 1.56 0.78 1.61 0.81 1.61 0.82 1.60 0.78 Bear Stern 1.32 0.91 1.44 0.00 1.36 0.96 1.34 0.92 Bell South 2.09 0.30 2.51 0.01 1.79 0.78 1.65 1.35 Bank America 2.21 0.33 2.55 0.51 2.99 0.41 2.37 0.62 Boeing Co. 1.57 0.10 1.58 -0.08 1.49 0.05 1.68 0.06 Amr Corp 1.92 0.74 2.00 0.70 2.11 0.53 1.97 0.80 AHE MHE AHE MHE AHE MHE AHE MHE Leverage B/M Ratio Stock Price Interest Rate Panel B of Table 10: Delta-Neutral Hedging Errors Across Individual Firms
2 0 −2 −4 −6 −8 1 2 3 4 5 6 srorrE tnioP sisaB dleiY Mean Errors from All Bonds 0 −1 −2 −3 −4 −5 −6 1 2 3 4 5 6 srorrE tnioP sisaB dleiY Mean Errors from AA−Rated Bonds 2 0 −2 −4 −6 −8 1 2 3 4 5 6 srorrE tnioP sisaB dleiY Mean Errors from A−Rated Bonds 2 0 −2 −4 −6 −8 −10 1 2 3 4 5 6 srorrE tnioP sisaB dleiY Mean Errors from BBB−Rated Bonds
0 −5 −10 −15 −20 1 2 3 4 5 6 srorrE tnioP sisaB dleiY Mean Errors for Short−Term Bonds 3 2.5 2 1.5 1 0.5 0 1 2 3 4 5 6 srorrE tnioP sisaB dleiY Mean Errors for Medium−Term Bonds 1 0.5 0 −0.5 −1 −1.5 1 2 3 4 5 6 srorrE tnioP sisaB dleiY Mean Errors for Long−Term Bonds
T M p m d b w c T M p m d b w c F i g u r e 1 h i s g r a p h d o d e l 1 i s r o (cid:12) t a b i l i t y o d e l ; a n d i s c r e p a n c y y s o l v i n g t Y t ; (cid:28) h e r e ( o u p o n b o n d F i g u r h i s g r a p h d o d e l 1 i s r o (cid:12) t a b i l i t y o d e l ; a n d i s c r e p a n c y y s o l v i n g t Y t ; (cid:28) h e r e ( o u p o n b o n d : O u t - o f - S a m p l e M e a n E r r o r s b y C r e d i t R a t i n g ( i n B a s i s P o i n t m e a n y i e l d b a s i s p o i n t e r r o r s i s p l a y s t h e o u t - o f - s a m p l e f o r c r e d i t r i s k m M o d e l 2 M o d e l 3 t h e l e v e r a g e r a t i o m o d e l ; i s t h e B / M r a t i o m o d e l ; M o d e l 4 M o d e l 5 m o d e l ; i s t h e l a g g e d y i e l d s p r e a d m o d e l ; i s t h e s t o c k M o d e l 6 i s t h e i n t e r e s t o n l y m o d e l . T h e m e a n y i e l d b a s i s p o i n t e r r o r b e t w e e n t h e m a r k e t y i e l d a n d t h e m o d e l d e t e r m i n e d y i e l d . Y i e l d s a r e i n t (cid:28) + (cid:0) (cid:0) P t ; (cid:28) R c Y t ; (cid:28) u d u F Y t t h e f o l l o w i n g e q u a t i o n : ( ) = e x p [ ( ) ] + e x p [ ( P t ; (cid:28) ) i s t h e y i e l d - t o - m a t u r i t y a n d ( ) i s e i t h e r t h e m a r k e t o r m o d e l - d e t e r c r e d i t r a t i n g p r i c e s . T h e r e s u l t s a r e s h o w n b y . e 2 : O u t - o f - S a m p l e M e a n E r r o r s b y M a t u r i t y ( i n B a s i s P o i n t s ) m e a n y i e l d b a s i s p o i n t e r r o r s i s p l a y s t h e o u t - o f - s a m p l e f o r c r e d i t r i s k m M o d e l 2 M o d e l 3 t h e l e v e r a g e r a t i o m o d e l ; i s t h e B / M r a t i o m o d e l ; M o d e l 4 M o d e l 5 m o d e l ; i s t h e l a g g e d y i e l d s p r e a d m o d e l ; i s t h e s t o c k M o d e l 6 i s t h e i n t e r e s t o n l y m o d e l . T h e m e a n y i e l d b a s i s p o i n t e r r o r b e t w e e n t h e m a r k e t y i e l d a n d t h e m o d e l d e t e r m i n e d y i e l d . Y i e l d s a r e i n t (cid:28) + (cid:0) (cid:0) P t ; (cid:28) c Y t ; (cid:28) u d u F Y t R h e f o l l o w i n g e q u a t i o n : ( ) = t e x p [ ( ) ] + e x p [ ( P t ; (cid:28) ) i s t h e y i e l d - t o - m a t u r i t y a n d ( ) i s e i t h e r t h e m a r k e t o r m o d e l - d e t e r b o n d m a t u r i t y p r i c e s . T h e r e s u l t s a r e s h o w n b y . 1 s ) o d e l s . i s t h e p r i c e i s t h e v e r t e d ; (cid:28) (cid:28) ) ] , m i n e d o d e l s . i s t h e p r i c e i s t h e v e r t e d ; (cid:28) (cid:28) ) ] , m i n e d
Cite this document
Gurdip Bakshi, Dilip Madan, & and Frank Zhang (2001). Investigating the Sources of Default Risk: Lessons from Empirically Evaluating Credit Risk Models (FEDS 2001-15). Board of Governors of the Federal Reserve System, Finance and Economics Discussion Series. https://whenthefedspeaks.com/doc/feds_2001-15
@techreport{wtfs_feds_2001_15,
author = {Gurdip Bakshi and Dilip Madan and and Frank Zhang},
title = {Investigating the Sources of Default Risk: Lessons from Empirically Evaluating Credit Risk Models},
type = {Finance and Economics Discussion Series},
number = {2001-15},
institution = {Board of Governors of the Federal Reserve System},
year = {2001},
url = {https://whenthefedspeaks.com/doc/feds_2001-15},
abstract = {From a credit risk perspective, little is known about the distress factors -- economy-wide or firm-specific -- that are important in explaining variations in defaultable coupon yields. This paper proposes and empirically tests a family of credit risk models. Empirically, we find that firm-specific distress factors play a role (beyond treasuries) in explaining defaultable coupon bond yields. Credit risk models that take into consideration leverage and book-to-market are found to reduce out-of-sample yield fitting errors (for the majority of firms). Moreover, the empirical evidence suggests that interest rate risk may be of first-order prominence for pricing and hedging. Measured by both out-of-sample pricing and hedging errors, the credit risk models perform relatively better for high grade bonds. Controlling for credit rating, the model performance is generally superior for longer maturity bonds compared to its shorter maturity counterparts. Using equity as an instrument reduces hedging errors. This paper provides an empirical investigation of credit risk models using observable economic factors.},
}