Vol. LV (2014) PL ISSN 0071-674X
A NEW ICAPM APPROACH TO MULTIFACTOR STOCK
PRICING USING BOOTSTRAP
S T A N I S Ł A W U R B A Ń S K I
AGH University of Science and Technology, Faculty of Management
e-mail: surbansk@zarz.agh.edu.pl
J A C E K L E Ś K O W
Cracow University of Technology, Institute of Mathematics
e-mail: jleskow@pk.edu.pl
A B S T R A C T
The aim of this w ork is the use of bootstrap m ethods for assessing of returns and risk of stock described by a small-to-moderate time series data. The paper presents the possibility of using bootstrap for testing the selected ICAPM application. We estimate systematic risk and risk premium components, depending on the fundam ental risk factors. We compare bootstrap and classical asymptotic GLS results.
The authors analyze quarterly returns of stocks listed on Warsaw Stock Exchange in 1995-2010. The full-sample observations are divided into two separate sub-periods: 1995-2004, the years preceding Poland's accession to the EU, and 2005-2010, the years of Poland's membership in the UE.
The com ponents of risk prem ium change in the second sub-period. Also, w e test the multifactor-efficiency (ME) of the generated portfolios. GRS and asymptotic Wald tests reject ME. However, the bootstrapped Wald test does not reject ME for the tested cases. Using the tested ICAPM application to forming ME portfolios makes it possible to offer a num ber of useful guidelines for portfolio managers.
S T R E S Z C Z E N IE
S. Urbański, J. Leśkow. Nowa aplikacja ICAPM do wieloczynnikowej wyceny akcji z zastosowaniem metod
bootstrap. Folia Oeconomica Cracoviensia 2014, 55: 15-33.
Celem niniejszej pracy jest zastosowanie m etod bootstrap do oszacowania stóp zwrotu i ryzyka akcji opisanych krótkimi szeregami czasowymi. Artykuł prezentuje możliwość zastosowania m etod bootstrap do testowania wybranej aplikacji ICAPM. My szacujemy składowe ryzyka syste matycznego i premii za ryzyko, w zależności od fundam entalnych czynników ryzyka. Porównu jemy wyniki otrzymane metodam i bootstrap i klasyczną uogólnioną m etodą najmniejszych kw a dratów.
Analizie poddajem y kwartalne stopy zw rotu akcji notow anych na Giełdzie Papierów Warto ściowych w Warszawie w latach 1995-2010. Wszystkie obserwacji dzielimy na dw a podokresy: 1995-2004 (okres poprzedzający wejście Polski do Unii Europejskiej) oraz 2005-2010 (okres człon kostwa Polski w Unii Europejskiej). Składowe premii za ryzyko ulegają zmianie w drugim pod- okresie. My testujemy również wieloczynnikową efektywność (ME) generowanych portfeli. Test GRS oraz asymptotyczny test Walda odrzuca ME. Natomiast bootstrapowy test Walda, w żadnym badanym przypadku nie odrzuca ME.
Zastosowanie testowanej aplikacji ICAPM do budow y portfeli wieloczynnikowo efektywnych pozwala na wyciągnięcie wielu użytecznych wskazówek dla zarządzających portfelami inwesty cyjnymi.
KEY WORDS — SŁOWA KLUCZOWE
asset pricing, bootstrap m ethod, return changes, systematic risk, multifactor efficiency wycena aktywów, metoda bootstrap, zmiany stop zwrotu, ryzyko systematyczne,
wieloczynnikowa efektywność
I N T R O D U C T I O N
T e s tin g t h e s t o c k p r i c i n g t h a t c o u l d b e o b s e r v e d i n t h e c o n d i t i o n s o f I C A P M v a l i d i t y c a n b e r e f e r r e d to a n a n a ly s i s o f m u ltif a c to r - e f f ic ie n c y (M E ) o f a g i v e n p o r t fo lio . F o r t h i s p u r p o s e , y o u c a n u s e t h e W a ld s ta tis tic s o f t h e a s y m p t o t i c d i s t r i b u t i o n x 2. W a ld t e s t t e n d s to o v e r - r e je c t t h e M E p o r t f o l i o h y p o t h e s i s ( s e e C h o u a n d Z h o u (2006), p . 221) f o r f in ite s a m p le s . H o w e v e r , a s m a l l - s a m p l e c a s e c a n b e a n a l y z e d w i t h t h e u s e o f t h e G R S t e s t — s e e G i b b o n s e t a l (1989) — o n c o n d i t i o n o f t h e n o r m a l i t y o f t h e s a m p le . T h e r e f o r e , f o r n o n - n o r m a l s m a l l s a m p l e s o n e s h o u l d c o n s i d e r a l t e r n a t i v e s c e n a r i o s lik e t h e b o o t s t r a p m e t h o d . O n e o f t h e p u r p o s e s o f t h i s a r ti c le is to s h o w t h e v a l i d i t y o f s u c h a n a p p r o a c h . T h e W a ld t e s t c a n b e a p p l i e d f o r l a r g e s a m p l e s a n d u n d e r t h e a s s u m p t i o n o f i n d e p e n d e n c e . H o w e v e r , t h e I C A P M a p p l i c a t i o n s i n e m e r g i n g m a r k e t s c a n b e t e s t e d w i t h t h e h e l p o f s a m p l e s o f a s m a l l - t o - m o d e r a t e s iz e f o r w h i c h o n l y ii d c o n d i t i o n s c a n b e a s s u m e d , b u t n o r m a l i t y is u s u a l l y r e je c t e d . O n e n e v e r k n o w s w h a t is t h e t r u e d i s t r i b u t i o n o f t h e r e t u r n s , t h e r e f o r e t h e r e is a n e e d to c o n s i d e r g o o d a p p r o x i m a t i o n s . C o n t e m p o r a r y s ta tis tic a l i n f e r e n c e p r o v i d e s r e s a m p l i n g a n d b o o t s t r a p m e t h o d s to c r e a te c o n f i d e n c e i n t e r v a l s f o r c a s e s o f s m a ll n o n - n o r m a l s a m p le s . R e c e n t r e s e a r c h p r o v i d e s a ls o r e s a m p l i n g t o o l s f o r ti m e - s e r i e s d a t a . F o r m o r e i n f o r m a t i o n , t h e r e a d e r is r e f e r r e d to L e ś k o w e t a l (2008, 2014). C h o u a n d Z h o u (2006) p r e s e n t t h e p o s s i b ilit y o f u s i n g t h e b o o t s t r a p m e t h o d t o t e s t t h e M E o f F a m a - F r e n c h (FF) p o r t f o l i o s a n d t h e p o r t f o l i o r e p r e s e n t i n g t h e C R S P i n d e x f o r t h e U .S . m a r k e t . R e s e a r c h w o r k s o n t e s t i n g t h e c la s s ic C a p i t a l A s s e t P r ic in g M o d e l a n d o t h e r s t u d i e s o n t h e P o lis h m a r k e t a r e p r e s e n t e d , a m o n g o t h e r s , b y O s iń s k a a n d R o m a ń s k i (1994), J a j u g a (2000), B o lt a n d M ił o b ę d z k i (2002), O s i e w a l s k i a n d P i p i e ń (2004), G u r g u l a n d M a j d o s z (2007) a n d Z a r z e c k i e t al. (2 0 0 4 -2 0 0 5 ).
I n t h i s w o r k w e t e s t t h e a p p l i c a t i o n o f t h e IC A P M f o r t h e W a r s a w S to c k E x c h a n g e (W S E ) d a t a i n 1 9 9 5 -2 0 1 0 . T h e a b o v e a p p r o a c h w a s p r o p o s e d b y U r b a ń s k i (2011). W e u s e b o o t s t r a p p r o c e d u r e s i n s t o c k p r i c i n g s i m u l a t e d b y t h e a g g r e g a t e d t h r e e - f a c t o r m o d e l. T h e a i m o f o u r r e s e a r c h is to c o n s i d e r a n a p p r o a c h f o r p r i c i n g o f s to c k s , d e t e r m i n e d b y t h e a s s e s s m e n t o f t h e s y s t e m a tic r i s k a n d r i s k p r e m i u m c o m p o n e n t s . A s a r e s u l t , t h e m u l t i f a c t o r e ff ic ie n c y o f t h e t e s t e d p o r t f o l i o c a n b e e v a l u a t e d . S e c tio n 1 d i s c u s s e s t h e o r e t i c a l m e t h o d s f o r t e s t i n g t h e m u ltif a c to r - e f f ic ie n c y o f a g i v e n p o r tfo lio . S e c tio n 2 p r o p o s e s t h e p o s s ib le u s e o f t h e b o o t s t r a p m e t h o d i n f in a n c e . S e c tio n 3 p r e s e n t s s e v e r a l p r o c e d u r e s f o r d a t a p r e p a r a t i o n i n o r d e r to u s e t h e s t u d i e d a l g o r i th m s . S e c tio n 4 s h o w s t h e r e s u l t s o f c a lc u la tio n s . S e c tio n 5 i n c l u d e s a s u m m a r y a n d c o n c lu s i o n s . 1. M U L T I F A C T O R - E F F I C I E N C Y R E S T R I C T I O N S M u lt if a c t o r a p p l i c a t i o n o f s t o c k p r i c i n g i n l i g h t o f t h e I C A P M c a n b e d e s c r i b e d b y t h e f o l l o w i n g e q u a t i o n : E ( R t ) = f i E ( f, ) , (1) w h e r e
R
t — ( r 1 tri t rN
t)' is N - v e c t o r o f t h e e x c e s s r e t u r n s o v e r t h e r is k - fr e e r a t e o n s t o c k i i n p e r i o d t, P = ( ^ l v . . , , . . . , P N )' a n d f t is t h e k - v e c to r o f f a c to rs . P o r tf o li o s s a t i s f y i n g t h e e q u a t i o n (1) a r e M E . A s t a t i s t i c a l m o d e l t e s t i n g a g e n e r a l f o r m o f t h e I C A P M c a n b e d e s c r i b e d b y t h e r e g r e s s i o n s (2) a n d (3) o f t h e f o l l o w i n g t w o - s t e p p r o c e d u r e : r , = a i + P i f , + e t , , V i = 1 - N ; t = 1 ,- > (2) r,t = T o + Y iP t + £ „ , * = 1, . . . , N ; t = 1, ^ , T (3) w h e r e g 1 is t h e k - v e c to r o f t h e s e c o n d p a s s r e g r e s s i o n p a r a m e t e r s a n d eit a n d e it a r e e r r o r c o m p o n e n t s . H e r e , N is t h e n u m b e r o f a s s e ts , a n d T is t h e n u m b e r o f o b s e r v a tio n s . P r ic in g i n l i g h t o f t h e IC A P M a i m s to e s t i m a t e t h e p a r a m e t e r s o f r e g r e s s i o n s (2) a n d (3), a s w e ll a s to p r o v e t h a t g e n e r a t e d p o r t f o l i o s a r e M E . T h e p r i c i n g r e s t r i c t i o n o f M E p o r t f o l i o s c a n b e f o r m u l a t e d a s t h e h y p o t h e s i s t e s t i n g p r o b l e m : H 0: a - 0 , w h e r e a = ( a 1, . . . , a N ) '.S u c h a n u l l h y p o t h e s i s c a n b e t e s t e d u s i n g t h e a s y m p t o t i c x 2 d i s t r i b u t i o n c o r r e s p o n d i n g t o t h e f o l l o w i n g W a ld s ta tis tic : W - a ' v a r [ a ] 1a , ~ ■ (4) If t h e e r r o r s eit d e f i n e d i n (2) a r e iid , t h e n (4) is o f t h e f o r m ( C o c h r a n e (2001), p p . 2 1 7 -2 1 9 ): W = --- --- ex' t e _1a , (5) 1 + E ( f )' v a r [ f J 1 E ( f t ) e ( ) w h e r e £ e = e e / ( T — k - 1), a n d e is t h e T x N m a t r i x o f r e s id u a ls . I n p r a c t i c e , a p p l y i n g t h e W a ld t e s t o r G R S m e t h o d r e q u i r e s e s t i m a t i n g m a tr ix S e. T h is , i n t u r n , i n d u c e s i m p o s i n g t h e n o r m a l i t y a s s u m p t i o n o n t h e r a n d o m e r r o r t e r m s i n (2) a n d (3) to e n s u r e t h a t t h e s ta tis tic t = 6?. /s e ( 6?.) h a s a t - S t u d e n t d i s t r i b u t i o n . 1 I n r e a lity , h o w e v e r , t h e e x a c t d i s t r i b u t i o n o f t is n o t k n o w n . T h e b o o t s t r a p m e t h o d c a n o v e r c o m e t h i s p r o b l e m . 2. R E S A M P L I N G M E T H O D A P P R O A C H C o n t e m p o r a r y s ta tis tic a l in f e r e n c e p r o v i d e s to o l s to d e a l w i t h s m a ll a n d n o n - n o r m a l s a m p le s . W e a r e n o w a b le to a p p r o x i m a t e t h e f in ite s a m p l e d i s t r i b u t i o n o f t h e e s t i m a t o r s w i t h o u t i n v o k i n g t h e n o r m a l i t y a s s u m p t i o n o r la r g e s a m p l e d i s t r i b u t i o n s . E x te n s iv e s u r v e y s o f b o o t s t r a p m e t h o d s c a n b e f o u n d f o r e x a m p le , i n m o n o g r a p h te x ts b y P o litis (1999) a n d L a h ir i (2003). T im e s e r ie s a p p l i c a t i o n s o f b o o t s t r a p a n d o t h e r r e s a m p l i n g m e t h o d s c a n b e f o u n d e.g. i n L e s k o w (2008, 2014). F o r s m a ll s a m p l e s , m o s t o f t h e r e s a m p l i n g m e t h o d s p r o v i d e m o r e r e lia b le r e s u l t s t h a n t h e n o r m a l a p p r o x i m a t i o n . F o r r e g r e s s i o n - t y p e m o d e l s , w e s t u d y s m a l l - s a m p l e d i s t r i b u t i o n s o f t h e e s t i m a t e s v i a b o o t s t r a p p i n g t h e r e s i d u a l s . I n s u c h a c a s e , t h e m o d e l e r r o r s a r e i i d a n d t h e f a c to r s a r e t r e a t e d a s fix e d c o n s ta n t s . I n t h i s c a s e , t h e f i t t e d r e s i d u a l s a r e r e s a m p l e d . 2 I n s u c h a s c e n a r io , t h e b o o t s t r a p p r o c e d u r e c a n b e d e s i g n e d i n t h e f o l l o w i n g w a y : 1) E s t i m a t e t h e p a r a m e t e r s o f r e g r e s s i o n s (2) a n d (3) b y a c h o s e n a s y m p t o t i c m e t h o d . T h e s e r e g r e s s i o n s i n b o o t s t r a p p r o c e d u r e a r e r e f e r r e d t o i n t h i s p a p e r a s " n u l l" r e g r e s s i o n s . U n d e r s u c h " n u ll" r e g r e s s i o n it is n e c e s s a r y to: a ) D e t e r m i n e t h e m o d e l r e s i d u a l s eit. b ) C a lc u la t e t h e W a ld s ta tis tic : 1 se ((?;) is a standard error of 6t.
W
= a ' v a r [cc] 1 a . (6) 2) R e p e a t t h e f o l l o w i n g p r o c e d u r e a l a r g e n u m b e r o f tim e s . a) D r a w t h e r e s i d u a l s e it, £ = 1 , . . T f r o m eit w i t h r e p l a c e m e n t . b ) G e n e r a t e t h e b o o t s t r a p r e t u r n s a s fo llo w s : K = a t + P J t + e* . (7) c) E s ti m a t e t h e b o o t s t r a p p a r a m e t e r s , o f t h e firs t p a t h o f t h e m o d e l , a i a n d P * o f t h e f o l l o w i n g r e g r e s s i o n : rt = « * + P i f + e t . (8) d ) E s ti m a t e t h e b o o t s t r a p p a r a m e t e r s , o f t h e s e c o n d p a t h o f t h e m o d e l , a n d Xi o f t h e f o l l o w i n g r e g r e s s i o n : * * * £ ,* rIt ~ To + 7 i fi t + £ u . (9)e) C a lc u la t e t h e b o o t s t r a p p e d W a ld sta tis tic :
W* = --- --- i--- ( a * ) ' t (10) 1 + E ( f t )' v a r [ f t ] - 1 E ( f t ) K J e f) C a lc u la t e t h e p e r c e n t a g e o f a * ' s a n d f3 * 's a n d /0*'s a n d y** 's , a n d W *'s t h a t a r e g r e a t e r t h a n a a n d b i a n d g 0 a n d g 1, a n d W. T h e p e r c e n t a g e s a r e t h e p - v a lu e s o f t h e b o o t s t r a p te st. O n e o f t h e m a i n c o n c e r n s w h i l e u s i n g t h e b o o t s t r a p m e t h o d is c o n s is te n c y , i.e. c o n c o r d a n c e b e t w e e n t h e q u a n t i l e s d e r i v e d f r o m t h e b o o t s t r a p d i s t r i b u t i o n a n d t h e a s y m p t o t i c o n e . T h e b o o t s t r a p q u a n t i l e s c a n b e d e r i v e d u s i n g a c o m p u t e r a l g o r i t h m d e s c r i b e d a b o v e . I n t h i s c a s e , t h e c o n s i s t e n c y o f b o o t s t r a p is p r e s e n t e d i n t h e m o n o g r a p h o f D a v i s o n a n d H i n k l e y (1999). 3. D A T A I n t h i s s e c t i o n w e a n a l y z e t h e q u a r t e r l y r e t u r n s o f s to c k s l i s t e d o n W S E in 1995- 2 0 1 0 . T h e f u l l - s a m p l e o b s e r v a t i o n s a r e d i v i d e d i n t o t w o s e p a r a t e s u b - p e r i o d s : 1 9 9 5 -2 0 0 4 , t h e y e a r s p r e c e d i n g P o l a n d 's a c c e s s io n to t h e E U , a n d 2 0 0 5 -2 0 1 0 , t h e y e a r s o f P o l a n d 's m e m b e r s h i p i n t h e U E . D a t a r e f e r r i n g t o t h e
f u n d a m e n t a l r e s u l t s o f t h e i n s p e c t e d c o m p a n i e s a r e t a k e n f r o m t h e d a t a b a s e d r a w n u p b y N o t o r i a S e r w i s S p. z o .o . D a t a f o r d e f i n i n g r e t u r n s o n s e c u r itie s a r e p r o v i d e d b y t h e W a r s a w S to c k E x c h a n g e . T h e d a t a p r e s e n t e d b y U r b a ń s k i (2012b) i n d i c a t e t h a t t h e W S E is a m o n g t h e a v e r a g e - s i z e d E u r o p e a n s t o c k e x c h a n g e s . I t ju s t if ie s t h e c h o ic e o f t h e W S E a s a n a r e a f o r r e s e a r c h i n g t h e r e t u r n s i n C e n t r a l E u r o p e 's e m e r g i n g m a r k e ts . T h e e n t i r e s a m p l e c o m p r i s e s 56 q u a r t e r l y i n v e s t m e n t p e r i o d s f r o m M a y 10, 1996 to M a y 12, 2010. T h e f irs t s u b - p e r i o d c o v e r s 36 q u a r t e r s f r o m M a y 1 0 , 1996 to M a y 19, 2005. T h e s e c o n d s u b - p e r i o d c o v e r s 2 0 q u a r t e r s f r o m M a y 19, 2005 to M a y 15, 2010. A r a p i d i n c r e a s e i n t h e n u m b e r o f W S E c o m p a n i e s is r e c o r d e d a f te r 2004, fo l l o w i n g P o l a n d 's a c c e s s io n to t h e E U . H o w e v e r , i t h a s b e e n a c c o m p a n i e d b y a n in c r e a s e i n t h e n u m b e r o f s p e c u l a t i v e s to c k s w h o s e r e t u r n s a r e n o t li n k e d to t h e i r f in a n c ia l r e s u l t s ; s e e U r b a ń s k i (2012a). C o n s e q u e n t l y , t h e t e s t s a r e p e r f o r m e d f o r t w o m o d e s . M o d e 1 c o n s i d e r s all W S E s to c k s e x c e p t o f c o m p a n i e s c h a r a c t e r i z e d b y a n e g a t i v e b o o k v a l u e . I n m o d e 2, w e e l i m i n a t e s p e c u l a t i v e s to c k s m e e t i n g o n e o f t h e f o l l o w i n g b o u n d a r y c o n d i tio n s : a) M V / B V > 100, b ) R O E < 0 a n d B V > 0 a n d M V / B V > 30 a n d rit > 0, w h e r e M V is t h e s t o c k m a r k e t v a l u e , R O E is t h e r e t u r n o n b o o k v a l u e (BV). T h e s p e c u l a t i v e s to c k s a p p e a r f r o m Q 1 o f 2005. T h e n u m b e r o f a n a l y z e d c o m p a n i e s d e c r e a s e d f r o m 10% i n 2005 to 30% i n 2010, a f t e r e x c lu s io n o f s p e c u l a t i v e s to c k s . A ll s t o c k r e t u r n s a r e c a l c u l a t e d i n e x c e s s o f 9 1 - d a y P o lis h T r e a s u r y b ill r e t u r n (R F ). T h e b o o t s t r a p q u a n t i l e is b a s e d o n 10,000 r e s a m p l e s o f t h e d a t a . T h e i n s p e c t e d s e c u r itie s a r e d i v i d e d i n t o q u i n t i l e p o r t f o l i o s b u i l t o n t h e b a s is o f f u n d a m e n t a l f u n c t i o n a l F U N , p r e s e n t e d i n e q u a t i o n (11), a n d N U M a n d D E N f u n c t i o n s c o n s t i t u t i n g t h e n u m e r a t o r a n d d e n o m i n a t o r o f F U N , r e s p e c ti v e ly .3
F U N - nor(RO E)*nor(AP)*nor(AZO )*nor(AZN )
nor(M V/E)*nor(M V/BV)
’ ( 1 1 ) w h e r e X S (Q t) X Z O ( Q ,) R O E = F 1;A P = F 2 = - P ;A Z O = F 3 = -X S (n Q t) X Z O ( n Q ) t=1 t=1 t=13 The tested securities are divided into quintile portfolios in one direction; 5 portfolios are formed on FUN, 5 on NUM and 5 on DEN.
ź Z N (Q t) A Z N = F 4 = - f 1--- ,M V/£ = F i ;M V /B V = F 6. Ź Z N (n Q t) t=1 V a ria b le s Fj (j = 1, ..., 6) a r e t r a n s f o r m e d to s t a n d a r d i z e d a r e a s r a n g i n g < a j ; b j> , i n k e e p i n g w i t h E q u a t i o n (12): F - c * F mm n o r ( F , ) = f a , + ( b , - a j * --- ^---J— ^~.--- 1. (12) d * F max — c * F mm + e I n E q u a t i o n s (11) a n d (12), t h e c o r r e s p o n d i n g i n d i c a t i o n s a r e a s f o llo w s . i i i R O E is r e t u r n o n b o o k e q u i t y ; X S ( Q , ) , £ Z O ( Q , ) , £ Z N ( Qt) a r e v a l u e s t=1 t=1 t=1 t h a t a r e a c c u m u l a t e d f r o m t h e b e g i n n i n g o f t h e y e a r a s n e t s a le s r e v e n u e (S), o p e r a t i n g p r o f i t (Z O ) a n d n e t p r o f i t ( Z N ) a t t h e e n d o f "i" q u a r t e r (Q,); i ____________ i i _ X
S(nQ t),
XZO (nQ t),
XZN (nQ t)
a r e a v e r a g e v a l u e s , a c c u m u l a t e d f r o m t h e t=1 t=1 t=1 b e g i n n i n g o f t h e y e a r a s S, Z O a n d Z N a t t h e e n d o f Q , o v e r t h e la s t n y e a r s ;4 M V / E is t h e m a r k e t - t o - e a r n i n g v a l u e r a ti o ; M V / B V is t h e m a r k e t - t o - b o o k v a l u e r a ti o ; aj, bj, Cj, dj, ej a r e v a r i a t i o n p a r a m e t e r s . C a lc u la t io n s p r o v e , t h a t i n m o d e l i n g e q u i l i b r i u m o n t h e s t o c k m a r k e t , i t is p o s s i b le to a s s u m e i d e n t i c a l v a l u e s f o r all p a r a m e t e r s ; s e e U r b a ń s k i (2011). T h e f u n c t i o n s Fj (j = 1, . .. ,6 ) a r e t r a n s f o r m i n t o e q u a l n o r m a l i z e d a r e a < 1 ; 2 > . 5 I n c o m p a r i s o n w i t h t h e w o r k c o n d u c t e d b y F F (1995) a n d C o c h r a n e (2001), i t is a s s u m e d t h a t F U N m a y c o n s t i t u t e p o s it iv e c h a r a c te r is t ic s a s a b a s is f o r t h e g e n e r a l d e s c r i p t i o n o f r e t u r n s . T h e f u n c t i o n N U M r e p r e s e n t s a n i n v e s t o r f o r m i n g a p o r t f o l i o w h i c h c o n s is ts o f t h e b e s t f u n d a m e n t a l c o m p a n i e s . W h e r e a s D E N r e p r e s e n t s a n i n v e s t o r p o r t f o l i o w h i c h c o n s is ts o f t h e u n d e r v a l u e d s to c k s . S im i la rly , F U N r e p r e s e n t s a n i n v e s t o r f o r m i n g a p o r t f o l i o w h i c h c o n s is ts o f t h e b e s t f u n d a m e n t a l a n d s i m u l t a n e o u s l y u n d e r v a l u e d sto c k s . F U N , N U M a n d D E N a r e c a l c u l a t e d f o r all a n a l y z e d s e c u r itie s a t t h e b e g i n n i n g o f e a c h i n v e s t m e n t p e r i o d i n w h i c h t h e r e t u r n is to b e c a lc u la t e d . F U N , N U M a n d D E N f o r p o r t f o l i o s c o n s ti4 The present research assumes that n = 3 years.
i i i ____________ i ____________
5 If X Z N (Q t) , X Z O ( Q t) , X Z N (n Q t) or £ Z O ( n Q t) in equation (11) is negative, the
func-t=1 t=1 t=1 t=1
t u t e a v e r a g e a r i t h m e t i c a l v a l u e s o f t h e s e f u n c t i o n s o f v a r i o u s p o r t f o l i o s e c u r itie s . R e t u r n s o n g i v e n p o r t f o l i o s a r e a v e r a g e s t o c k r e t u r n s w e i g h t e d b y m a r k e t c a p i ta liz a ti o n s . T h e f a c to r s f t a r e a s s i g n e d to c o m p a n y p o r tf o lio s . 4 . R E S U L T S W e t e s t t h e a g g r e g a t e d t h r e e - f a c t o r m o d e l p r e s e n t e d b y U r b a ń s k i (2011). T h is m o d e l a n a l y s e s t h e i n f l u e n c e o f e x c e s s m a r k e t r e t u r n s (R M )6 a n d f a c to r s f HMLN a n d f LMHD o n r e t u r n s i n t h e a n a l y z e d p o r tf o l io s . f HMLN ( h i g h m i n u s lo w ) is t h e d if f e r e n c e b e t w e e n t h e r e t u r n s f r o m t h e p o r t f o l i o w i t h t h e h i g h e s t a n d l o w e s t N U M t v a l u e s i n t h e p e r i o d t; f LMHD ( lo w m i n u s h i g h ) is t h e d if f e r e n c e b e t w e e n t h e r e t u r n s f r o m t h e p o r t f o l i o w i t h t h e l o w e s t a n d h i g h e s t D E N t v a l u e s i n t h e p e r i o d t . A b s o lu te v a l u e s o f c o r r e l a t i o n c o e f f ic ie n t b e t w e e n t h e r e s p o n s e v a r ia b le a n d e x p l a n a t o r y v a r ia b le s r a n g e f r o m 0.05 to 0.92. A b s o lu te v a l u e s o f t h e c o r r e l a t i o n c o e f f ic ie n t b e t w e e n f a c to r s a r e r e a c h i n g t h e le v e ls o f 0.23 f o r f u l l - s a m p l e o b s e r v a t i o n s , 0.37 f o r s u b - p e r i o d 1 9 9 5 -2 0 0 5 , a n d 0.23 f o r s u b - p e r i o d 2 0 0 5 -2 0 1 0 , r e s p e c tiv e ly . F o r t h e firs t a n d s e c o n d s u b - p e r i o d t h e c o r r e l a t i o n b e t w e e n R M t- R F t a n d f H M m is e q u a l to 0.24 a n d 0.18, r e s p e c ti v e ly , a n d b e t w e e n R M t - R F t a n d f ^ MHD is -0 .3 7 a n d -0.16. I t is p o s s i b le , t h e r e f o r e , to d u p l i c a t e i n f o r m a t i o n . T h e o r t h o g o - n a l i z e d m a r k e t f a c to r s a r e d e f i n e d u s i n g t h e f o l l o w i n g r e g r e s s i o n : R M t - R F t = a + P H M L N f tHMLN + P LM H D f r HD + e t ; t = 1, . . . , T, (13) w h e r e : M o d e 1; f u ll s a m p l e a = -0.01; @HMLN = ° . 29; b LMHD = - 0 .27; R 2 = 6.32% (76.20% ) (15.87% ) (13.86% ) M o d e 2; f u ll s a m p l e a = -0.02; @HMLN = 0 .33; b LMHD = - 0 .010; R 2 = 4.97% (50.66% ) (11.38% ) (58.35% ) M o d e 1; f ir s t s u b - p e r i o d a = -0.01; b HMLN = 0 .40; b LMHD = - 0 .59; R 2 = 25.18% (82.42% ) (5.43% ) (0.90% )
M o d e 1; s e c o n d s u b - p e r i o d a = 0 .0 3; Ph m l n = - 1 .05; Pl m h d = 0.48; R 2 = 20.82 % (63.57% ) (8 .0 1 % ) (7.24% ) M o d e 2; s e c o n d s u b - p e r i o d a = -° . ° 1; - @h m l n = 0 .32; Pl m h d = 0 .57; R2 = 29 .° 7 % (82.36% ) (51.28% ) (1.72% ) U n d e r t h e r e g r e s s i o n m o d e l (13) t h e v a l u e s o f v a r ia b le l o a d i n g s a r e i n c l u d e d f o r all t e s t e d p e r i o d s . T h e c o r r e s p o n d i n g p - v a lu e s a p p e a r i n b r a c k e ts . R e g r e s s io n (13), e s p e c ia lly f o r s u b - p e r i o d s , c o n t a i n s h i g h e r e x p l a n a t o r y p o w e r . T h e v a l u e o f t h e o r t h o g o n a l i z e d m a r k e t f a c to r is d e f i n e d a s f o llo w s :7 f M = a + ^ . (14) T h e r e s p o n s e v a r i a b l e a n d t h e e x p l a n a t o r y v a r ia b le s a r e s u b j e c t to s ta tio n a r - it y t e s t s w h o s e h y p o t h e s i s is b a s e d o n t h e D i c k e y - F u l l e r te s t. D ic k e y - F u ll e r te s ts a n d a u g m e n t e d D i c k e y - F u l l e r t e s t s c o n f i r m la c k o f u n i t r o o t f o r e a c h t e s t c a s e a t 1% s ig n if ic a n c e le v e l.8 T h is l e a d s to c o n c l u s i o n s r e g a r d i n g t h e s t a t i o n a r i t y o f t h e a n a l y z e d v a r ia b le s . W e t e s t t h e a g g r e g a t e d t h r e e - f a c t o r m o d e l i n t w o p a s s e s : r„ - R F t = a t + + P UMHj r HD + P f + e ; = ; V t= 1 ,...,1 5 , (15) r it ~ R F t ~ 7 o + y HM LN f i t , HMLN + 7 lM H D fit,IM H D + J M o P tM O + £ t t ’ t ~ ^ ~ (16) B e ta v a l u e s a r e e s t i m a t o r s o f t h e s y s t e m a t i c risk . T h e s e c o n d p a s s e s t i m a t e s t h e b e t a l o a d i n g s w h i c h d e f i n e r i s k p r e m i u m s . R e g r e s s i o n p a r a m e t e r s i n (15) a n d (16) a r e e s t i m a t e d v i a G L S — f o l l o w i n g P r a i s - W i n s t e n p r o c e d u r e , a n d b y t h r e e b o o t s t r a p m e t h o d s : q u a n t i l e b o o t s t r a p , B C a b o o t s t r a p , a n d t - b o o t s t r a p ; s e e E f r o n a n d T ib s h i r a n i (1993). H o m o s k e d a s t i c i t y o f t h e r e s i d u a l s is c o n f i r m e d u s i n g
7 A similar procedure concerning the orthogonalization of the market factor is applied by Fama and French (1993), p. 27-31, for the five-factor model. The loadings of all of the tested HML, SMB, TERM and DEF variables differ significantly from zero. The determination coefficient of the analyzed regression (by FF) is R2 = 38%.
8 Dickey-Fuller tests are carried out for the three tested periods. 18 tested cases include the re sponse variable for 5 portfolios formed on FUN, NUM and DEN and 3 explanatory variables: f tMO
f HMLN and f LMHD . The augmented Dickey-Fuller tests are carried out for lag, defined on the basis of minimizing the modified Akaike criterion, assuming that maximum lag equals 4. Test findings are avail able from the authors on request.
W h i t e a n d B r e u s c h - P a g a n m e t h o d s . T h e r e f o r e , t h e h e t e r o s c e d a s c i t y c o r r e c t i o n is n o t r e q u i r e d . 9 T h e p a r a m e t e r s o f t h e s e c o n d p a s s c a n b e e s t i m a t e d b y t h r e e v a r ia n ts : 1) t h e p o o l e d ti m e - s e r i e s a n d c r o s s - s e c ti o n e s tim a te , 2) t h e " p u r e c r o s s - s e c ti o n a l" e s tim a te , o n t h e b a s is o f t i m e s e r ie s a v e r a g e s , 3) t h e F a m a - M a c B e t h p r o c e d u r e t h a t m e a n s r u n n i n g a c r o s s - s e c ti o n a l r e g r e s s i o n a t e a c h p o i n t i n tim e ; e s t i m a t e d p a r a m e t e r s y 0 a n d y x a r e a v e r a g e c r o s s s e c t i o n a l e s t i m a t e s o f y 0t a n d y 1t .T h e ti m e - s e r i e s s t a n d a r d d e v i a t i o n s o f y 0t a n d f lt a r e u s e d to e s t i m a t e t h e s t a n d a r d e r r o r o f y 0 a n d y 1 .10 If t h e e x p l a n a t o r y v a r i a b l e s o f r e g r e s s i o n (16) d o n o t v a r y o v e r ti m e , a n d if t h e e r r o r s a r e c r o s s - s e c ti o n a ll y c o r r e l a t e d b u t n o t c o r r e l a t e d o v e r ti m e , t h e n t h e p o o l e d t i m e - s e r i e s a n d c r o s s - s e c ti o n a l O L S e s t im a te , t h e " p u r e c r o s s - s e c ti o n a l" O L S e s tim a te , a n d t h e t h e F a m a - M a c B e t h p r o c e d u r e a r e i d e n tic a l; s e e C o c h r a n e (2001), p p . 2 4 7 -2 5 0 . W E e s t i m a t e t h e r i s k p r e m i u m v e c t o r u s i n g t h e p o o l e d tim e - s e r ie s a n d c r o s s - s e c t i o n d a t a . I n d e p e n d e n t v a r i a b l e s ( b e ta s ) r e m a i n p e r m a n e n t f o r all p e r i o d s , w h i l e d e p e n d e n t v a r i a b l e s c o n s t i t u t e t h e r e t u r n s w h i c h s h o u l d b y n a t u r e b e r a n d o m ; s e e C o c h r a n e (2001), p . 247. T h e r e f o r e , w e a s s u m e t h e la c k o f a u t o c o r r e l a t i o n o f t h e r e s i d u a l c o m p o n e n t . T h e i m p a c t o f h e t e r o s k e d a s t i c i t y is t a k e n i n t o a c c o u n t b y m e a n s o f t h e c h a n g e o f v a r ia b le s m e t h o d . 11 Table 1
The Parameter Values of Time-Series Regression of Excess Stock Returns on the Orthogonalized Stock-Market Factor, fMO and the Mimicking Returns for the N U M Value (f HMLN) and D EN Value
(f LMHD) Factors
r , -
RFt
= a , + + P.,Mof t
MO +e
tt;t
= ; Vi = 1, ..., 15Mode 1. The sample period is from 1995 to 2010, T=56 Quarters
Portfolio „i" Quantile bootstrap, 0 BCa bootstrap, 0 t-bootstrap „null" regression
^
0.025^
0.975^
0.025^
0.975 p-valuea e " p-valuea R2 % 0* = Pi H M L N Q ~ P i,H M L N 1 -0.531 -0.180 -0.531 -0.174 0.002 -0.357 0.000 88.60 5 0.331 0.715 0.331 0.715 0.000 0.520 0.000 85.01 6 -0.716 -0.266 -0.702 -0.248 0.002 -0.472 0.000 83.489
The co-variance matrix of regression coefficients is also estimated by means of the Newey-West estimator where standard errors are corrected for autocorrelation and heteroskedasticity. The results are qualitatively similar. They are readily available upon request.10
X l is the vector f \ V f H M L N , 7 LM H D , 7 M O ] .11
See footnote 9.10 0.398 0.767 0.381 0.751 0.000 0.591 0.000 85.49 11 -0.056 0.297 -0.054 0.298 0.230 0.120 0.210 83.91 12 0.088 0.410 0.047 0.380 0.000 0.250 0.005 87.98 13 0.143 0.510 0.122 0.499 0.004 0.330 0.002 78.67 14 0.089 0.433 0.087 0.431 0.000 0.273 0.004 86.02 15 -0.065 0.366 -0.039 0.421 0.248 0.138 0.220 86.12 = P t M O
0
=P
m o 1 1.011 1.251 1.019 1.260 0.000 1.125 0.000 88.60 5 0.878 1.118 0.895 1.139 0.000 1.002 0.000 85.01 6 0.996 1.309 1.006 1.313 0.000 1.149 0.000 83.48 10 0.876 1.100 0.878 1.102 0.000 0.989 0.000 85.49 11 0.851 1.075 0.847 1.074 0.000 0.964 0.000 83.91 15 0.979 1.242 0.984 1.246 0.000 1.111 0.000 86.12 ^ _ P i,L M H D ^ _ P i,L M H D 1 -0.722 -0.426 -0.742 -0.444 0.002 -0.572 0.000 88.60 5 -0.477 -0.172 -0.502 -0.191 0.002 -0.331 0.000 85.01 6 -0.667 -0.288 -0.665 -0.278 0.002 -0.469 0.000 83.48 10 -0.549 -0.254 -0.574 -0.269 0.002 -0.404 0.000 85.49 11 0.024 0.327 0.017 0.325 0.030 0.180 0.031 83.91 12 0.069 0.345 0.069 0.345 0.006 0.203 0.008 87.98 13 -0.324 -0.010 -0.294 0.007 0.058 -0.163 0.062 78.67 14 -0.897 -0.602 -0.903 -0.604 0.002 -0.752 0.000 86.02 15 -1.038 -0.672 -1.065 -0.691 0.002 -0.872 0.000 86.12Regression parameters for all bootstrap iterations and "null" regression are estimated by GLS. Portfolios for i = 1 and i = 5 are formed on minimal and maximal values of FUN. Portfolios for
i = 6 and i = 10 are formed on minimal and maximal values of NUM . Portfolios for i = 11 and
i = 15 are formed on minimal and maximal values of DEN. 0 oo25 is the bootstrapped value of the estimator for the 2,5% level and, similarly, 9 0 915 is the bootstrapped value of the estimator for the 97,5% level. The bootstrap quantile is based on 10,000 data resamples. Negative-By stocks are excluded from the portfolios. The errors-in-variables are adjusted and follow Shanken (1992). a Corresponds to the significance test for model parameters in the null hypotheses. Bold type — the param eter is significantly different from zero at the level of 5%.
T h e i m p a c t o f e s t i m a t i o n e r r o r s o f t h e t r u e b e t a v a l u e s i n t h e f irs t p a s s is c o n s i d e r e d b y c o r r e c t i n g t h e s t a n d a r d e r r o r s o f b e t a l o a d i n g s e s t i m a t e d i n t h e s e c o n d p a s s . W i t h t h i s p u r p o s e i n m i n d , S h a n k e n 's e s t i m a t o r is a p p l i e d ; s e e S h a n k e n (1992).
T able 1 p r e s e n t s t h e v a l u e s o f p a r a m e t e r s o f r e g r e s s i o n (15) f o r t h e f u l l - s a m p l e a n d f o r t h e p o r t f o l i o s o f m o d e 1 t y p e . 12 T h e r e g r e s s i o n p a r a m e t e r s e s t i m a t e d i n " n u ll" r e g r e s s i o n s f o r t h e firs t a n d s e c o n d s u b - p e r i o d s a r e s u b j e c t to C h o w 's s ta b ili ty te s ts . I n m o s t c a s e s , t h e r e s u l t s c o n f i r m t h e s ta b ili ty o f t h e p a r a m e t e r s a t t h e le v e l o f 5 % . T h e r e g r e s s i o n p a r a m e t e r s f o r t e s t c a s e s , e s t i m a t e d i n " n u ll" r e g r e s s i o n a n d t h r e e b o o t s t r a p m e t h o d s a r e sim ila r. A lso , t h e c r o s s - s e c ti o n c h a n g e s o f s y s t e m a t i c r i s k c o m p o n e n t , f o r t h e p o r t f o l i o s f o r m e d o n m o d e 1 a n d m o d e 2 a r e sim ila r.
F o r p o r t f o l i o s f o r m e d o n F U N a n d N U M , t h e s y s t e m a t i c r i s k c o m p o n e n t
bi,HMLN i n c r e a s e s m o n o t o n i c a l l y f r o m n e g a t i v e v a l u e s f o r t h e s m a lle s t F U N a n d
N U M q u i n t i l e s to p o s it iv e v a l u e s f o r t h e l a r g e s t q u in tile s . H o w e v e r , t h e r i s k c o m p o n e n t b i,LMHD a s s u m e s n e g a t i v e v a l u e s f o r all q u in tile s .
F o r p o r t f o l i o s f o r m e d o n D E N , t h e s y s t e m a t i c r i s k c o m p o n e n t b i,LMHD d e c r e a s e s m o n o t o n i c a l l y f r o m p o s it iv e v a l u e s f o r t h e s m a lle s t D E N q u i n t i l e s to n e g a tiv e v a l u e s f o r t h e l a r g e s t q u in tile s . T h e r i s k c o m p o n e n t bi,HMLN a s s u m e s p o s it iv e v a l u e s f o r all q u in tile s .
T h e s c h e m e s o f r e t u r n c h a n g e s o n p o r t f o l i o s f o r m e d o n F U N a n d D E N (fo r t h e f u l l - s a m p l e a n d f o r t h e p o r t f o l i o s o f m o d e 1 t y p e ) a r e p r e s e n t e d i n F ig u r e 1 a n d F ig u r e 2.
a) b)
Figure 1. Influence off HMLN factor on returns of portfolios formed on FUN and D E N a
F ig u r e 1 s h o w s t h e in f l u e n c e o f f HMLN o n r e t u r n s o f p o r t f o l i o s f o r m e d o n F U N ( F ig u r e 1a) a n d D E N ( F ig u r e 1b). P o rtf o lio f o r i = 1 is f o r m e d o n m i n i m a l v a l u e o f F U N o r D E N . P o rtf o lio f o r i = 5 is f o r m e d o n m a x im a l v a l u e o f F U N o r D E N . N e g a t i v e - B V s to c k s a r e e x c l u d e d f r o m t h e p o r tf o lio s . T h e s a m p l e p e r i o d is f r o m 1995 to 2010, 56 Q u a r t e r s .
a) b )
Figure 2. Influence off LMHD factor on returns of portfolios formed on FUN and D E N a
T h is f ig u r e s h o w s t h e i n f l u e n c e o f f LMHD o n r e t u r n s o f p o r t f o l i o s f o r m e d o n
F U N ( F ig u r e 2 a) a n d D E N ( F ig u r e 2b). P o rtf o lio f o r i = 1 is f o r m e d o n m i n i m a l v a l u e o f F U N o r D E N . P o rtf o lio f o r i = 5 is f o r m e d o n m a x i m a l v a l u e o f F U N o r
D E N . N e g a tiv e - B V s to c k s a r e e x c l u d e d f r o m t h e p o r tfo lio s . T h e s a m p l e p e r i o d is f r o m 1995 to 2010, 5 6 Q u a r t e r s . T h e c o n d u c t e d r e s e a r c h i n d i c a t e s t h a t l o n g i n v e s t m e n t s i n c o m p a n i e s w i t h la r g e F U N o r N U M v a l u e s l e a d to h i g h e r r e t u r n s f o r g r o w i n g f HMLN a n d d e c r e a s i n g f LMHD v a l u e s . L o n g i n v e s t m e n t s i n c o m p a n i e s w i t h la r g e D E N ( lo w B V / M V a n d E / M V ) d e m o n s t r a t e h i g h e r r e t u r n s f o r g r o w i n g f HMLN a n d d e c r e a s i n g f LMHD v a l u e s . H o w e v e r, l o n g i n v e s t m e n t s i n c o m p a n i e s w i t h s m a ll D E N v a l u e s ( h i g h B V / M V a n d E / M V ) d e m o n s t r a t e h i g h e r r e t u r n s f o r g r o w i n g f HMLN a n d f LMHD v a l u e s . T h e v a l u e s o f t h e R 2 c o e f f ic ie n t r e a c h h i g h v a l u e s a t 90% .
C r o s s - s e c tio n c h a n g e s o f r i s k c o m p o n e n t s bi,HMLN a n d b i,LMHD a r e s im ila r f o r t h e w h o l e s a m p l e a n d t h e firs t s u b - p e r i o d . B e t a d i s t r i b u t i o n s i n t h e s e c o n d s u b p e r i o d , f o r p o r t f o l i o s f o r m e d o n D E N , a r e sim ila r, w h i l e t h e s e c h a n g e s f o r p o r t f o lio s f o r m e d o n F U N a n d N U M a r e m o r e d if f ic u lt to i n t e r p r e t . T h e v a l u e s o f p a r a m e t e r s o f r e g r e s s i o n (16) a r e p r e s e n t i n g i n T able 2. C o e f f ic ie n ts g 1 = gMO, g 2 = gHMLN a n d g 3 = g LMHD c o n s t i t u t e s y s t e m a t i c r i s k p r e m i u m i n t e r m s o f t h e f a c t o r c o n n e c t e d w i t h a m a r k e t p o r t f o l i o a n d t h e f HMLN a n d f LMHD fa c to rs . T h e r e s u l t s f o r t h e w h o l e s a m p l e a r e a s f o llo w s : t h e r i s k p r e m i u m s g HMLN a n d g LMHD e s t i m a t e d b y t h r e e b o o t s t r a p m e t h o d s a r e s ig n i f i c a n t ly h i g h e r t h a n z e r o ; t h e s e r e s u l t s f o r t h e p o r t f o l i o s f o r m e d o n m o d e 1 a n d m o d e 2 a r e s im i la r; h o w e v e r , if s p e c u l a t i v e s to c k s a r e n o t e x c l u d e d f r o m c o n s i d e r a t i o n , g LMHD e s t i m a t e d i n " n u ll" r e g r e s s i o n ( a ls o , i n t h e s e c o n d s u b - p e r i o d ) is e q u a l to z e r o .
T h e c o m p o n e n t s g HMLN a n d g LMHD e s t i m a t e d b y b o o t s t r a p a r e s ig n i f i c a n t ly h i g h e r t h a n z e r o i n t h e b o t h s u b - p e r i o d s . T h e g LMHD c o m p o n e n t r a n g e s f r o m 4% i n 1 9 9 6 -2 0 0 5 to 8% i n 2 0 0 5 -2 0 1 0 . H o w e v e r , g HMLN r a n g e s f r o m 6% i n 1 9 9 6 -2 0 0 5 to 2% i n 2 0 0 5 -2 0 1 0 . T h e c o m p o n e n t g MO e s t i m a t e d i n " n u ll" r e g r e s s i o n is in s i g n i f i c a n t l y d i f f e r e n t f r o m z e r o f o r all t h e t e s t e d p e r i o d s . T h e c o r r e s p o n d i n g p - v a lu e s a r e h i g h e r t h a n 0.19. H o w e v e r , t h e b o o t s t r a p e s t i m a t i o n s f o r t h e f u ll s a m p l e a n d f irs t s u b - p e r i o d i n d i c a t e t h e s ig n i f i c a n t p o s it iv e r e s u lt s . Table 2
The risk premium vector (g) values estimated from the second-pass regression for the aggregated three-factor model
r it ~ R F f ~ J o + JhM L N P i.H M L N + JlM H D fii.L M H D + JmO fii.M O + £ i t ’ i ~ 1,
Mode
Quantile bootstrap, 9 BCa bootstrap, 9
t-boot-strap
"null" regression Para
meter
^2.5%
^97.5%@
2.5%6
97.5% p-value a 0 p-value aThe sample period is from 1995 to 2010,
T = 56 Quarters To -0.12 -0.02 -0.09 -0.09 0.00 -0.09 0.14 1 Yhmln 0.03 0.06 0.03 0.08 0.00 0.05 0.02 Y mo 0.01 0.11 0.05 0.18 0.00 0.07 0.21 Ylmhd 0.01 0.06 0.02 0.08 0.00 0.04 0.11 Yo -0.15 0.01 -0.10 -0.12 0.01 -0.08 0.32 2 Yhmln 0.03 0.08 0.04 0.10 0.00 0.05 0.01 Ymo -0.04 0.14 0.01 0.33 0.07 0.06 0.48 Ylmhd 0.03 0.07 0.04 0.09 0.00 0.05 0.02
The sample period is from 1995 to 2005,
T = 36 Quarters Yo -0.14 -0.02 -0.11 -0.12 0.00 -0.09 0.12 1 Yhmln 0.03 0.08 0.04 0.11 0.00 0.06 0.03 Ymo 0.01 0.12 0.04 0.20 0.00 0.08 0.19 Ylmhd 0.01 0.06 0.02 0.09 0.00 0.04 0.13
The sample period is from 2005 to 2010, T = 20 Quarters 7 o -0.06 0.04 -0.02 -0.03 0.50 -0.01 0.85 1 7 HMLN 0.00 0.03 0.01 0.00 0.14 0.01 0.64 7 MO -0.05 0.09 0.00 -0.04 0.46 0.02 0.82 7 LMHD -0.01 0.04 0.01 0.00 0.15 0.02 0.60 T o -0.17 0.09 -0.10 -0.19 0.34 -0.06 0.67 2 7 HMLN -0.01 0.05 0.01 -0.02 0.03 0.02 0.42 7 MO -0.12 0.14 -0.05 0.18 0.63 0.02 0.87 7 LMHD 0.04 0.11 0.07 0.15 0.00 0.08 0.00
Regression parameters for all bootstrap iterations and "null" regression are estimated by GLS. Portfolios for i = 1-5 are formed on FUN. Portfolios for i = 6-10 are formed on NUM . Portfolios for i = 11-15 are formed on DEN. 0 25V is the bootstrapped value of the estimator for the 2,5% level and, similarly, 0 97 5% is the bootstrapped value of the estimator for the 97,5% level. The bootstrap quantile is based on 10,000 data resamples. In m ode 1 negative-BV stocks are excluded from the portfolios. In m ode 2 speculative stocks are excluded from the portfolios. It is assumed that speculative stocks meet one of the following two conditions: 1) M V /BV > 100 and rit > 0, 2)
ROE < 0 and M V /B V > 30 and rit > 0, w here M V is the stock market value, ROE is the return on book value (BV), rit is the return of portfolio i in period t. a Corresponds to the significance test for m odel parameters in the null hypotheses. Bold type — the param eter is significantly different from zero at the level of 5%. Italic type — the param eter is significantly different from zero at the level of 10%.
T h e v a l u e o f y MO f o r t h e s e c o n d s u b - p e r i o d is e q u a l to z e r o a ls o f o r b o o t s t r a p a n d " n u ll" e s t i m a t i o n s p o i n t i n g t o w a r d t h e d e c is iv e i m p a c t o f r i s k i n t e r m s o f t h e f HMLN a n d f LMHD f a c to r s o n c r o s s - s e c ti o n r e t u r n s . T h is i n d i c a t e s t h a t f MO d o e s n o t a p p e a r to b e i m p o r t a n t f a c to r i n I C A P M c o n f i r m i n g t h e p r e v i o u s s t u d i e s (see , f o r e x a m p le , F a m a a n d F r e n c h (1992), J a g a n n a t h a n a n d W a n g (1996), L e t t a u a n d L u d v i g s o n (2001) a n d P e k o v a (2006)).
Table 3
The results of multifactor efficiency tests
r -
RFt
= a, +
P H M L N fr *
+
h LM H D frHD
+
PlMo f MO
+ e,/t - 1, . . T ; Vi = 1,
15
Mode Quantile bootstrap, W* W GRSw5;
W
i ^ p-value (X2) Statistic value p-value (X2) Statistic value p-value (F) Panel A: The sample period is from 1995 to 2010, T = 56 Quarters1 141.16 124.47 0.99 36.24 0.00 1.77 0.08
2 150.65 134.24 0.97 43.18 0.00 2.10 0.03
The sample period is from 1995 to 2010, T = 36 Quarters
1 197.16 167.03 0.97 42.97 0.00 1.61 0.17
The sample period is from 2005 to 2010, T = 20 Quarters
1 1140.63 787.38 0.65 166.53 0.00 1.39 0.50
2 2754.80 1825.96 0.99 57.43 0.00 0.48 0.84
Panel B: Chou and Zhou (2006), Fama-French's Factors
Period: 1964-1993 0.03 <0.01 0.01
Panel C: Chou and Zhou (2006), CRSP index
Period: 1926-1995 0.07 0.01 0.03
Period: 1986-1995 0.38 0.21 0.28
Panel A; H 0 : a,. = 0, V, = 1,...,«. W is the statistic of Wald. GRS is the F-statistic of Gibbons et al (1989). In m ode 1 negative-BV stocks are excluded from the portfolios. In m ode 2 speculative stocks are excluded from the portfolios. It is assumed that speculative stocks m eet one of the following two conditions: 1) M V /B V > 100 and rit > 0, 2) ROE < 0 and M V /B V > 30 and rit > 0, w here M V is the stock market value, ROE is the return on book value (BV), rit is the return of portfolio i in period t.
In Panel B the authors examine the joint efficiency of the Fama-French's factors in: r it - R F t = & i + P i,H M L f t H ,M + P i,S M B f t SMB + P i,M O ( R M t - R F t) + e i t, w here rit' s are m onthly
returns on 25 Fama-French's portfolios and R M t-R F t is the excess return on a market index. In Panel C the authors examine the efficiency of the CRSP value-weighted index in the standard market model:
R
t =a
+ firpt+
et
, w here R t is a vector of returns on 10 CRSP size decile portfolios in excess of the 30-day T-bill rate. The bootstrap quantile is based on 10,000 data resamples.M E is t e s t e d u n d e r t h e a s s u m p t i o n t h a t e r r o r s o f t h e r e g r e s s i o n (15) a r e iid . A lso , w e t e s t t h e n o r m a l i t y o f r e s i d u a l s . 13 W e e m p l o y t h r e e e f f ic ie n c y te s ts , t h e G R S te s t, t h e a s y m p t o t i c W a ld t e s t a n d b o o t s t r a p te s ts . T h e e m p ir ic a l r e s u l t s a r e r e p o r t e d i n T able 3. U n d e r ii d a s s u m p t i o n , t h e a s y m p t o t i c W a ld t e s t r e j e c t M E o f t h e t e s t e d p o r tf o l io s f o r all t h e i n v e s t i g a t e d p e r i o d s a t t h e s ig n if ic a n c e le v e l b e l o w 1% . T h e G L S t e s t r e je c t s M E f o r t h e w h o l e s a m p l e f o r p o r tf o l io s f o r m e d u n d e r a s s u m p t i o n m o d e 1 a t t h e 8% s ig n ific a n c e le v e l, a n d u n d e r a s s u m p t i o n m o d e 2 a t t h e 4% s ig n if ic a n c e le v e l. H o w e v e r , t h e b o o t s t r a p p e d W a ld te s t, W*, d o e s n o t r e j e c t e f f ic ie n c y f o r i n v e s t i g a t e d p e r i o d s . W e m a y c o n c l u d e t h a t t h e a g g r e g a t e d tr e e - f a c t o r m o d e l g e n e r a t e s M E p o r t f o l i o s o n t h e W S E w h e n s t o c k r e t u r n s a r e a s s u m e d to c o m e f r o m ii d m o d e ls . M o r e o v e r , w e a ls o c o m p a r e o u r M E r e s u l t s t o o t h e r p r o c e d u r e s i m p l e m e n t e d o n A m e r i c a n m a r k e t ; s e e C h o u a n d Z h o u (2006). T h e r e s u l t s a r e p r e s e n t e d i n P a n e l B a n d C o f T able 3. T h e p - v a l u e s o b t a i n e d s u g g e s t a s t r o n g r e je c t io n . I t is, n e v e r t h e l e s s , q u i t e i n t e r e s t i n g to o b s e r v e t h a t t h e b o o t s t r a p d e r i v e d p - v a l u e s a r e g r e a t e r t h a n t h e n o n - b o o s t r a p o n e s . 5. C O N C L U S I O N S T h e u s a g e o f b o o t s t r a p to t e s t t h e I C P M a p p l i c a t i o n p r o p o s e d b y U r b a ń s k i (2011) is p r e s e n t e d f o r W S E s to c k s . T h e c o n d u c t e d r e s e a r c h l e a d s to t h e f o l l o w i n g c o n c lu s io n s : 1. T h e u s e o f b o o t s t r a p p r o c e d u r e s a l lo w s f o r a n a c c u r a t e a s s e s s m e n t o f r e t u r n c h a n g e s a s c o m p a r e d w i t h c la s s ic a l a s y m p t o t i c m e t h o d s . 2. L o n g i n v e s t m e n t s i n c o m p a n i e s w i t h la r g e F U N o r N U M d e m o n s t r a t e h i g h e r r e t u r n s f o r g r o w i n g f HMLN a n d d e c r e a s i n g f LMHD v a l u e s . 3. L o n g i n v e s t m e n t s i n c o m p a n i e s w i t h la r g e D E N ( lo w B V / M V a n d E / M V ) d e m o n s t r a t e h i g h e r r e t u r n s f o r g r o w i n g f HMLN a n d d e c r e a s i n g f LMHD v a l u e s . 4. L o n g i n v e s t m e n t s i n c o m p a n i e s w i t h s m a ll D E N v a l u e s ( h i g h B V / M V a n d E / M V ) r e c o r d h i g h e r r e t u r n s f o r g r o w i n g f HMLN a n d f LMHD v a l u e s . 5. E s ti m a t e s o f s y s t e m a tic r i s k c o m p o n e n t s f o r t e s t c a s e s u s i n g c la s s ic a l p r o c e d u r e s a n d b o o t s t r a p m e t h o d s a r e sim ila r. 6. T h e c r o s s - s e c t i o n c h a n g e s o f s y s t e m a t i c r i s k c o m p o n e n t , f o r t h e p o r t f o l i o s f o r m e d o n t h e b a s is o f all a n a l y z e d s to c k s ( M o d e 1) a n d s to c k s w i t h t h e ex c e p t i o n t h e s p e c u l a t i v e s to c k s ( M o d e 2), a r e sim ila r. 7. T h e r i s k p r e m i u m c o m p o n e n t s e s t i m a t e d b y b o o t s t r a p a r e s ig n i f i c a n t ly d if fe r e n t f r o m z e r o i n all t e s t e d c a se s.
13 The Shapiro-Wilk tests confirm the residuals normality for the whole sample in 9 out of 15 tested portfolios.
8. If s p e c u l a t i v e s to c k s a r e n o t e x c l u d e d f r o m c o n s i d e r a t i o n , r i s k p r e m i u m c o m p o n e n t , g LMHD e s t i m a t e d i n " n u l l " r e g r e s s i o n is in s i g n if ic a n tly d i f f e r e n t f r o m z e r o i n all t e s t e d p e r i o d s . 9. T h e r i s k p r e m i u m g HMLN ( d e t e r m i n i n g t h e i n v e s t o r s e n s iti v ity to f in a n c ia l r e s u lts ) e q u a l s a p p r o x . 6% p e r q u a r t e r i n t h e f irs t s u b - p e r i o d a n d d e c r e a s e s to 1% i n t h e s e c o n d s u b - p e r i o d . 10. T h e r i s k p r e m i u m g LMHD ( d e t e r m i n i n g t h e i n v e s t o r s e n s i t i v i t y to t h e v a l u e ) e q u a l s a p p r o x . 4% p e r q u a r t e r i n t h e firs t s u b - p e r i o d a n d g r o w s i n t h e s e c o n d s u b - p e r i o d to 8 % , a f te r t h e e l i m i n a t i o n o f s p e c u l a t i v e sto c k s. 11. G R S a n d a s y m p t o t i c W a ld te s t s r e j e c t M E o f t h e m o s t p o r t f o l i o s s i m u l a t e d b y t h e t e s t e d IC A P M a p p l ic a tio n . H o w e v e r , t h e b o o t s t r a p p e d W a ld t e s t d o e s n o t r e j e c t e f f ic ie n c y f o r t h e t e s t e d c a se s. R E F E R E N C E S
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