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A preliminary examination of the no arbitrage property in the Canadian security market Hung, Reynold 1984

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A PRELIMINARY EXAMINATION OF THE NO ARBITRAGE PROPERTY I N THE CANADIAN SECURITY MARKET By REYNOLD HUNG B.Comm.(Hon.), Queen's U n i v e r s i t y , 1982 A THESIS SUBMITTED IN P A R T I A L FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN BUSINESS ADMINISTRATION i n THE FACULTY OF GRADUATE STUDIES F a c u l t y o f Commerce a n d B u s i n e s s A d m i n i s t r a t i o n We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA M a r c h 1984 © R e y n o l d Hung, 1984 In presenting/ t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department o f COMMERCE AND BUSINESS ADMINISTRATION The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date March 2 7 , 1 9 8 4 ABSTRACT T h i s paper examines the e m p i r i c a l i m p l i c a t i o n s o f the no a r b i t r a g e p r o p e r t y u s i n g the Canadian common stock s e c u r i t y market. In a d d i t i o n , an a l t e r n a t i v e s e c u r i t y r i s k measure t o the market "beta" i s i n t r o d u c e d and t e s t e d . The model f o r t e s t i n g i s d e r i v e d from the s t a t e p r e f e r -ence model which s t a t e s t h a t t h e r e e x i s t s a s e t of p o s i t i v e p r i m i t i v e s e c u r i t y p r i c e s which i s c o n s i s t e n t with the observed set o f complex s e c u r i t y p r i c e s . A s e t of market k e r n e l s i s estimated from the Canadian common stock data on which the t e s t i n g of the no a r b i t r a g e p r o p e r t y i s based. The e m p i r i c a l evidence s t r o n g l y suggests t h a t there i s a d i f f e r e n c e i n market c o n d i t i o n between the U.S. and Canada, and the no a r b i t r a g e p r o p e r t y should not be r e j e c t e d i n Canada. Furthermore, the u s e f u l n e s s of the a l t e r n a t i v e s e c u r i t y r i s k measure cannot be e s t a b l i s h e d based on the r e s u l t s o f t h i s paper. i i i T a b l e o f C o n t e n t s Page A b s t r a c t i i L i s t o f T a b l e s i v A c k n o w l e d g e m e n t v i i 1. I n t r o d u c t i o n 1 2. The S t a t e P r e f e r e n c e M o d e l 3 3. M e t h o d o l o g y 6 a) The A n a l y s i s 6 b) The E s t i m a t i o n o f t h e M a r k e t K e r n e l 8 c) The T e s t i n g o f t h e No A r b i t r a g e P r o p e r t y i n C a n a d a 12 d) A l t e r n a t i v e E s t i m a t e s o f S e c u r i t y R i s k P r e m i u m 16 4. D a t a 20 5. R e s u l t s 21 a) The No A r b i t r a g e P r o p e r t y 21 b) The A p p l i c a b i l i t y o f U s i n g cov(X , Z ) a s a R i s k M e a s u r e • 26 6. C o n c l u s i o n 30 7. B i b l i o g r a p h y 58 i v L i s t o f T a b l e s T a b l e Page 1 F - t e s t a n d C h i - s q u a r e t e s t s by u s i n g f o r J a n u a r y 1956 - December 1960 32 2 F - t e s t a n d C h i - s q u a r e t e s t s by u s i n g R f o r J a n u a r y 1956 - December 1960 33 3 F - t e s t a n d C h i - s q u a r e t e s t s by u s i n g R^ f o r J a n u a r y 1961 - J u n e 1968 . 34 4 F - t e s t a n d C h i - s q u a r e t e s t s by u s i n g R f o r J a n u a r y 1961 - J u n e 1968 ™ 35 5 30 s e c u r i t i e s s u b s a m p l e s s e l e c t e d f r o m t h e B r e n n a n & Thompson p a p e r f o r c o m p a r i s o n p u r p o s e . 36 6 A l t e r n a t i v e e s t i m a t e s o f m a r k e t k e r n e l , J a n u a r y 1956 - December 1960 37 7 A l t e r n a t i v e e s t i m a t e s o f m a r k e t k e r n e l , J a n u a r y 1961 - J u n e 1968 38 8 C o r r e l a t i o n m a t r i x o f k e r n e l s , J a n u a r y 1956 -December 196 0 39 9 C o r r e l a t i o n m a t r i x o f k e r n e l s , J a n u a r y 1961 -J u n e 1968 40 10 C o r r e l a t i o n m a t r i x o f k e r n e l s , J a n u a r y 1956 -J u n e 196 8 41 11 C o v a r i a n c e s b e t w e e n t h e 118 s e c u r i t i e s r i s k p r e m i u m a nd t h e CAPM k e r n e l Zm f o r t h e p e r i o d J a n u a r y 1956 - December 196 0 42 12 C o v a r i a n c e s b e t w e e n t h e 118 s e c u r i t i e s r i s k p r e m i u m a n d t h e CAPM k e r n e l Zm f o r t h e p e r i o d J a n u a r y 1956 - J u n e 1958 43 13 C o v a r i a n c e s b e t w e e n t h e 118 s e c u r i t i e s r i s k p r e m i u m a n d t h e CAPM k e r n e l Zm f o r t h e p e r i o d J u l y 1958 - December 1960 44 14 C o v a r i a n c e s b e t w e e n t h e 118 s e c u r i t i e s r i s k p r e m i u m a n d t h e CAPM k e r n e l Zm f o r t h e p e r i o d J a n u a r y 1961 - J u n e 196 8 . 45 V L i s t o f T a b l e s ( c o n t i n u e d )  T a b l e Page 15 C o v a r i a n c e s b e t w e e n t h e 118 s e c u r i t i e s r i s k p r e m i u m a n d t h e CAPM k e r n e l Zm f o r t h e p e r i o d J a n u a r y 1961 - S e p t e m b e r 1964 46 16 C o v a r i a n c e b e t w e e n t h e 118 s e c u r i t i e s r i s k p r e m i u m a n d t h e CAPM k e r n e l Zm f o r t h e p e r i o d O c t o b e r 1964 - J u n e 1968 47 2 17 R e g r e s s i o n c o e f f i c i e n t , R a n d t - s t a t i s t i c s o f t h e r e g r e s s i o n b e t w e e n c o v a r i a n c e s o f t h e 118 s e c u r i t i e s a n d Zm o v e r d i f f e r e n t p e r i o d s o f t i m e 4 8 2 18 R e g r e s s i o n c o e f f i c i e n t , R a n d t - s t a t i s t i c s o f t h e r e g r e s s i o n b e t w e e n c o v a r i a n c e s o f t h e 118 s e c u r i t i e s a n d Zm o v e r d i f f e r e n t p e r i o d s o f t i m e 49 19 C o v a r i a n c e s b e t w e e n t h e 118 s e c u r i t i e s r i s k p r e m i u m a n d t h e u n c o n s t r a i n e d k e r n e l Zu f o r t h e p e r i o d J a n u a r y 1956 - December 1960 50 2 0 C o v a r i a n c e s b e t w e e n t h e 118 s e c u r i t i e s r i s k p r e m i u m a n d t h e u n c o n s t r a i n e d k e r n e l Zu f o r t h e p e r i o d J a n u a r y 1956 - J u n e 195 8 51 21 C o v a r i a n c e s b e t w e e n t h e 118 s e c u r i t i e s r i s k p r e m i u m a n d t h e u n c o n s t r a i n e d k e r n e l Zu f o r t h e p e r i o d J u l y 1958 - December 1960 52 22 C o v a r i a n c e s b e t w e e n t h e 118 s e c u r i t i e s r i s k p r e m i u m a n d t h e u n c o n s t r a i n e d k e r n e l Zu f o r t h e p e r i o d J a n u a r y 1961 - J u n e 1968 53 23 C o v a r i a n c e s b e t w e e n t h e 118 s e c u r i t i e s r i s k p r e m i u m a n d t h e u n c o n s t r a i n e d k e r n e l Zu f o r t h e p e r i o d J a n u a r y 1961 - S e p t e m b e r 1964 . . . . 54 24 C o v a r i a n c e s b e t w e e n t h e 118 s e c u r i t i e s r i s k p r e m i u m a n d t h e u n c o n s t r a i n e d k e r n e l Zu f o r t h e p e r i o d O c t o b e r 1964 - J u n e 1968 55 2 25 R e g r e s s i o n c o e f f i c i e n t , R a n d t - s t a t i s t i c s o f t h e r e g r e s s i o n b e t w e e n c o v a r i a n c e s o f t h e 118 s e c u r i t i e s a n d Zu o v e r d i f f e r e n t p e r i o d s o f t i m e 5 6 v i L i s t o f T a b l e s ( c o n t i n u e d ) T a b l e Page 2 26 R e g r e s s i o n c o e f f i c i e n t , R a n d t - s t a t i s t i c s o f t h e r e g r e s s i o n b e t w e e n c o v a r i a n c e s o f t h e 118 s e c u r i t i e s a n d Zu o v e r d i f f e r e n t p e r i o d s o f t i m e 5 7 v i i A c k n o w l e d g e m e n t I w i s h t o t h a n k my M.Sc. C o m m i t t e e - Rex Thompson, P h i l i p p e J o r i o n a n d R o b e r t J o n e s - e s p e c i a l l y Rex Thompson, my c o m m i t t e e c h a i r m a n , f o r t h e i r h e l p f u l comments and s u g g e s t i o n s . N a t u r a l l y , a l l e r r o r s a r e my r e s p o n s i b i l i t y . 1 1. INTRODUCTION D u r i n g t h e p a s t t w e n t y y e a r s , a number o f m o d e l s f o r t h e d e t e r m i n a t i o n o f r i s k y a s s e t p r i c e s i n s e c u r i t y m a r k e t s h a v e b e e n p r o p o s e d , t h e most p r o m i n e n t o f w h i c h a r e t h e c a p i t a l a s s e t p r i c i n g m o d e l (CAPM) [8] a n d t h e a r b i t r a g e p r i c i n g m o d e l ( A P M ) , [ 7 ] . Amongst a l l t h e s e m o d e l s , t h e r e i s t h e common a s s u m p t i o n t h a t t h e s e c u r i t y m a r k e t i s i n e q u i l i b r i u m . When e q u i l i b r i u m i s c o m b i n e d w i t h t h e a s s u m p t i o n o f n o n -s a t i a t i o n ( i . e . t h e f i r s t d e r i v a t i v e o f t h e u t i l i t y f u n c t i o n i s g r e a t e r t h a n z e r o ) , t h e no a r b i t r a g e p r o p e r t y o f s e c u r i t y m a r k e t s i s o b t a i n e d . The no a r b i t r a g e p r o p e r t y s t a t e s t h a t i t i s n o t p o s s i b l e t o f o r m a p o r t f o l i o w i t h z e r o n e t i n v e s t m e n t w h i c h w i l l h a v e a n e g a t i v e r e t u r n w i t h z e r o p r o b a b i l i t y a n d a p o s i t i v e r e t u r n w i t h n o n - z e r o p r o b a b i l i t y . I t i s t h e r e f o r e a v e r y b a s i c p r o p e r t y t h a t h a s b e e n u s e d by a l l f i n a n c e m o d e l b u i l d e r s . I f t h i s p r o p e r t y i s u l t i m a t e l y p r o v e n t o be f a l s e , t h e n a l m o s t a l l o f t h e p r e s e n t a s s e t v a l u a t i o n m o d e l s w i l l h a v e t o be r e v i s e d . The p u r p o s e o f t h i s p a p e r i s t o p e r f o r m a p r e l i m i n a r y t e s t i n g o f t h e no a r b i t r a g e p r o p e r t y w i t h C a n a d i a n s e c u r i t y m a r k e t d a t a . T h i s t e s t i s b a s e d on U.S. k e r n e l s e s t i m a t e d i n B r e n n a n a nd Thompson [1983] a n d i s an e x t e n s i o n o f t h e i r t e s t i n g p r o c e d u r e t o t h e C a n a d i a n c a p i t a l m a r k e t . M o r e o v e r , an a l t e r n a t i v e r i s k m e a s u r e i s i n t r o d u c e d a n d a t e s t c o n c e r n -i n g i t s e x p l a n a t o r y a n d p r e d i c t i v e power o v e r t i m e i s p e r -2 f o r m e d . I n s e c t i o n t w o , t h e t h e o r e t i c a l m o d e l on w h i c h t h i s e m p i r i c a l t e s t i n g i s b a s e d i s d i s c u s s e d . I n s e c t i o n t h r e e , p a r t o f t h e B r e n n a n a n d Thompson m e t h o d o l o g y t o g e t h e r w i t h t h e e x t e n s i o n i s d e s c r i b e d . S e c t i o n f o u r d e s c r i b e s t h e d a t a u s e d i n p e r f o r m i n g t h i s e m p i r i c a l t e s t , a n d r e s u l t s a r e d i s c u s s e d i n s e c t i o n f i v e . L a s t l y , i n s e c t i o n s i x , c o n c l u -s i o n s c o n c e r n i n g t h e r e s u l t s o b t a i n e d i n p r e v i o u s s e c t i o n s a n d r e c o m m e n d a t i o n s f o r f u t u r e r e s e a r c h a r e p r e s e n t e d . 3 2. THE STATE PREFERENCE MODEL S e c u r i t i e s h a v e a t i m e d i m e n s i o n . The s e c u r i t i e s i n -v e s t m e n t d e c i s i o n s o f i n d i v i d u a l s a r e d e c i s i o n s r e g a r d i n g t h e t i m i n g o f c o n s u m p t i o n o v e r some f u t u r e t i m e i n t e r v a l . The p a s s a g e o f t i m e i n v o l v e s u n c e r t a i n t y a b o u t t h e f u t u r e , a n d h e n c e t h e f u t u r e v a l u e o f a s e c u r i t y i n v e s t m e n t . T h i s u n c e r t a i n t y c a n be r e p r e s e n t e d a s a v e c t o r o f p r o b a b l e p a y o f f s a t some f u t u r e d a t e , a n d an i n d i v i d u a l ' s p o r t f o l i o o f i n v e s t m e n t s i s a m a t r i x o f p r o b a b l e p a y o f f s on t h e d i f -f e r e n t s e c u r i t i e s c o m p r i s i n g t h e p o r t f o l i o . ( [ 5 ] , C h a p t e r 5 ) . I n t h e l a n g u a g e o f s t a t e p r e f e r e n c e , u n c e r t a i n t y t a k e s t h e f o r m o f n o t k n o w i n g w h a t t h e s t a t e o f t h e w o r l d w i l l be a t some f u t u r e d a t e . However, g i v e n t h e s t a t e , t h e p a y o f f o f t h e s e c u r i t y i s assumed t o be known w i t h c e r t a i n t y . T h u s , t o t h e i n v e s t o r , a s e c u r i t y i s a s e t o f c e r t a i n p a y o f f s , e a c h a s s o c i a t e d w i t h u n c e r t a i n s t a t e s o f t h e w o r l d . M o re f o r m a l l y , t h e s t a t e p r e f e r e n c e m o d e l l i s t s a l l s t a t e s i n n a t u r e t h a t a r e m u t u a l l y e x c l u s i v e a n d e x h a u s t i v e , a n d a s s i g n s a c a s h f l o w f o r e v e r y s t a t e . I t r e q u i r e s a com-p l e t e m a r k e t w h i c h r e q u i r e s t h a t t h e r e a r e a s many l i n e a r l y i n d e p e n d e n t s e c u r i t i e s a s t h e r e a r e s t a t e s o f n a t u r e . G i v e n a c o m p l e t e m a r k e t , a n i n d i v i d u a l c a n t h e o r e t i c a l l y r e d u c e t h e u n c e r t a i n t y a b o u t t h e v a l u e o f h i s f u t u r e w e a l t h t o z e r o . A l l t h a t r e m a i n s i s u n c e r t a i n t y a b o u t w h i c h s t a t e o f t h e w o r l d w i l l a c t u a l l y o c c u r . T h a t i s , by d i v i d i n g h i s 4 w e a l t h among a l l t h e a v a i l a b l e s e c u r i t i e s , t h e i n v e s t o r c o u l d , i f he c h o o s e s , c o n s t r u c t a p o r t f o l i o w h i c h w o u l d h a v e t h e same p a y o f f i n e v e r y s t a t e , e v e n t h o u g h t h e p a y o f f s o f i n d i v i d u a l s e c u r i t i e s v a r y o v e r s t a t e s . The a n a l y s i s o f t h e s e c u r i t y m a r k e t s w i t h i t s m o d e l i s f a c i l i t a t e d b y t h e c o n c e p t o f a p r i m i t i v e s e c u r i t y . A p r i m i -t i v e s e c u r i t y i s d e f i n e d a s a s e c u r i t y w h i c h p a y s a r e t u r n o f $1 i f a g i v e n s t a t e o c c u r s a n d n o t h i n g i f a n y o t h e r s t a t e o c c u r s . T h i s a l l o w s t h e l o g i c a l d e c o m p o s i t i o n o f m a r k e t s e c u r i t i e s i n t o p o r t f o l i o s o f p r i m i t i v e s e c u r i t i e s . T h u s , e v e r y r e a l s e c u r i t y may be c o n s i d e r e d a c o m b i n a t i o n o f v a r i o u s p r i m i t i v e s e c u r i t i e s . M a t h e m a t i c a l l y , i t c a n be r e p r e s e n t e d a s f o l l o w s ([5] C h a p t e r 5). P ± = I P (w) Y ± (w) i s I (1) w h e r e (w) = t h e r e t u r n o n s e c u r i t y i i n s t a t e w = t h e i n i t i a l p r i c e o f t h e s e c u r i t y p (w) = a s e t o f n o n - n e g a t i v e p r i m i t i v e s t a t e p r i c e s A d i r e c t o b s e r v a t i o n f r o m (1) i s t h a t i f i n e v e r y s t a t e w , t h e p a y o f f Y^ i s a l w a y s t h e same amount, t h e n P^ = £p(w) d e n o t e s t h e p a y o f f o f a r i s k l e s s s e c u r i t y a n d l/£p(w) i s a l w a y s one p l u s t h e r i s k l e s s r a t e o f r e t u r n : — - — = 1 + r f IP(W) 5 w h e r e r f i s t h e r i s k l e s s r a t e o f r e t u r n . N o t e a l s o t h a t £p(w) i s a l w a y s s m a l l e r t h a n u n i t y o w i n g t o t h e t i m e v a l u e o f money. I t h a s b e e n w e l l - r e c o g n i s e d t h a t t h i s s t a t e p r e f e r e n c e i s o f g r e a t v a l u e when a n a l y z i n g p r o b l e m s , b u t i s n o t s u b j e c t t o e m p i r i c a l t e s t i n g . B r e n n a n and Thompson, h o w e v e r , h a v e d e r i v e d a p r o c e d u r e f o r t r a n s f o r m i n g t h e s t a t e p r i m i t i v e p r i c e s i n t o an e m p i r i c a l l y t e s t a b l e f o r m . I t h a s b e e n shown t h a t p r i m i t i v e p a y o f f s c a n be c o m b i n e d t o d u p l i c a t e c o m p l e x s e c u r i t y p a y o f f s . By u s i n g t h i s r e s u l t , t h e no a r b i t r a g e p r o p e r t y i s e q u i v a l e n t t o t h e e x i s t e n c e o f a n o t n e c e s s a r i l y u n i q u e s e t o f n o n - n e g a t i v e p r i m i t i v e s e c u r i t y p r i c e s w h i c h i s c o n s i s t e n t w i t h t h e o b s e r v e d s e t o f c o m p l e x s e c u r i t y p r i c e s . I n l i g h t o f t h i s , t h e y t e s t e d f o r t h e e x i s t e n c e o f s u c h a s e t o f p r i m i t i v e s e c u r i t y p r i c e s . P a r t o f t h e i r m e t h o d o l o g y i s t h e s u b j e c t o f t h e n e x t s e c t i o n . 6 3. METHODOLOGY a) The A n a l y s i s I n s e c t i o n 2, i t i s shown t h a t t h e no a r b i t r a g e p r o p e r t y i m p l i e s t h e e x i s t e n c e o f a s e t o f n o n - n e g a t i v e p r i m i t i v e s t a t e p r i c e s p(w) s u c h t h a t e q u a t i o n (1) h o l d s . D i v i d i n g b o t h s i d e s o f (1) by y i e l d s : I p(w) R ± ( w ) = 1 i G I (2) wh e r e R.(w) = Y . ( w ) / P . o r one p l u s t h e r a t e o f r e t u r n o n s e c u r i t y i . D e f i n e Z (w) = p(w ) / r r(w) w h e r e rr (w) i s t h e p r o b a b i l i t y t h a t s t a t e w o c c u r s . Then e q u a t i o n (2) becomes: I TT(W) Z (w) R ± ( w ) = E [ Z ( w ) R ± ( w ) ] = 1 (3) wh e r e E [ ] d e n o t e s t h e e x p e c t a t i o n o p e r a t i o n , a n d Z ( « ) > 0. ( S i n c e p (w) a n d TT (w) a r e b o t h > 0.) To t e s t (3) e m p i r i c a l l y i t i s n e c e s s a r y t o make some a s s u m p t i o n s a b o u t t h e s t a t i o n a r i t y o f t h e r e t u r n s a n d he n c e o f t h e random v a r i a b l e s Z ( w ) . We know t h a t Z(w) c a n n o t be s t a t i o n a r y b e c a u s e E [ Z (w) ] = I p(w) = 1 / 1 + r f (4) and t h e r i s k l e s s i n t e r e s t r a t e v a r i e s o v e r t i m e . T h e r e f o r e , B r e n n a n a n d Thompson c o n s i d e r two a l t e r n a t i v e s p e c i f i c a t i o n s o f s t a t i o n a r i t y a n d u s e them t o d e r i v e t h e e m p i r i c a l c o u n t e r -p a r t o f ( 3 ) . One o f t h e s e s p e c i f i c a t i o n s w i l l be d e s c r i b e d 7 n e x t . F r o m (3) we know t h a t t h e a b s e n c e o f a r b i t r a g e o p p o r -t u n i t i e s i n p e r i o d t ( s i n c e (3) i s d e r i v e d f r o m ( 1 ) ) i m p l i e s t h e e x i s t e n c e o f a random v a r i a b l e Z t ( w ) > 0 s u c h t h a t E [ Z t ( w ) R ± t ( w ) ] = 1 1 6 I (5) E [ Z t ( w ) ] = V l + r f t (6) D e f i n e Z(w) by Z(w) = (1 + r f t ) Z t ( w ) (7) Then s u b s t i t u t i n g f o r Z^(w) i n (5) y i e l d s E [ Z ( w ) R ± t ( w ) ] = (1 + r f t ) (8) L e t R„. = 1 + r f t , t h e n (6) and (7) i m p l i e s E [ Z ( w ) ] = 1 (9) From (8) a n d ( 9 ) , we g e t E [ Z ( w ) R ± t ( w ) ] = E [ z ( w ) R p t ] (10) o r e q u i v a l e n t l y : E [ Z ( R ± t - R p t ) ] = 0 (11) E q u a t i o n (11) i s known a s t h e r i s k p r e m i u m s t a t i o n a r i t y a s s u m p t i o n w h i c h s t a t e s t h a t i f t h e d i s t r i b u t i o n o f e x c e s s r a t e o f r e t u r n ( R i t ~ R F t ^ """s s t a t i o n a r y f ° r t = l , . . . , T , t h e n t h e no a r b i t r a g e p r o p e r t y i m p l i e s t h e e x i s t e n c e o f a random v a r i a b l e Z > 0 s u c h t h a t e q u a t i o n (11) h o l d s a n d Z h a s a s t a t i o n a r y d i s t r i b u t i o n . M o r e o v e r , e q u a t i o n (11) i s 8 c o n v e n i e n t l y w r i t t e n a s : where E[Z x..] = 0 i e i , t e T (12) i t X i t " ( R i t " R F t } F u r t h e r m o r e , t h e f i r s t o r d e r c o n d i t i o n s o f t h e C a p i t a l A s s e t P r i c i n g M o d e l o f p e r i o d t c a n be w r i t t e n i n a s i m i l a r f o r m a t : E [ ( a t - R m t ) ( R i t - R F t ) ] = 0 (13) w h e r e R t i s t h e r a t e o f r e t u r n on t h e m a r k e t p o r t f o l i o a n d ( a t - R- m t) i s p r o p o r t i o n a l t o t h e m a r g i n a l u t i l i t y o f t h e r e p r e s e n t a t i v e i n v e s t o r w i t h q u a d r a t i c u t i l i t y f u n c t i o n s . A s s u m i n g r i s k p r e m i u m s t a t i o n a r i t y h o l d s and n e g l e c t i n g t h e c h a n g i n g c o m p o s i t i o n o f t h e m a r k e t p o r t f o l i o , (13) i m p l i e s t h a t t h e r e e x i s t s a c o n s t a n t , a , s u c h t h a t E [ ( a - ( R m t - R p t ) ) . ( R ± t - R p t ) ] = 0 (14) C o m p a r i n g (14) t o ( 1 1 ) , we s e e t h a t t h e CAPM i m p o s e s t h e c o n s t r a i n t t h a t random v a r i a b l e Z i s a l i n e a r f u n c t i o n o f t h e m a r k e t r i s k premium. b) The E s t i m a t i o n o f t h e M a r k e t K e r n e l ( Z(w)) I t f o l l o w s f r o m o u r a n a l y s i s t h a t t h e m a r k e t k e r n e l Z m u s t s a t i s f y t h e f o l l o w i n g t h r e e c o n d i t i o n s : 9 P r ( Z < 0) = 0 E [ Z ] - 1 (15) E[Z x. t] w h e r e X . , = R., - R„, i s t h e e x c e s s r e t u r n o n i t i t F t s e c u r i t y i i n p e r i o d t . C o n s i d e r a s a m p l e o f T m o n t h l y e x c e s s r e t u r n s on N s e c u r i t i e s w h e r e N > T. U n d e r t h e a s s u m p t i o n o f r i s k p r e m i u m s t a t i o n a r i t y , t h e e x c e s s r e t u r n v e c t o r f o r a g i v e n month, X t , c o r r e s p o n d s t o t h e r e a l i z a t i o n o f a p a r t i c u l a r s t a t e . M o r e -o v e r , u n d e r t h e n u l l h y p o t h e s i s o f t h e no a r b i t r a g e p r o p e r t y , t h e r e e x i s t s a p a r t i c u l a r v a l u e o f t h e m a r k e t k e r n e l c o r -r e s p o n d i n g t o t h a t s t a t e , w h i c h we d e n o t e b y Z^ _. S i n c e s u c c e s s i v e s t a t e s a r e i n d e p e n d e n t , s o a r e s u c c e s s i v e v a l u e s o f Z t . L e t e.. = Z, X... Then u n d e r t h e n u l l h y p o t h e s i s i t t i t J t r E [ e ^ t ] = 0 and i s s e r i a l l y i n d e p e n d e n t f o r e a c h s e c u r i t y i . I f t h e v a r i a n c e o f e., i s assumed t o be f i n i t e , t h e n t h e i t c o v a r i a n c e m a t r i x o f £., a c r o s s t h e s e c u r i t i e s c a n be w r i t t e n i t a s Q w h e r e aN -CTjN 2 a N < (16) I f we d e f i n e e. = £ e i f - ^ f o l l o w s t h a t e w i t h 1 t = l 10 e l e m e n t e^ i s a s y m p t o t i c a l l y n o r m a l l y d i s t r i b u t e d w i t h mean z e r o a n d c o v a r i a n c e m a t r i x £ = TO, . U n d e r t h e a s s u m p t i o n t h a t t h e c o n s t r a i n t i s n o t b i n d i n g , t h e m a r k e t k e r n e l c a n be e s t i m a t e d t h r o u g h a L a g r a n g i a n m i n i m i z a t i o n p r o b l e m . M i n L = e'£ e - A ( Z ' l - T) H G Z w h e r e Z = a t v e c t o r w i t h t y p i c a l e l e m e n t Z^ _ e = X ' Z x = L x x ' x 2 ' • • * X N ] T X N X i s a L a g r a n g e m u l t i p l i e r T The l i n e a r c o n s t r a i n t i m p o s e d i s £ Z. = T w h i c h f o l l o w s t = l d i r e c t l y f r o m E (Z) = 1. The q u a d r a t i c f o r m o n t h e L.H.S. i s n o t a f f e c t e d by t h e s c a l e o f t h e 1^. The l i n e a r c o n s t r a i n t s i m p l y s e t s t h e s c a l e t o a c o n v e n i e n t l e v e l . The s o l u t i o n f o r Z i s Z u = T ( 1 ' V - 1 ! ) - 1 V - 1 1 w h ere V = X I'1 X 1 Z^ = e s t i m a t e o f Z w i t h o u t r e g a r d t o t h e n o n - n e g a t i v i t y c o n s t r a i n t . By i g n o r i n g t h e n o n - n e g a t i v i t y c o n s t r a i n t , p r o b l e m s a s s o c i a t e d w i t h q u a d r a t i c p r o g r a m m i n g c a n be a v o i d e d . T h i s p r o c e d u r e i s s a t i s f i e d p r o v i d e d t h a t t h e e s t i m a t e d m a r k e t k e r n e l s do n o t v i o l a t e t h e n o n - n e g a t i v i t y c o n s t r a i n t . Un-11 f o r t u n a t e l y , i t t u r n s o u t t h a t 14 o f t h e 90 e s t i m a t e s o f t h e Z u a r e n e g a t i v e a n d t h u s q u a d r a t i c p r o g r a m m i n g h a s t o be e m p l o y e d a s w e l l . The r e s u l t s a r e d e n o t e d W^, t h e c o n s t r a i n e d e s t i m a t i o n o f t h e m a r k e t k e r n e l s s u c h t h a t t h e y must be g r e a t e r t h a n o r e q u a l t o z e r o . I n a d d i t i o n , t h e CAPM m a r k e t k e r n e l i s e s t i m a t e d f r o m e q u a t i o n ( 1 4 ) . D e f i n e X , = R . - R„. , t h e n (14) becomes: ^ mt mt F t (a - X .) X.. mt i t = 0 (18) P r e m u l t i p l y i n g (18) by h ^ , t h e f r a c t i o n o f t h e m a r k e t p o r t f o l i o a c c o u n t e d f o r by s e c u r i t y . i , a n d summing o v e r i , we g e t (a - X . ) x . mt mt = 0 (19) A l i t t l e a l g e b r a w i l l show t h a t a = E [X ] / E [X . ] mt ' mt T hen t h e m a r k e t k e r n e l u n d e r t h e CAPM Z i s g i v e n by up t o a s c a l a r m u l t i p l e by Z = E [ X ] / E [X ] - X . mt mt ' mt mt (20) o r "mt V a r t ( m ) + (E ( X ^ ) ) - X m t ( 2 1 ) U s i n g t h e a b o v e p r e c e d e n c e s d e s c r i b e d , Z , Z a n d Z ' ^ u c m a r e e s t i m a t e d by B r e n n a n a n d Thompson u s i n g t h e U.S. NYSE d a t a . W i t h t h e u s e o f t h e s e m a r k e t k e r n e l s and e q u a t i o n ( 1 1 ) , 12 t h e y show t h a t t h e no a r b i t r a g e p r o p e r t y i s n o t r e f u t e d i n t h e U.S. m a r k e t . N o t i c e t h a t t h e t e s t i n g u s i n g Z i s a c t u a l l y a j o i n t t e s t o f w h e t h e r t h e CAPM k e r n e l i s t h e m a r k e t k e r n e l a n d t h e no a r b i t r a g e p r o p e r t y , s i n c e t h e CAPM i m p o s e s t h e c o n s t r a i n t t h a t Z i s a l i n e a r f u n c t i o n o f t h e m a r k e t r i s k premium. T h e r e f o r e , r e j e c t i o n o f t h e h y p o t h e s i s c a n e i t h e r come f r o m r e j e c t i o n o f t h e CAPM k e r n e l a s t h e m a r k e t k e r n e l , o r f r o m r e j e c t i o n o f t h e no a r b i t r a g e p r o p e r t y , g i v e n t h e s t a t i o n a r i t y a s s u m p t i o n o f t h e r i s k p r e m i u m . c) The T e s t i n g o f t h e No A r b i t r a g e P r o p e r t y i n Canada T h i s p a p e r e x t e n d s t h e w o r k o f B r e n n a n a nd Thompson by a p p l y i n g t h e i r m o d e l t o t h e C a n a d i a n s e c u r i t y m a r k e t . T h i s i s p e r f o r m e d by u s i n g t h e i r t h r e e d i f f e r e n t e s t i m a t e d m a r k e t k e r n e l s , Z , Z a n d Z . I f t h e r e i s no s i g n i f i c a n t d i f -u c m ^ f e r e n c e b e t w e e n m a r k e t c o n d i t i o n s i n t h e U.S. a n d C a n a d a , i t i s e x p e c t e d t h a t t h e m a r k e t k e r n e l s c a n be u s e d i n Ca n a d a a n d t h e no a r b i t r a g e p r o p e r t y w i l l h o l d a s w e l l . However, i f t h e r e s u l t s p r o v e c o n t r a d i c t o r y , i t may be c a u s e d by d i f f e r e n c e s i n m a r k e t i m p e r f e c t i o n s b e t w e e n t h e two c o u n t r i e s ( e . g . t r a n s a c t i o n c o s t s ) , o r t a x s t r u c t u r e d i f -f e r e n c e s , a n d n o t n e c e s s a r i l y b e c a u s e o f t h e a r b i t r a g e o p p o r t u n i t i e s i n C a n a d a . B e f o r e m o v i n g o n t o t h e a c t u a l t e s t i n g p r o c e d u r e , a w o r d o f c a u t i o n m u s t be s t a t e d . The p r o c e d u r e t o be u s e d i g n o r e s e x c h a n g e r a t e f l u c t u a t i o n s b e t w e e n Canada a nd t h e 13 U.S. I t assumes t h a t t h e e x c h a n g e r a t e i s c o n s t a n t d u r i n g t h e p e r i o d o f i n t e r e s t . I n r e a l i t y , t h e a c t u a l r a t e f l u c t u -a t e s b e t w e e n 0.9 5 a n d 1.07 Can/U.S. T h i s f l u c t u a t i o n may c r e a t e a b i a s i n t h e e s t i m a t i o n o f b o t h t h e u n c o n s t r a i n e d and t h e CAPM k e r n e l a n d t h e r e f o r e t h e r e s u l t s o f t h e t e s t . H o w e v e r , s i n c e t h e c o v a r i a n c e b e t w e e n e x c h a n g e r a t e s and e x c e s s r e t u r n o f i n d i v i d u a l s t o c k s a r e s m a l l e m p i r i c a l l y , t h i s s i m p l i f i e d p r o c e d u r e s h o u l d n o t c r e a t e s e r i o u s p r o b l e m s . L e t = ~ R F t ^ f F r o m e q u a t i o n (11) , t h e e x p e c t e d v a l u e o f w i l l be z e r o . T h e s e E ^ ' S a r e e s t i -m a t e d w i t h t h e u s e o f t h e u n c o n s t r a i n e d Z a n d f o r c o n v e n i e n c e u a r e d e n o t e d e., u. S i m i l a r l y , e.^ a n d e. m a r e a l s o e s t i -l t J i t i t m a t e d w i t h t h e u s e o f t h e c o n s t r a i n e d m a r k e t k e r n e l (Z Q ) a n d t h e CAPM k e r n e l (Z ) . I n a d d i t i o n , t h e means o f a l l m t h r e e e ^ ' s o v e r t i m e a r e a l s o c o m p u t e d f o r e a c h o f t h e s e c u r i t i e s u s e d i n t h e d a t a s e t . They a r e d e n o t e d £ ^ U / e.° a n d e. m r e s p e c t i v e l y . 1 1 C J. A f t e r t h e ab o v e c o m p u t a t i o n , 30 s t o c k s a r e t a k e n f r o m t h e 15 0 s t o c k s a m p l e a t a t i m e and t h e f o l l o w i n g a r e c o m p u t e d : - u'„ -1 - u - c ' " -1 - c - m'" -1 - m e. Q, z. , e. 0, e. , e. e. (22) I U l i e 1 1 m I ~ -1 ~ -1 ~ -1 wh e r e Q , 0, and Q a r e t h e i n v e r s e o f t h e u c m u c e s t i m a t e d v a r i a n c e - c o v a r i a n c e m a t r i x o f t h e , a n d m e. . I 14 T h i r t y s t o c k s a r e u s e d e a c h t i m e b e c a u s e o n l y a l i m i t e d p e r i o d o f m a r k e t k e r n e l s i s a v a i l a b l e . I f t h e number o f s t o c k s u s e d i s g r e a t e r t h a n one h a l f o f t h e t i m e p e r i o d s when t h e s e m a r k e t k e r n e l s a r e a v a i l a b l e , t h e n t h e e s t i m a t e d v a r i a n c e - c o v a r i a n c e m a t r i c e s become u n s t a b l e , r e s u l t i n g i n l a r g e e r r o r s i n t h e e s t i m a t e s . By m u l t i p l y i n g e a c h o f t h e e x p r e s s i o n s i n (22) by T , w h e r e T i s t h e number o f p e r i o d s i n v o l v e d , t h e r e s u l t i n g s t a t i s t i c s w i l l f o l l o w a c h i - s q u a r e d i s t r i b u t i o n w i t h 30 d e g r e e s o f f r e e d o m . The o u t c o m e c a n t h e n be c o m p a r e d w i t h t h e f i g u r e i n t h e c h i - s q u a r e t a b l e t o d e t e r m i n e w h e t h e r t h e no a r b i t r a g e p r o p e r t y h o l d s , e v e n w i t h t h e u s e o f U.S. m a r k e t k e r n e l s . T h i s t e s t w i l l s u b s e q u e n t l y be r e f e r r e d t o a s t h e c h i - s q u a r e t e s t . The c h i - s q u a r e t e s t , h o w e v e r , r e l i e s on 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 o f t h e s t a t i s t i c s . T h i s i s t e n u o u s b e c a u s e t h e s m a l l s a m p l e p r o p e r t i e s c o u l d d i f f e r c o n s i d e r a b l y . A more c o n s e r v a t i v e a p p r o a c h i s t o compare t h e s t a t i s t i c s t o a F d i s t r i b u t i o n i n t h e f o l l o w i n g way: U n d e r t h e a s s u m p t i o n t h a t i s j o i n t l y n o r m a l , t h e s t a t i s t i c , w h i l e h a v i n g an a s y m p t o t i c x? d i s t r i b u t i o n , u h a s an e x a c t s m a l l s a m p l e d i s t r i b u t i o n as w e l l . I t c a n be shown t h a t T ~ J c ~ F J ( T - 1) D J , T - J 15 where S = the s t a t i s t i c T = number of time series observations J = number of s e c u r i t i e s F j T _ j = F - s t a t i s t i c s with J, T-J degrees of ' freedom Another extension of t h i s paper comes from equation (10). It i s a c t u a l l y true that the following w i l l hold: E (Z " R ± t) = E ( Z R j t) V i , j (23) and R F£ i s simply a special case of Rj^« Therefore, an addi-t i o n a l proxy i s used for Rj t» namely R m t f the return on the Canadian market p o r t f o l i o . Thus equation (11) becomes: Z (R. - R ,) =0 (24) i t m t j A s i m i l a r estimation and computation procedure w i l l be used to determine i f equation (24) holds. I f i t holds, i t can then be concluded that equation (2 3) holds and equation (11) becomes more general since i t w i l l no longer be constrained to R_.. Ft F i n a l l y , the Canadian Z , and Z . are also estimated, J ut mt The former i s estimated from the Lagrangian minimization pro-cedure while the l a t t e r from the use of equation (21). The Z £ 1 s are then transformed such that they have a mean of unity consistent with the other market kernels. 16 U s i n g t h e ab o v e m e t h o d o l o g y a n d p e r f o r m i n g a s i m i l a r c o m p u t a t i o n w i t h t h e u s e o f t h e C a n a d i a n Z ^ and Z ^ , i t c a n be shown w h e t h e r t h e no a r b i t r a g e p r o p e r t y h o l d s i n C a n a d a a n d w h e t h e r t h e CAPM k e r n e l i s i n f a c t t h e t r u e m a r k e t k e r n e l . d) A l t e r n a t i v e E s t i m a t e s o f S e c u r i t y R i s k P r e m i u m E q u a t i o n (12) i m p l i e s t h a t : E(i) = ~ C O V ( X ' Z ) (25) E (Z) U s i n g t h e c o n v e n t i o n t h a t E (Z) = 1, e q u a t i o n (25) becomes E ( X ) = - c o v (X ,Z ) (26) wh e r e E ( X ) = E ( R . ) - R i s t h e i n d i v i d u a l s t o c k r i s k i t i t p r e m i u m . E q u a t i o n (26) o f f e r s a new m e a s u r e o f r i s k p r e m i u m w i t h t h e u s e o f t h e no a r b i t r a g e p r o p e r t y . I n t h i s s e n s e , i t r e q u i r e s a much w e a k e r s e t o f a s s u m p t i o n s a s c o m p a r e d t o t h e CAPM b e t a . I f t h e e m p i r i c a l e v i d e n c e o f t h i s new r i s k m e a s u r e i s a s g o o d a s t h e CAPM b e t a , t h e n t h i s r i s k m e a s u r e w i l l be c o m p a r a b l e t o t h e CAPM b e t a as a u s e f u l r i s k m e a s u r e . M o r e o v e r , CAPM e s t i m a t e s o f t h e r i s k p r e m i u m c a n a l s o be o b t a i n e d b y u s i n g t h e CAPM k e r n e l e s t i m a t e i n e q u a t i o n ( 2 6 ) . As t h i s new r i s k p r e m i a m e a s u r e o n l y r e q u i r e s t h e a s s u m p t i o n o f t h e no a r b i t r a g e p r o p e r t y , i t i s e s s e n t i a l t o 17 t e s t f o r i t s e x p l a n a t o r y and p r e d i c t i v e power over time, and to determine i f the r e l a x a t i o n o f the r e s t r i c t i v e assumptions o f CAPM have any d e t r i m e n t a l e f f e c t s on the new r i s k measure. I f the answer to the above q u e s t i o n i s n e g a t i v e , one can conclude t h a t a l o t o f the r e s t r i c t i v e assumptions i n the CAPM can be r e l a x e d . The t e s t i n g procedure i s as f o l l o w s : Only 118 firms i n the 1956-1960 and 1961-1968 subsamples have o b s e r v a t i o n s across the e n t i r e p e r i o d o f time. These fi r m s were s e l e c t e d f o r the t e s t i n g procedure. A r e g r e s s i o n i s run between the market k e r n e l and each i n d i v i d u a l f i r m ' s excess r e t u r n . Since the CAPM k e r n e l (Z ) i s h i g h l y n e g a t i v e l y c o r r e l a t e d w i t h the r e t u r n on the market measure ( R m ) , i t i s expected t h a t the c o r r e l a t i o n 2 of d e t e r m i n a t i o n (R ) between each of these r e g r e s s i o n s w i l l be h i g h . Moreover, i t i s t h e o r e t i c a l l y sound t h a t the un-2 c o n s t r a i n e d k e r n e l (? u) has a h i g h R when re g r e s s e d a g a i n s t the excess r e t u r n o f the i n d i v i d u a l f i r ms as w e l l . The second step c a l c u l a t e s c o v a r i a n c e s between the excess r e t u r n and the market k e r n e l f o r each i n d i v i d u a l f i r m f o r d i f f e r e n t s u b p e r i o d s . T h i s c o v a r i a n c e , o f course, i s the r i s k premia measure. I f the CAPM k e r n e l (Z ) i s used c m as a proxy f o r the market k e r n e l , the r e s u l t i s e s s e n t i a l l y a m u l t i p l e o f the CAPM beta. On the o t h e r hand, i f the un-c o n s t r a i n e d k e r n e l (Z ) i s used, the covar i a n c e i s the new u ' r i s k premia measure. 18 B o t h p r o x i e s a r e u s e d i n t h e two d i f f e r e n t r i s k p r e m i u m c a l c u l a t i o n s . The r e s u l t i n g r i s k p remiums a r e u s e d t o p e r -f o r m t i m e s e r i e s r e g r e s s i o n t o d e t e r m i n e i f t h e i r e x p l a n a t o r y 2 a n d p r e d i c t i v e p o w e r s a r e h i g h o v e r t i m e . The R i n t h e s e r e g r e s s i o n s w i l l show t h e e x p l a n a t o r y power o f t h e r i s k m e a s u r e s w h e r e a s t h e s i g n i f i c a n c e o f t h e t - s t a t i s t i c s w i l l show t h e i r p r e d i c t i v e p o w e r . P a s t h i s t o r y o f e m p i r i c a l r e s e a r c h e s h a s shown t h a t t h e CAPM b e t a h a s b o t h a h i g h e x p l a n a t o r y a n d p r e d i c t i v e p o w e r . E l t o n a n d G r u b e r [ 1 9 8 1 ] , f o r e x a m p l e , r e c o r d e d t h e a s s o c i a t i o n o f b e t a s b e t w e e n two a d j a c e n t t i m e p e r i o d s . T h e y l o o k e d a t B l u m e ' s [3] r e p r e s e n t a t i v e r e s u l t s w h e r e he c o m p u t e d b e t a s u s i n g t i m e s e r i e s r e g r e s s i o n s o n m o n t h l y d a t a f o r n o n - o v e r l a p p i n g s e v e n y e a r p e r i o d s . Blume g e n e r a t e d b e t a s on s i n g l e s t o c k p o r t f o l i o s , 2 s t o c k p o r t f o l i o s , 4 s t o c k p o r t f o l i o s a n d so f o r t h up t o 5 0 s t o c k p o r t f o l i o s . F o r e a c h s i z e p o r t f o l i o , he e x a m i n e d how h i g h l y c o r r e l a t e d t h e b e t a s f r o m one p e r i o d w e r e w i t h t h e b e t a s f o r a s e c o n d p e r i o d . The p e r i o d s o f i n t e r e s t a r e J u l y 1954 - J u n e 1961 a n d J u l y 1961 - J u n e 1968, w h i c h a r e f a i r l y c l o s e t o t h e p e r i o d s e x a m i n e d i n t h i s p a p e r . M o r e o v e r , 118 s i n g l e s t o c k p o r t f o l i o s a r e e s s e n t i a l l y f o r m e d i n t h i s p a p e r and t h e r e -f o r e t h e r e s u l t s a r e d i r e c t l y c o m p a r a b l e t o t h e c o e f f i c i e n t 2 o f d e t e r m i n a t i o n (R ) o f 0.36 f r o m t h e t i m e s e r i e s r e g r e s s i o n o f t h e s i n g l e s t o c k p o r t f o l i o b e t a s r e c o r d e d i n E l t o n and 2 G r u b e r . One w o u l d t h e r e f o r e e x p e c t a s i m i l a r l y h i g h R 19 o v e r t i m e when t h e CAPM r i s k p r e m i a i s r e g r e s s e d o v e r t i m e , s i n c e t h i s r i s k p r e m i a i s a m u l t i p l e o f t h e CAPM b e t a . No p r i o r e x p e c t a t i o n c a n be f o r m e d b e f o r e t h e t i m e s e r i e s r e g r e s s i o n o f t h e no a r b i t r a g e r i s k p r e m i a . I f t h e 2 r e s u l t i n g R i s h i g h a n d t h e t - s t a t i s t i c i s s i g n i f i c a n t , o n e may c o n c l u d e t h a t t h e no a r b i t r a g e r i s k p r e m i a h a s h i g h e x p l a n a t o r y a n d p r e d i c t i v e power o v e r t i m e . I f t h e r e s u l t i s o t h e r w i s e , t h e no a r b i t r a g e r i s k p r e m i a w i l l n o t be a u s e f u l r i s k m e a s u r e a n d t h e r e f o r e one s h o u l d c o n t i n u e t o r e l y o n t h e CAPM b e t a as t h e r e l e v a n t r i s k m e a s u r e . 20 r 4. DATA The d a t a comes f r o m t h e CAPM t a p e f i l e w h i c h r e c o r d s d a t a o f m a r k e t r e t u r n , r i s k f r e e r a t e and d i v i d e n d a d j u s t e d r e t u r n s o f 391 i n d i v i d u a l s t o c k s f r o m t h e p e r i o d J a n u a r y 1950 t o December 1980. T h e s e r e t u r n d a t a w e r e c o m p u t e d u s i n g t h e Wood Gundy 1981 t a p e w h i c h r e c o r d e d t h e p r i c e s o f t h e 391 s t o c k s f o r t h i s p e r i o d [ 5 ] . The p e r i o d o f i n t e r e s t i n t h i s s t u d y i s f r o m J a n u a r y 1956 t o J u n e 1968. T h i s p e r i o d i s f u r t h e r d i v i d e d i n t o two s u b p e r i o d s - J a n u a r y 1956 t o December 1960 a n d J a n u a r y 1961 t o J u n e 1968. T h i s d i v i s i o n was t o t a l l y a r b i t r a r y and i t s p u r p o s e was t o d e t e r m i n e i f t h e r e s u l t s w e r e c o n s i s t e n t o v e r t i m e . Of t h e 391 s t o c k s , o n l y 121 h a v e a c t u a l r e t u r n s r e -c o r d e d i n t h e f i r s t s u b p e r i o d a n d 15 7 h a v e a c t u a l r e t u r 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 c o n v e n i e n c e , o n l y t h e f i r s t 120 s t o c k s w e r e u s e d i n t h e f i r s t s u b p e r i o d a n d 150 s t o c k s i n t h e s e c o n d p e r i o d . The r i s k f r e e r a t e comes f r o m t h e C a n a d i a n 90 d a y s T r e a s u r y B i l l r a t e f o r t h e same p e r i o d . F i n a l l y , t h e U.S. m a r k e t k e r n e l f i g u r e s f o r 1961-1968 w e r e d e r i v e d f r o m t h e e s t i m a t e s o f B r e n n a n and Thompson, and t h e C a n a d i a n u n c o n s t r a i n e d m a r k e t k e r n e l was e s t i m a t e d by Thompson. A l l U.S. s e c u r i t y r e t u r n d a t a w e r e t a k e n f r o m CRSP [ 4 ] . 21 5. RESULTS a) The No Arbitrage Property Before going into actual d e t a i l s of the chi-square test and the F-test, a comparison was done between the s t a t i s t i c s of Brennan and Thompson [1983] and those of t h i s paper. The s i m i l a r i t y between the two papers indicates that t h i s paper supports the res u l t s of Brennan and Thompson. Table 5 shows some of the representative r e s u l t s from Brennan and Thompson [1983] . The s t a t i s t i c s computed are based on 30 security subsamples with 30 degrees of freedom. These s t a t i s t i c s were chosen primarily for comparison pur-poses since a sim i l a r number of s e c u r i t i e s are used i n the subsample o f - t h i s paper. The f i r s t column of table 5 i s within sample s t a t i s t i c s with the use of the CAPM kernel. This implies that the CAPM kernel estimates and the excess return i n equation (11) comes from the same set of data over the same period of time. In the Brennan and Thompson paper, the set of data i s U.S. common stock returns. An exact comparison can therefore be drawn between t h e i r set of s t a t i s t i c s and the set of s t a t i s t i c s i n tables 1 and 3 under the column of Can Zm. These s t a t i s t i c s are d i r e c t l y comparable to Brennan and Thompson because they are e s t i -mated with the use of the CAPM kernel. Moreover, the CAPM kernel estimate i t s e l f and the excess return i n equation (11) 22 comes f r o m a s i m i l a r s e t o f C a n a d i a n common s t o c k r e t u r n s f o r t h e same p e r i o d o f t i m e . The s e c o n d c o l u m n o f t a b l e 5 i s o u t - o f - s a m p l e e s t i m a t e s f r o m B r e n n a n a n d Thompson w i t h t h e u s e o f t h e CAPM k e r n e l . T h i s c o l u m n i s c o m p a r a b l e t o t a b l e s 1 and 3 u n d e r t h e c o l u m n o f Zm, s i n c e t h e u s e o f U.S. Zm, e s t i m a t e d w i t h t h e U.S. m a r k e t d a t a t o t e s t t h e no a r b i t r a g e p r o p e r t y i n C a n a d a , i s an o u t - o f - s a m p l e t e s t . A b r i e f e x a m i n a t i o n o f t h e r e l e v a n t c o l u m n s r e v e a l s t h a t t h e m a g n i t u d e o f t h e s t a t i s t i c s i n b o t h c a s e s i s q u i t e s i m i l a r a n d t h u s one c a n c o n c l u d e t h a t t h i s p a p e r s u p p o r t s t h e f i n d i n g s o*f B r e n n a n a n d Thompson [1983] . P r i o r t o d r a w i n g a n y c o n c l u s i o n s as t o w h e t h e r t h e no a r b i t r a g e p r o p e r t y s h o u l d be r e j e c t e d i n C a n a d a , one must l o o k a t t h e i m p l i c a t i o n s o f d i f f e r e n t d i s t r i b u t i o n a s s u m p t i o n s o f t h e e t ° t h e r e j e c t i o n r e g i o n s . A s m e n t i o n e d e a r l i e r , r e l i a n c e o n 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 o f t h e s t a t i s t i c s i s d a n g e r o u s b e c a u s e t h e r e i s no g u a r a n t e e t h a t t h e s m a l l s a m p l e w i l l f o l l o w t h e d i s t r i b u t i o n . As a r e s u l t , r e l y i n g s o l e l y o n t h e c h i - s q u a r e r e j e c t i o n r e g i o n w i t h 30 d e g r e e s o f f r e e d o m i s n o t a v e r y g o o d p r a c t i c e . On t h e o t h e r h a n d , i f i s assumed t o be j o i n t l y n o r m a l , a more c o n s e r v a t i v e a p p r o a c h c a n be u s e d t o compare t h e r e s u l t i n g s t a t i s t i c s t o a F - d i s t r i b u t i o n . T h i s p u t s t h e 5% r e j e c t i o n r e g i o n f o r t h e 90 p e r i o d c a s e a t 23 F J , T-J x JCT-1) T-J 89 = 1.65 x ^ = 73.425 and the 5% r e j e c t i o n r e g i o n f o r the 60 p e r i o d case at F x J ( T - l ) 1.84 x 59 108.56 J , T - J T-J An attempt t o determine the d i s t r i b u t i o n of u s i n g one of the f i r m s i n the 150 f i r m sample with 90 p e r i o d s i s per-formed. The f i r m i s drawn randomly from the sample where has a l r e a d y been c a l c u l a t e d . The mean and v a r i a n c e of e^ _^ of t h i s f i r m i i s then computed. Using the mean and v a r i a n c e of t h i s f i r m i , a new data s e t wi t h 90 o b s e r v a t i o n s having the same mean and v a r i a n c e and with the n o r m a l i t y d i s -t r i b u t i o n i s generated. A comparison between the two s e t s of data i s then performed and the Kolmogorov-Smirnov s t a t i s t i c s i s computed. T h i s s t a t i s t i c r e v e a l s the l a r g e s t d i f f e r e n c e be-tween the two sample cumulative d i s t r i b u t i o n f u n c t i o n s . For t h i s p a r t i c u l a r f i r m , the Kolmogorov-Smirnov s t a t i s t i c i s com-puted t o be 0.25 [1]. Since t h i s , i s w e l l above the 5% r e j e c t i o n r e g i o n o f 0.15, the n o r m a l i t y assumption must be r e j e c t e d . A f u r t h e r examination o f the histogram of t h i s sample f i r m shows t h a t the d i s t r i b u t i o n o f i s f a t - t a i l e d , i m p l y i n g t h a t there are more o b s e r v a t i o n s i n the r e j e c t i o n r e g i o n than would have been under the n o r m a l i t y assumption. T h i s i m p l i e s t h a t the s i g n i f i c a n c e l e v e l should be h i g h e r than t h a t i m p l i e d by the F-t e s t . In ot h e r words, the 5% r e j e c t i o n r e g i o n should be 24 w e l l a bove 73.425 i n t h e 90 p e r i o d c a s e a n d 108.56 i n t h e 60 p e r i o d c a s e . A l l t h e s e e x t r a p o l a t i o n s , o f c o u r s e , d e p e n d on t h e a s s u m p t i o n t h a t t h e f i r m c h o s e n i s a r e p r e s e n t a t i v e member o f t h e o t h e r f i r m s i n t h e s a m p l e . Armed w i t h t h e above i n s i g h t , one c a n now d e t e r m i n e w h e t h e r o r n o t t h e no a r b i t r a g e p r o p e r t y s h o u l d be r e j e c t e d i n C a n a d a . I n many c a s e s , t h e no a r b i t r a g e p r o p e r t y i s r e -j e c t e d u n d e r t h e c h i - s q u a r e r e j e c t i o n r e g i o n o f 43.77. T h i s r e j e c t i o n r e g i o n , h o w e v e r , s h o u l d n o t be t a k e n t o o s e r i o u s l y , b a s e d on t h e r e a s o n s d e s c r i b e d a b o v e . I n s t e a d , one s h o u l d be more i n c l i n e d t o u s e t h e F - t e s t r e j e c t i o n r e g i o n . V i e w e d i n t h i s l i g h t , o n l y one 30 s t o c k s u b s a m p l e w i t h t h e u s e o f t h e u n c o n s t r a i n e d U.S. k e r n e l (Zu) c a n be r e j e c t e d i n one t i m e p e r i o d ( 1 9 6 1 - 1 9 6 8 ) . No o t h e r s c a n be r e j e c t e d i n a n y t i m e p e r i o d f o r any s u b s a m p l e . I f t h i s a r g u m e n t i s t a k e n t o t h e e x t r e m e , g i v e n t h e a s s u m p t i o n t h a t t h e e x a m i n e d f i r m above i s r e p r e s e n t a t i v e o f a l l f i r m s i n t h e s a m p l e i n t e r m s o f d i s t r i b u t i o n , one c a n c o n c l u d e t h a t i n none o f t h e c a s e s s h o u l d t h e no a r b i -t r a g e p r o p e r t y be r e j e c t e d . Of a l l t h e p r o x i e s f o r t h e m a r k e t k e r n e l u s e d , t h e C a n a d i a n u n c o n s t r a i n e d k e r n e l (Can Zu) seems t o be t h e b e s t p r o x y f o r t h e C a n a d i a n m a r k e t k e r n e l , f o l l o w e d by t h e C a n a d i a n CAPM k e r n e l (Can Zm) and t h e U.S. CAPM k e r n e l (Zm), t h e U.S. c o n s t r a i n e d k e r n e l (Zc) and l a s t l y t h e U.S. u n -25 c o n s t r a i n e d k e r n e l (Zu). The c r i t e r i a used t o judge t h e i r r e l a t i v e a t t r a c t i v e n e s s as being the best proxy i s the t o t a l of the s t a t i s t i c s . The s m a l l e r the t o t a l , the b e t t e r a proxy i t i s f o r the Canadian market k e r n e l . These s t a t i s t i c s are recorded i n t a b l e s 1 t o 4, as w e l l as the d i f f e r e n t p r o x i e s which are estimated i n t h i s paper f o r Canadian market k e r n e l s from January 1956 t o June 1968. They are presented i n t a b l e s 6 and 7. To complete the a n a l y s i s , the c o r r e l a t i o n m a t r i c e of d i f f e r e n t p r o x i e s over the e n t i r e p e r i o d and both sub-pe r i o d s are r e p o r t e d i n t a b l e s 8 t o 10. Some a d d i t i o n a l i n s i g h t s can be drawn from these s t a t i s -t i c s . For example, although one can conclude t h a t the Canadian market i s d i f f e r e n t from the U.S. market i n terms o f market i m p e r f e c t i o n s , the d i f f e r e n c e i s not s u f f i c i e n t f o r i n d i v i d u a l s t o make a r b i t r a g e p r o f i t s across the border. T h i s o b s e r v a t i o n can be confirmed by l o o k i n g a t the s t a t i s t i c s recorded i n t a b l e s 1 t o 4. Although the s t a t i s t i c s are markedly g r e a t e r i n magnitude when u s i n g the U.S. p r o x i e s f o r the Canadian market k e r n e l and thus show evidence of a d i f f e r e n c e i n market c o n d i t i o n s , the no a r b i t r a g e p r o p e r t y i s not r e j e c t e d once a more a p p r o p r i a t e 5% r e j e c t i o n r e g i o n ~\ i s assumed. Another i n t e r e s t i n g o b s e r v a t i o n i s t h a t although the Canadian CAPM k e r n e l i s not the bes t proxy f o r the Canadian market k e r n e l , the in h e r e n t assumption t h a t there i s no a r b i t r a g e o p p o r t u n i t y under the CAPM cannot be r e j e c t e d as the t e s t s t a t i s t i c s are w e l l w i t h i n the F - t e s t 5% r e j e c t i o n 26 r e g i o n . F i n a l l y , t h e t e s t i n g o f e q u a t i o n (23) a s a more g e n e r a l f o r m o f e q u a t i o n (11) c a n n o t be r e j e c t e d , b a s e d o n t h e t e s t r e s u l t s i n t a b l e s 2 a n d 4. A l t h o u g h t h e s t a t i s t i c s v a r y c o n s i d e r a b l y i n m a g n i t u d e , when e q u a t i o n (24) i s u s e d i n p l a c e o f e q u a t i o n ( 1 1 ) , u n d e r no s u b s a m p l e o v e r b o t h t i m e p e r i o d s i s t h e r e a c a s e w here t h e s t a t i s t i c s l i e o u t s i d e t h e 5% r e j e c t i o n r e g i o n a s d e f i n e d b y t h e F - t e s t . b) The A p p l i c a b i l i t y o f U s i n g C o v ( X , Z ) a s a R i s k M e a s u r e The C a n a d i a n CAPM k e r n e l (Zm) i s t h e f i r s t k e r n e l u s e d i n t h e r i s k m e a s u r e o f C o v ( X , Z m ) . B e f o r e t h i s r i s k p r e m i a i s c a l c u l a t e d , r e g r e s s i o n s b e t w e e n t h e i n d i v i d u a l s t o c k s ' e x c e s s r e t u r n a n d t h e CAPM k e r n e l a r e p e r f o r m e d . As t h e CAPM k e r n e l i s h i g h l y n e g a t i v e l y c o r r e l a t e d w i t h t h e r e t u r n o n t h e m a r k e t (Rm) a n d t h e Rm m e a s u r e i s h i g h l y c o r r e l a t e d w i t h e a c h i n d i v i d u a l s t o c k s ' e x c e s s r e t u r n , one w o u l d e x p e c t a h i g h n e g a t i v e c o r r e l a t i o n b e t w e e n t h e CAPM k e r n e l a n d t h e s t o c k s ' e x c e s s r e t u r n . The r e s u l t s o f t h e r e g r e s s i o n s a r e a s e x p e c t e d . The s l o p e c o e f f i c i e n t o f t h e r e g r e s s i o n s a r e h i g h l y n e g a t i v e l y s i g n i f i c a n t o v e r t i m e i n a l m o s t a l l o f t h e c a s e s . The a v e r a g e 2 r e g r e s s i o n R i s a r o u n d 0.35. T h i s p r o c e d u r e i s p e r f o r m e d t o v e r i f y t h a t t h e e x c e s s r e t u r n s o f i n d i v i d u a l s t o c k s a n d t h e CAPM k e r n e l d o , i n f a c t , come f r o m t h e same p e r i o d o f t i m e . 27 The c o v a r i a n c e b e t w e e n t h e i n d i v i d u a l s e c u r i t i e s ' e x c e s s r e t u r n a n d t h e CAPM k e r n e l ( i . e . c o v ( X , Z m ) ) i s t h e n c a l c u l a t e d f o r d i f f e r e n t s u b p e r i o d s o f t i m e . T h e y a r e shown i n t a b l e s 11 t o 1 6 . N o t i c e t h a t t h e s e c o v a r i a n c e s a r e i n f a c t m u l t i p l e s o f t h e CAPM b e t a m e a s u r e . V i e w e d i n t h i s l i g h t , one w o u l d e x p e c t t h a t when t h e s e c o v a r i a n c e s a r e r e g r e s s e d a c r o s s t i m e , 2 t h e c o e f f i c i e n t o f d e t e r m i n a t i o n (R ) w o u l d be f a i r l y h i g h a n d c o m p a r a b l e t o t h e r e s u l t s o f B l u m e , a s r e c o r d e d i n E l t o n 2 and G r u b e r . H o w e v e r , i t t u r n s o u t t h a t R i s c o n s i s t e n t l y l o w a n d t h e t - s t a t i s t i c s a r e o n l y s i g n i f i c a n t when t h e r e g r e s -s i o n s a r e p e r f o r m e d f o r a d j a c e n t t i m e p e r i o d s o v e r a s h o r t i n t e r v a l . As a w h o l e , t h e e x p l a n a t o r y power o f t h e r e g r e s -s i o n o v e r t h e e n t i r e p e r i o d i s c l o s e t o z e r o a n d t h e t - s t a t i s t i c i s i n s i g n i f i c a n t . T h e s e r e s u l t s a r e r e p o r t e d i n T a b l e 1 7. I t a p p e a r s t h a t e i t h e r s o m e t h i n g p e c u l i a r h a s h a p p e n e d t o t h e d a t a s e t w h i c h b i a s e s t h e s e r e g r e s s i o n s o r t h a t t h e CAPM b e t a i s n o t a u s e f u l r i s k m e a s u r e o v e r t h i s d a t a s e t o f s e c u r i t y r e t u r n s . F u r t h e r e x a m i n a t i o n o f t h e d a t a s e t r e v e a l s t h a t o u t o f t h e 118 s e c u r i t i e s , t h e r e a r e two p e c u l i a r s t o c k s , L a k e D u f a u l t a n d N o r t h g a t e E x p l o r a t i o n ( o b s e r v a t i o n s 9 and 1 0 ) , w h i c h h a v e e x c e s s r e t u r n s o f 1100% and 250% r e s p e c t i v e l y . S i n c e t h e o b s e r v a t i o n s a r e s e v e r a l s t a n d a r d d e v i a t i o n s f r o m 2 t h e mean, t h e r e g r e s s i o n R a n d t - s t a t i s t i c s a r e g r e a t l y a f f e c t e d . 28 A s e c o n d r e g r e s s i o n b e t w e e n t h e c o v a r i a n c e s o v e r t i m e i s a g a i n p e r f o r m e d , t h i s t i m e w i t h o u t t h e two o u t l y i n g o b s e r v a -t i o n s w h i c h h a v e p h e n o m e n a l e x c e s s r e t u r n s . The r e s u l t s 2 a r e shown i n t a b l e 18. N o t i c e t h a t t h e R h a s m a r k e d l y i m -p r o v e d and a l l t - s t a t i s t i c s a r e s i g n i f i c a n t o v e r t i m e . O v e r 2 t h e w h o l e i n t e r v a l , t h e R i s 0.226 and t h e t - s t a t i s t i c i s h i g h l y s i g n i f i c a n t . One c a n t h e r e f o r e c o n c l u d e t h a t a l t h o u g h 2 t h e R i s n o t a s h i g h a s t h a t o f Blume o v e r a s i m i l a r t i m e i n t e r v a l , t h e CAPM b e t a i s s t i l l a u s e f u l r i s k m e a s u r e f o r t h e C a n a d i a n s t o c k s w i t h i n t h e p e r i o d o f i n t e r e s t i n t h i s s t u d y . E x a c t l y t h e same p r o c e d u r e i s r e p e a t e d , b u t w i t h t h e u s e o f t h e C a n a d i a n u n c o n s t r a i n e d k e r n e l (Zu) a s t h e k e r n e l u s e d i n t h e r i s k m e a s u r e o f c o v ( X , Z u ) . The r e s u l t s a r e n o t s a t i s -f a c t o r y . I n g e n e r a l , r e g r e s s i o n s b e t w e e n t h e e x c e s s r e t u r n o f d i f f e r e n t s e c u r i t i e s a n d t h e Zu a r e i n s i g n i f i c a n t . The c o v a r i a n c e b e t w e e n t h e i n d i v i d u a l s e c u r i t i e s ' e x c e s s r e t u r n a n d t h e u n c o n s t r a i n e d k e r n e l ( i . e . c o v ( X , Z u ) ) i s t h e n c a l c u -l a t e d f o r d i f f e r e n t s u b p e r i o d s o f t i m e . T h i s i s e s s e n t i a l l y t h e new r i s k p r e m i a m e a s u r e . ( t a b l e s 19-24) . The t i m e s e r i e s r e g r e s s i o n b e t w e e n t h e c o v a r i a n c e s , w i t h a nd w i t h o u t t h e two h i g h e x c e s s r e t u r n s t o c k s , i s shown i n t a b l e s 25 and 26. 2 A s one c a n s e e , t h e R 's a r e c l o s e t o z e r o a n d t h e t - s t a t i s t i c s a r e i n s i g n i f i c a n t . An e x p l a n a t i o n t h a t i s c o n s i s t e n t w i t h t h e a b o v e o b s e r v a -t i o n s i s t h a t t h e u n c o n s t r a i n e d k e r n e l (Zu) i s s a m p l e s e n s i t i v e . 29 S i n c e t h e two o u t l y i n g s e c u r i t i e s w i t h h i g h e x c e s s r e t u r n s h a v e b e e n e n t e r e d i n t o t h e c o m p u t a t i o n o f Zu, t h e e s t i m a t i o n h a s b e e n b i a s e d a n d t h e r e f o r e t h e a b o v e r e s u l t s a r e o b s e r v e d . An i n t e r e s t i n g e x t e n s i o n i s t o r e p e r f o r m t h e t e s t w i t h a new s e t o f Zu, e x c l u d i n g t h e two s e c u r i t i e s w i t h h i g h e x c e s s r e t u r n s i n t h e c o m p u t a t i o n s . One c a n s p e c u l a t e t h a t t h e r e s u l t s w i l l i m p r o v e . 30 6. CONCLUSION W i t h r e s p e c t t o t h e r e s u l t s , i t a p p e a r s t h a t t h e no a r b i t r a g e p r o p e r t y s h o u l d n o t be r e j e c t e d i n C a n a d a . I t seems t h a t t h e no a r b i t r a g e p r o p e r t y i n C a n a d a c a n o n l y be r e j e c t e d i n some c a s e s i f we assume 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 o f t h e s t a t i s t i c s . Once t h e n o r m a l d i s t r i b u t i o n i s a s s u m e d , t h e no a r b i t r a g e p r o p e r t y c a n o n l y be r e j e c t e d i n one t i m e p e r i o d f o r one 30 s t o c k s u b s a m p l e i f t h e U.S. u n c o n -s t r a i n e d k e r n e l i s u s e d a s a p r o x y f o r t h e C a n a d i a n m a r k e t k e r n e l . No o t h e r s c a n be r e j e c t e d i n a n y t i m e p e r i o d f o r any s u b s a m p l e . N o t o n l y i s t h e r e no a r b i t r a g e o p p o r t u n i t y w i t h i n C a n a d a , t h e no a r b i t r a g e p r o p e r t y seems t o h o l d a c r o s s t h e b o r d e r a s w e l l . T h i s i m p l i e s t h a t a U.S. i n v e s t o r c a n n o t e a r n an a b n o r m a l r e t u r n b y i n v e s t i n g i n C a n a d a , a s s u m i n g t h a t t h e e x c h a n g e r a t e i s c o n s t a n t o v e r t i m e . M o r e o v e r , one c a n c o n c l u d e t h a t t h e C a n a d i a n u n c o n s t r a i n e d k e r n e l i s t h e b e s t p r o x y f o r t h e C a n a d i a n m a r k e t k e r n e l . On t h e o t h e r h a n d , a l t h o u g h t h e CAPM k e r n e l i s n o t t h e b e s t p r o x y f o r t h e C a n a d i a n m a r k e t k e r n e l , one c a n n o t r e f u t e t h e no a r b i t r a g e a s s u m p t i o n i n t h e CAPM b a s e d on t h e r e s u l t i n g s t a t i s t i c s i n t h i s p a p e r . F i n a l l y , a s f o r t h e v a l i d i t y o f u s i n g t h e c o v a r i a n c e s b e t w e e n t h e e x c e s s r e t u r n o f i n d i v i d u a l s t o c k s a n d the. u n -c o n s t r a i n e d k e r n e l a s a new r i s k m e a s u r e , t h e r e s u l t i s i n -31 c o n c l u s i v e . U n f o r t u n a t e l y , t h i s m e a s u r e i s e x t r e m e l y d a t a s e n s i t i v e a n d t h e r e a r e two o u t l y i n g o b s e r v a t i o n s i n t e r m s o f a b n o r m a l l y h i g h e x c e s s r e t u r n i n t h e s a m p l e s e t . T h i s o b s c u r e s t h e r e s u l t s o f t h e e x p e r i m e n t . As t h i s new r i s k p r e m i a m e a s u r e d o e s n o t r e q u i r e t h e s t r o n g a s s u m p t i o n s , a s i n t h e c a s e o f t h e CAPM b e t a , a n d t h u s i s more t h e o r e t i c a l l y s o u n d , more t h o r o u g h t e s t i n g s h o u l d be p e r f o r m e d i n t h e f u t u r e w i t h r e s p e c t t o t h i s m e a s u r e . One o b v i o u s e x t e n s i o n i s t o r e p e r f o r m t h e t e s t i n g p r o c e d u r e w i t h a new s e t o f Zu w h i c h i s e s t i m a t e d w i t h o u t u s i n g t h e two s e c u r i t i e s w i t h e x t r e m e l y h i g h e x c e s s r e t u r n s . T h i s e x t e n s i o n w i l l be l e f t f o r f u t u r e i n t e r e s t e d i n d i v i d u a l s . Zm Can Zm Can Zu 1 s t 30 stocks 2nd 30 stocks 3 r d 30 stocks 4 t h 30 stocks 39.5039 35.9834 52.3820* 27.1945 42.4921 34.2566 47.5471* 28.2151 40.3003 28.2166 25.8718 12.0332 Total 155.0438 152.5109 106.4219 32 Chi-square t e s t with 30 degrees of freedom @ a = 0.05 = 43.77 No arbitrage property rejected under the chi-square t e s t . F-test with 30 s e c u r i t i e s and 60 periods . @ a = 0.05 = 108.56 t No arbitrage property rejected under the F-t e s t s . Table 1 F-test and Chi-square t e s t by using R for F January 1956 - December 1960 Zm Can Zm Can Zu 1st 30 stocks 2nd 30 stocks 3rd 30 stocks 4th 30 stocks 43.3581 37.7123 100.7400* 41.6479 45.3584* 36.8242 99.9700* 41.0963 48.1337* 31.5761 49.5979* 21.1218 Total 223.4383 223.2489 150.4294 3 3 Chi-square t e s t with 30 degrees of freedom @ a = 0.05 = 43.77 * No arbitrage property rejected under the chi-square t e s t . F - t e s t with 30 s e c u r i t i e s and 60 periods @ a = 0.05 = 108.56 t No arbitrage property rejected under the F - t e s t s . Table 2 F-tes t and Chi-square t e s t by using R for January 1956 - December 1960 34 Zu Zc Zm Can Zm Can Zu 1st 30 stocks 2nd 30 stocks 3rd 30 stocks 4th 30 stocks 5th 30 stocks 55.3063" 33.8524 66.9030* 81.9035*+ 53.6266* 50.6075 33.4258 65.905i" 72.1916" 57.5594* 43.5416 30.8356 36.6577 61.8829" 63.4275" 42.0168 31.8311 36.2457 69.8073" 61.0484" 18.1670 9.6170 19.4040 18.2704 17.6316 T o t a l 291.5918 279.6894 236.3453 240.9493 83.0900 Chi-Square t e s t with 30 degrees of freedom @ a = 0 .05 = 4 3 . 7 7 * No arbitrage property rejected under the chi-square t e s t . F-test with 30 s e c u r i t i e s and 90 periods @ a = 0 .05 = 73 .425 ^ No arbitrage property rejected under the F-test. Table 3 F-test and Chi-square t e s t by using f o r January 1961 - June 1968 35 Zu Zc Zm 1st 30 stocks 2nd 30 stocks 3rd 30 stocks 4th 30 stocks 5th 30 stocks 59.5018' 35.5870 66.7238* 74.6817* 46.2267* 52.2760 34.0172 64.8256* 63.4206* 55.1691" 44.5338 30.2311 36.8240 57.8114* 64.8513* Can Zm 41.7573 30.6456 36.8847 65.6502* 61.8867* Can Zu 23.7071 8.2553 21.7404 18.3011 14.8161 T o t a l 282.7210 269.7085 234.2516 236.8245 86.8200 Chi-Square t e s t with 30 degrees of freedom @ a = 0.05 = 43.77 * No arbitrage property rejected under the chi-square t e s t . F - t e s t with 30 s e c u r i t i e s and 90 periods @ a = 0.05 = 73.425 t No arbitrage property rejected under the F-test. Table 4 F-tes t and Chi-square t e s t by using f o r January 1961 - June 1968 36 30 Security Subsamples Within Sample Out of Sample Subsample Zm Zm 1 44.55 40.80 2 47.78 74.10 3 31.76 41.29 4 46.19 51.41 5 38.22 6 16.73 Within and Out-of sample Chi-square t e s t of the no arbitrage property i n U.S. with 30 degrees of freedom by using R p Table 5 30 security subsamples selected from the Brennan Si Thompson paper f o r comparison purpose 37 Month US ZM CAN ZU CAN ZM 1 1 .18525 1 .55040 1 .01728 2 0 .84690 1 .58268 0 .96350 3 0 .73497 1 .43148 0 .92230 4 1 .03221 2 .37984 1 .00822 5 1 .30427 2 .11710 1 .03064 6 0 .92188 2 .35374 0 .96667 7 0 .80809 0 .52326 0 .93164 8 1 .16204 1 .00164 1 .03686 9 1 .28025 1 .17204 1 .04970 10 1 .01600 1 .08150 1 .02378 11 0 .99077 0 .15258 1 .04918 12 0 .94668 1 .91526 0 .92854 13 1 .03806 1 .63434 0 .99633 14 1 .20299 0 .82308 1 .04229 15 0 .92092 1 .06578 0 .96872 16 0 .89883 1 .78032 0 .96459 17 0 .92609 0 .94518 0 .96840 18 1 .10274 •3 .03582 1 .02396 19 1 .02313 1 .79148 1 .03353 20 1 .36933 2 .64396 1 .08896 21 1 .38508 1 .36140 1 .07283 22 1 .47093 1 .53318 1 .07359 23 0 .93165 0 .48792 0 .96088 24 1 .36151 0 .18264 1 .05557 25 0 .39775 2 .18250 0 .96581 26 1 .12490 0 .67206 1 .01348 27 0 .82626 1 .59558 0 .97247 28 0 .86190 -0 .00816 1 .00620 29 0 .80601 2 .05824 0 .95496 30 0 .84838 1 .80126 0 .97107 Month US ZM CAN ZU CAN ZM 31 0 .71047 2 .73534 0 .95367 32 0 .86205 0 .75978 0 .98994 33 0 .72833 0 .82128 0 .96779 34 0 .85150 0 .48180 0 .99829 35 0 .79096 0 .04986 0 .99177 36 0 .81322 -1 .16658 0 .98523 37 0 .79939 -0 .00222 0 .97556 38 0 .88610 -0 .14064 0 .99709 39 0 .97656 -0 .50166 1 .00906 40 0 .89741 0 .73410 0 .99364 41 1 .01116 0 .18210 1 .00345 42 1 .03021 0 .65820 0 .99350 43 0 .87077 -0 .11514 0 .96417 44 1 .14963 0 .36990 1 .05145 45 1 .34700 1 .78140 1 .05413 46 0 .91680 0 .35004 1 .00586 47 0 .96764 0 .71094 1 .01143 48 0 .92771 1 .30818 0 .96748 49 1 .30831 2 .56254 1 .04980 50 1 .01161 1 .08276 1 .03834 51 1 .21607 0 .37842 0 .99519 52 1 .17645 1 .47282 1 .02234 53 0 .90191 -0 .57942 0 .97376 54 0 .92355 -0 .52752 1 .02300 55 1 .16515 0 .13848 1 .02508 56 0 .83610 1 .93428 0 .93723 57 1 .41748 1 .84788 1 .04337 58 1 .19954 -0 .28860 1 .00187 59 0 .76260 ' 0 .18594 0 .96257 60 0 .81868 -0 .07242 0 .95228 Table 6 Al t e r n a t i v e Estimates of Market Kernel January 1956 - December 1960 38 Month CAN ZU CAN ZM 1 0.86886 0.72828 2 0.71469 0.82480 3 -0.16506 0.90636 4 0.55125 0.79857 5 0.88488 0.88363 6 0.28629 1.01521 7 -0.29241 0.94077 8 0.41814 0.93043 9 1.06488 1.12137 10 0.86868 0.95918 11 0.24984 0.82790 12 0.31356 0.89883 13 0.59049 1.19288 14 -0.23427 0.98979 15 0.75528 1.05161 16 1.42191 1.18908 17 2.71395 1.49833 18 0.92889 1.42793 19 0.33138 0.93505 20 1.77300 0.90268 21 1.91682 1.26679 22 5.19282 0.98617 23 1.24245 0.65998 24 0.03366 0.91735 25 1.80486 0.79839 26 1.10934 1.21421 27 3.20661 0.85255 28 0.75924 0.79693 29 -1.12356 0.92272 30 -0.47322 1.25213 31 1.19916 1.18575 32 1.85445 1.00703 33 3.26043 0.85723 34 0.23598 1.00902 35 2.95146 1.06148 36 1.11465 0.81977 37 0.19791 0.90362 38 1.18179 1.08392 39 0.24687 0.78659 40 2.27772 0.85197 41 1.67868 0.83047 42 4.28130 1.05897 43 2.78361 0.90811 44 1.44621 1.06195 45 -0.89757 0.81422 Month CAN ZU CAN ZM 46 1.48716 1.00288 47 0.18612 0.99482 48 1.66401 0.99751 49 -0.42426 0.75148 50 0.66321 1.05231 51 1.71666 1.09157 52 0.78381 0.92354 53 3.20967 1.05587 54 3.24738 1.43003 55 2.63016 1.10822 56 1.54134 0.88346 57 2.37555 0.93990 58 -0.57096 0.94171 59 2.82393 1.11880 60 0.97758 0.99832 61 -1.54386 0.83491 62 0.50850 1.15339 63 0.61641 1.14252 64 1.01916 0.98249 65 1.16208 1.23880 66 1.93761 1.05132 67 0.58221 1.15076 68 2.01429 1.44844 69 2.25288 1.17226 70 2.97945 0.92950 71 -0.53064 1.05447 72 2.43594 0.89678 73 2.09736 0.65198 74 -0.72783 0.98255 75 -1.38015 0.83076 76 -0.16245 0.91805 77 1.19646 1.24815 78 0.53820 0.86161 79 1.32759 0.86015 80 0.89388 1.10320 81 0.83295 0.92237 82 -0.92034 1.37301 83 -0.94734 0.88708 84 -0.56565 0.99172 85 1.02987 1.22005 86 0.94275 1.29162 87 0.42282 1.24406 88 -1.57356 0.51684 89 -0.36558 1.08713 90 0.08955 0.73383 Table 7 Al t e r n a t i v e Estimates of Market Kernel January 1961 - June 1968 39 CZU 1.0000 ZM .1685 1.0000 CZM .0596 .8068 1.0000 CZU ZM CZM Table 8 Co r r e l a t i o n matrix of kernels January 1956 - December 1960 40 z u 1 . 0 0 0 0 z c . 9 7 1 1 1 . 0 0 0 0 ZM . 3 2 0 2 . 3 4 0 6 1 . 0 0 0 0 CZU . 3 5 4 4 . 3 4 9 9 . 2 6 8 1 1 . 0 0 0 0 CZM . 3 1 1 4 . 3 3 3 1 . 8 0 6 9 . 2 1 3 1 1 . 0 0 0 0 ZU ZC ZM CZU CZM Table 9 C o r r e l a t i o n Matrix of Kernels January 1 9 6 1 - June 1 9 6 8 czu 1.0000 ZM .2396 1.0000 CZM .1855 .7384 1.0000 CZU ZM CZM T a b l e 10 C o r r e l a t i o n matrix of kernels January 1956 - June 1968 42 :curity Cov(X ,Z ) i t mt Security Cov(X ,Z ) i t mt Security Cov(X. .Z i t ml 1 -0.00193 40 -0.00171 79 -0.00159 2 -0.00134 41 -0.00131 80 -0.00127 3 -0.00224 42 -0.00065 81 -0.00161 4 -0.00129 43 -0.00072 82 -0.00196 5 -0.00175 44 -0.00026 83 -0.00208 6 -0.00154 45 -0.00097 84 -0.00147 7 -0.00182 46 -0.00107 85 -0.00142 8 -0.00218 47 -0.00073 86 -0.00036 9 -0.00377 48 -0.00094 87 -0.00061 10 -0.00276 49 -0.00067 88 -0.00046 11 -0.00299 50 0.00002 89 -0.00022 12 -0.00032 51 -0.00136 90 -0.00070 13 -0.00108 52 -0.00074 91 -0.00068 14 -0.00041 53 -0.00080 92 -0.00105 15 -0.00140 54 -0.00071 93 -0.00148 16 -0.00125 55 -0.00068 94 -0.00069 17 -0.00115 56 -0.00094 95 -0.00094 18 -0.00180 57 -0.00129 96 -0.00138 19 -0.00147 58 -0.00094 97 -0.00181 20 -0.00034 59 -0.00111 98 -0.00094 21 -0.00030 60 -0.00169 99 -0.00155 22 -0.00036 61 -0.00096 100 -0.00133 23 -0.00240 62 -0.00157 101 -0.00169 24 -0.00246 63 -0.00139 102 -0.00151 25 -0.00243 64 -0.00242 103 -0.00072 26 -0.00237 65 -0.00121 104 -0.00042 27 -0.00144 66 -0.00113 105 -0.00071 28 -0.00193 67 -0.00180 106 -0.00031 29 -0.00304 68 -0.00060 107 -0.00086 30 -0.00172 69 -0.00195 108 -0.00110 31 -0.00234 70 0.00004 109 -0.00062 32 -0.00190 71 -0.00001 110 -0.00193 33 -0.00111 72 -0.00158 111 -0.00110 34 -0.00112 73 -0.00151 112 -0.00059 35 -0.00173 74 -0.00145 113 -0.00135 36 -0.00179 75 -0.00151 114 -0.00059 37 -0.00160 76 -0.00141 115 -0.00172 38 -0.00115 77 -0.00052 116 -0.00011 39 -0.00239 78 -0.00105 117 118 -0.00159 -0.00154 Table 11 Covariances between the 118 s e c u r i t i e s r i s k premium and the CAPM kernel Z m f o r the period January 1956 - December 1960 43 i c u r i t y Cov(X ,Z ) i t mt Security Cov(X. .Z ) i t mt Security Cov(X. .Z i t mt 1 -0.00242 40 -0.00155 79 -0.00237 2 -0.00165 41 -0.00192 80 -0.00215 3 -0.00292 42 -0.00112 81 -0.00254 4 -0.00187 43 -0.00117 82 -0.00229 5 -0.00237 44 0.00002 83 -0.00264 6 -0.00228 45 -0.00066 84 -0.00186 7 -0.00300 46 -0.00048 85 -0.00191 8 -0.00378 47 -0.00096 86 -0.00048 9 -0.00670 48 -0.00098 87 -0.00073 10 -0.00510 49 -0.00122 88 -0.00042 11 -0.00527 50 0.00026 89 -0.00065 12 -0.00049 51 -0.00183 90 -0.00055 13 -0.00207 52 -0.00105 91 -0.00073 14 -0.00035 53 -0.00135 92 -0.00117 15 -0.00151 54 -0.00081 93 -0.00207 16 -0.00163 55 -0.00050 94 -0.00128 17 -0.00186 56 -0.00053 95 -0.00150 18 -0.00217 57 -0.00153 96 -0.00188 19 -0.00195 58 -0.00116 97 -0.00299 20 -0.00031 59 -0.00107 98 -0.00106 21 -0.00079 60 -0.00239 99 -0.00199 22 -0.00042 61 -0.00144 100 -0.00163 23 -0.00328 62 -0.00260 101 -0.00234 24 -0.00309 63 -0.00189 102 -0.00178 25 -0.00379 64 -0.00353 103 -0.00056 26 -0.00333 65 -0.00190 104 -0.00037 27 -0.00209 66 -0.00211 105 -0.00068 28 -0.00254 67 -0.00243 106 -0.00009 29 -0.00371 68 -0.00093 107 -0.00100 30 -0.00245 69 -0.00281 108 -0.00139 31 -0.00421 70 -0.00037 109 -0.00082 32 -0.00255 71 -0.00010 110 -0.00262 33 -0.00152 72 -0.00170 111 -0.00158 34 -0.00211 73 -0.00193 112 -0.00104 35 -0.00253 74 -0.00170 113 -0.00216 36 -0.00240 75 -0.00231 114 -0.00034 37 -0.00211 76 -0.00156 115 -0.00249 38 -0.00170 77 -0.00084 116 0.00045 39 -0.00298 78 -0.00090 117 118 -0.00196 -0.00198 Table 12 Covariances between the 118 s e c u r i t i e s r i s k premium and the CAPM kernel Z m for the period January 1956 - June 1958 44 i c u r i t y Cov(X. .Z ) i t mt Security Cov(X ,Z ) i t mt Security Cov(X. .Z i t mt 1 -0.00140 40 -0.00183 79 -0.00080 2 -0.00098 41 -0.00069 80 -0.00040 3 -0.00152 42 -0.00016 81 -0.00067 4 -0.00070 43 -0.00030 82 -0.00161 5 -0.00110 44 -0.00053 83 -0.00152 6 -0.00079 45 -0.00127 84 -0.00107 7 -0.00066 46 -0.00157 85 -0.00096 8 -0.00058 47 -0.00048 86 -0.00022 9 -0.00081 48 -0.00086 87 -0.00046 10 -0.00028 49 -0.00015 88 -0.00049 11 -0.00075 50 -0.00020 89 0.00021 12 -0.00013 51 -0.00087 90 -0.00079 13 -0.00006 52 -0.00044 91 -0.00065 14 -0.00042 53 -0.00026 92 -0.00090 15 -0.00127 54 -0.00059 93 -0.00089 16 -0.00082 55 -0.00085 94 -0.00013 17 -0.00050 56 -0.00136 95 -0.00033 18 -0.00142 57 -0.00102 96 -0.00090 19 -0.00105 58 -0.00071 97 -0.00063 20 -0.00033 59 -0.00115 98 -0.00079 21 0.00021 60 -0.00097 99 -0.00108 22 -0.00032 61 -0.00042 100 -0.00102 23 -0.00158 62 -0.00046 101 -0.00104 24 -0.00190 63 -0.00083 102 -0.00120 25 -0.00109 64 -0.00132 103 -0.00086 26 -0.00150 65 -0.00055 104 -0.00045 27 -0.00078 66 -0.00018 105 -0.00071 28 -0.00136 67 -0.00114 106 -0.00049 29 -0.00248 68 -0.00026 107 -0.00070 30 -0.00102 69 -0.00109 108 -0.00080 31 -0.00052 70 0.00047 109 -0.00040 32 -0.00127 71 0.00013 110 -0.00124 33 -0.00067 72 -0.00142 111 -0.00061 34 -0.00005 73 • -0.00102 112 -0.00018 35 -0.00090 74 -0.00119 113 -0.00055 36 -0.00116 75 -0.00067 114 -0.00080 37 -0.00109 76 -0.00127 115 -0.00096 38 -0.00059 77 -0.00022 116 -0.00064 39 -0.00174 78 -0.00115 117 -0.00124 118 -0.00113 Table 13 Covariances between the 118 s e c u r i t i e s r i s k premium and the CAPM kernel Z m f o r the period July 1958 - December 1960 45 >. c u r i t y Cov(X. .Z J i t mt Security Cov(X..,Z ) i t mt Security Cov(X. .Z i t mi 1 -0.00811 40 -0.00689 80 -0.00574 2 -0.00553 41 -0.00729 81 -0.00588 3 -0.00643 42 -0.00139 82 -0.00774 4 -0.00529 43 -0.00635 83 -0.00307 5 -0.0065,3 44 -0.00445 84 -0.00521 6 -0.00517 45 -0.00436 85 -0.00558 7 -0.00633 46 -0.00444 86 -0.00288 8 -0.00462 47 -0.00474 87 -0.00394 9 0.00923 48 -0.00613 88 -0.00349 10 0.00079 49 -0.00399 89 -0.00357 11 -0.00662 50 -0.00270 90 -0.00550 12 -0.00798 51 -0.00749 91 -0.00397 13 -0.00648 52 -0.00440 92 -0.00626 14 0.00085 53 -0.00425 93 -0.00577 15 -0.00472 54 -0.00405 94 -0.00333 16 -0.00374 55 -0.00572 95 -0.00318 17 -0.00519 56 -0.00465 96 -0.00552 18 -0.00598 57 -0.00599 97 -0.00740 19 -0.00616 58 -0.00627 98 -0.00660 20 0.00094 59 -0.00685 99 -0.00682 21 0.00101 60 -0.00397 100 -0.00545 22 -0.00073 61 -0.00738 101 -0.00603 23 -0.00620 62 -0.00772 102 -0.00642 24 -0.00623 63 -0.00549 103 -0.00502 25 -0.00601 64 -0.00679 104 -0.00642 26 -0.00892 65 -0.00701 105 -0.00493 27 -0.00264 66 -0.00908 106 -0.00791 28 -0.00494 67 -0.00835 107 -0.00515 29 -0.00727 68 -0.00749 108 -0.00481 30 -0.00492 69 -0.00787 109 -0.00101 31 -0.00729 70 -0.00439 110 -0.00766 32 -0.00709 71 -0.00367 111 -0.00638 33 -0.00365 72 -0.00700 112 -0.00546 34 -0.00380 73 -0.00827 113 -0.00532 35 -0.00811 74 -0.00620 114 -0.00667 36 -0.00430 75 -0.00626 115 -0.00724 37 -0.00550 76 -0.00634 116 -0.00425 38 -0.00629 77 -0.00590 117 -0.00633 39 -0.00541 78. 79 -0.00566 -0.00680 118 -0.00203 Table 14 Covariances between the 118 s e c u r i t i e s r i s k premium and the CAPM kernel Z m for the period January 1961 - June 1968 46 i c u r i t y Cov(X. .Z ) i t mt Security Cov(X. .Z ) i t mt Security Cov(X., ,Z i t ml 1 -0.00745 40 -0.00452 79 -40.00742 2 -0.00494 41 -0.00651 80 -0.00665 3 -0.00697 42 0.00089 81 -0.00436 4 -0.00391 43 -0.00595 82 -0.00610 5 -0.00807 44 -0.00353 83 -0.00193 6 -0.00496 45 -0.00377 84 -0.00451 7 -0.00560 46 -0.00328 85 -0.00467 8 -0.00675 47 -0.00411 86 -0.00331 9 0.02656 48 -0.00433 87 -0.00325 10 0.01185 49 -0.00317 88 -0.00283 11 -0.00777 50 -0.00092 89 -0.00087 12 -0.00874 51 -0.00926 90 -0.00470 13 -0.00623 52 -0.00612 91 -0.00274 14 -0.00026 53 -0.00302 92 -0.00604 15 -0.00332 54 -0.00508 93 -0.00528 16 -0.00433 55 -0.00531 94 -0.00292 17 -0.00370 56 -0.00260 95 -0.00300 18 -0.00601 57 -0.00562 96 -0.00466 19 -0.00453 58 -0.00599 97 -0.00820 20 0.00144 59 -0.00411 98 -0.00511 21 0.00193 60 -0.00484 99 -0.00490 22 -0.00089 61 -0.00195 100 -0.00463 23 -0.00302 62 -0.01117 101 -0.00414 24 -0.00578 63 -0.00466 102 -0.00496 25 " -0.00431 64 -0.00587 103 -0.00464 26 -0.01081 65 -0.00564 104 -0.00442 27 -0.00287 66 -0.00804 105 -0.00434 28 -0.00352 67 -0.00624 106 -0.00765 29 -0.00521 68 -0.00413 107 -0.00529 30 -0.00411 69 -0.00642 108 -0.00541 31 -0.00664 70 -0.00165 109 0.00076 32 -0.00534 71 -0.00458 110 -0.00858 33 -0.00263 72 -0.00667 111 -0.00696 34 -0.00290 73 -0.00537 112 -0.00254 35 -0.00840 74 -0.00532 113 -0.00407 36 -0.00256 75 -0.00551 114 -0.00438 37 -0.00412 76 -0.00466 115 -0.00469 38 -0.00305 77 -0.00566 116 -0.00389 39 -0.00362 78 -0.00280 117 118 -0.00465 0.00044 Table 15 Covariances between the 118 s e c u r i t i e s r i s k premium and the CAPM kernel Z m for the period January 1961 - September 1964 47 Security Cov(X ,Z ) 1 -0.00859 2 -0.00548 3 -0.00562 4 -0.00629 5 -0.00482 6 -0.00486 7 -0.00684 8 -0.00203 9 -0.00156 10 -0.00793 11 -0.00527 12 -0.00676 13 -0.00661 14 0.00167 15 -0.00605 16 -0.00331 17 -0.00707 18 -0.00568 19 -0.00796 20 0.00019 21 -0.00010 22 -0.00113 23 -0.00929 24 -0.00688 25 -0.00788 26 -0.00649 27 -0.00217 28 -0.00635 29 -0.00936 30 -0.00568 31 -0.00803 32 -0.00903 33 -0.00413 34 -0.00431 35 -0.00690 36 -0.00559 37 -0.00624 38 -0.00899 39 -0.00666 Security Cov(X. ,Z ) xt mt 40 -0.00863 41 -0.00778 42 -0.00352 43 -0.00670 44 -0.00533 45 -0.00496 46 -0.00502 47 -0.00556 48 -0.00795 49 -0.00458 50 -0.00445 51 -0.00547 52 -0.00259 53 -0.00558 54 -0.00287 55 -0.00604 56 -0.00678 57 -0.00613 58 -0.00615 59 -0.00857 60 -0.00254 61 -0.01197 62 -0.00472 63 -0.00625 64 -0.00687 65 -0.00807 66 -0.00972 67 -0.00987 68 -0.00982 69 -0.00894 70 -0.00725 71 -0.00229 72 -0.00651 73 -0.00997 74 -0.00665 75 -0.00608 76 -0.00742 77 -0.00592 78 -0.00893 Table 16 Security Cov(X. ,Z ) xt mt 79 -0.00643 80 -0.00445 81 -0.00717 82 -0.00968 83 -0.00363 84 -0.00591 85 -0.00655 86 -0.00215 87 -0.00437 88 -0.00461 89 -0.00628 90 -0.00636 91 -0.00498 92 -0.00742 93 -0.00613 94 -0.00423 95 -0.00360 96 -0.00602 97 -0.00644 98 -0.00776 99 -0.00869 100 -0.00631 101 -0.00799 102 -0.00789 103 -0.00510 104 -0.00797 105 -0.00499 106 -0.00756 107 -0.00451 108 -0.00400 109 -0.00229 110 -0.00666 111 -0.00544 112 -0.00923 113 -0.00611 114 -0.00854 115 -0.00987 116 -0.00401 117 -0.00721 118 -0.00421 Covariances between the 118 s e c u r i t i e s r i s k premium and the CAPM kernel Z m f o r the period October 1964 - June 1968 48 Period Dependent Variable July 1958 Jan. 1961 Oct. 1964 to to to Dec. 1960 Sept. 1964 June 1968 Jan. 1956 to Dec. 1960 Jan. 1961 to Dec. 1968 Jan. 1956 to June 1958 b 0.16044 R 2 0.14251 t 4.3906 -0.60697 0.03092 -1.9239 0.38735 0.03590 2.0782 1.5546 0.89695 31.775 0.026938 0.00016 0.13752 <D r-i X! (8 •H M > -P C (U -d fl Q) ft 0) T3 fl H J u l y 1958 to Dec. 1960 Jan. 1961 to Sept. 1964 Oct. 1964 to June 1968 R t R t R t 1.4961 0.03393 2.0186 1.5922 0.10955 3.7777 0.17689 0.08919 3.3703 0.45616 0.42754 9.3077 -0.015059 0.00728 -0.92232 0.079646 0.07145 2.9876 1.5813 0.10146 3.6191 0.53024 0.75253 18.782 0.74449 0.52046 11.22 Jan. 1956 to Dec. 1960 R t 0.40324 0.01356 1.2626 Table 17 2 Regression c o e f f i c i e n t , R and t - s t a t i s t i c s of the regression between covariances of the 118 s e c u r i t i e s and Zm over d i f f e r e n t periods o f time 49 Dependent Variable Period July 1958 Jan. 1961 Oct. 1964 Jan. 1956 Jan. 1961 to to to to to Dec. 1960 Sept. 1964 June 1968 Dec. I960 Dec. 1968 Jan. 1956 to June 1958 IT t 0.22479 0.21577 5.6006 0.99464 0.19329 5.2263 0.64459 0.07884 3.1237 0.60581 0.88796 30.059 0.83421 0.18896 5.1536 0) H A rd •H U rd > •P C 0) T3 C 0) a <u Tl C H J u l y 1958 to Dec. 1960 Jan. 1961 to Sept. 1964 Oct. 1964 to June 1968 t R t R t 1.2103 0.06702 2.8616 1.6452 0.12028 3.9480 0.35709 0.12385 4.0143 0.97425 0.53779 11.517 0.12432 0.19141 5.1947 0.096456 0.11863 3.9171 1.4842 0.14007 4.3092 0.69291 0.66725 15.12 0.68532 0.67204 15.284 Jan. 1956 b to R2 Dec. 1960 t 1.4200 0.22630 5.7744 Table 18 o Regression c o e f f i c i e n t , R and t - s t a t i s t i c s of the regression between covariances of the 116 s e c u r i t i e s and Zm over d i f f e r e n t periods of time 50 scurity Cov(X. t,Z u t) Security Cov(X. .Z ) i t ut Security Cov(X. ,Z i t ul 1 0.00216 40 0.00259 79 -0.01416 2 0.00333 41 0.00268 80 0.00181 3 -0.00881 42 -0.01020 81 -0.00147 4 0.00372 43 -0.00033 82 0.00828 5 -0.01058 44 -0.00293 83 -0.01244 6 0.00160 45 -0.00806 84 -0.00709 7 -0.00364 46 -0.01074 85 -0.00741 8 -0.00449 47 -0.01059 86 -0.00227 9 0.01760 48 -0.00485 87 -0.00179 10 0.01050 49 0.00032 88 -0.00536 11 0.00264 50 -0.00402 89 -0.00592 12 -0.00165 51 -0.01232 90 -0.00513 13 0.01157 52 -0.00304 91 -0.00554 14 -0.00991 53 0.00169 92 -0.00410 15 0.00084 54 -0.00719 93 -0.00784 16 -0.00087 55 -0.00822 94 -0.00911 17 0.00005 56 -0.00733 95 -0.00680 18 0.00108 57 -0.00457 96 -0.00523 19 0.00670 58 -0.00496 97 -0.00155 20 -0.01261 59 -0.00913 98 -0.00863 21 -0.00721 60 -0.02727 99 -0.00724 22 0.00542 61 -0.00953 100 -0.00678 23 0.01087 62 0.00066 101 -0.00356 24 -0.00152 63 -0.00135 102 -0.00843 25 0.00934 64 0.00114 103 -0.00512 26 0.00638 65 0.00452 104 -0.00850 27 0.00556 66 -0.00048 105 -0.00842 28 0.00250 67 -0.00616 106 -0.00910 29 0.01881 68 -0.01113 107 -0.00262 30 -0.00020 69 -0.00434 108 -0.00203 31 0.01089 70 0.00728 109 -0.00655 32 -0.00315 71 -0.00541 110 -0.00366 33 -0.00499 72 -0.00878 111 0.00236 34 -0.00411 73 -0.01051 112 0.00459 35 0.00055 74 0.00822 113 0.00689 36 0.00084 75 0.00195 114 0.00205 37 -0.00256 76 0.00123 115 -0.00466 38 0.00316 77 0.00585 116 -0.00684 39 -0.00181 78 -0.02382 117 -0.00061 118 0.00340 Table 19 Covariances between the 118 s e c u r i t i e s r i s k premium and the -unconstrained .kernel Zu. f o r the period January 1956 - December 1960 51 i c u r i t y C o v<X i t , Z u t > Security Cov(X..,Z ) i t ut Security Cov(X. ,Z i t ut 1 0.01203 40 0.00847 79 -0.01198 2 0.01636 41 0.00833 80 0.00090 3 -0.00478 42 -0.00345 81 -0.00523 4 0.01079 43 0.00201 82 -0.01027 5 0.00011 44 -0.00313 83 -0.00875 6 0.00177 45 0.00058 84 0.00001 7 0.00280 46 0.01267 85 -0.01608 8 -0.00101 47 0.01379 86 -0.00280 9 0.04752 48 0.00698 87 0.00151 10 0.03346 49 -0.00205 88 -0.00224 11 0.00865 50 -0.00388 89 -0.01243 12 0.01191 51 -0.01596 90 -0.00478 13 0.02190 52 -0.00974 21 -0.00918 14 0.00225 53 0.00616 92 0.00951 15 0.00555 54 -0.00039 93 -0.00777 16 0.00540 55 0.00010 94 -0.00259 17 0.00451 56 -0.00928 95 -0.00447 18 0.00600 57 -0.00398 96 -0.00552 19 -0.00026 58 -0.00151 97 -0.00713 20 -0.01547 59 -0.00490 98 -0.00215 21 0.00008 60 -0.02013 99 0.00117 22 0.01424 61 0.00929 100 0.00315 23 0.00931 62 0.02058 101 -0.00589 24 -0.01371 63 0.01013 102 -0.00688 25 0.00774 64 0.00315 103 -0.00366 26 -0.00054 65 -0.00166 104 -0.00664 27 0.00051 66 -0.00466 105 0.00115 28 -0.00253 67 -0.00243 106 -0.00219 29 -0.00795 68 -0.00095 107 -0.00077 30 0.00016 69 -0.00584 108 -0.00480 31 0.00953 70 0.00857 109 -0.02043 32 0.00209 71 0.00931 110 ' 0.00549 33 0.00308 72 -0.00402 111 0.00097 34 0.00524 73 0.01227 112 -0.00752 35 0.01132 74 0.01373 113 0.00911 36 0.01069 75 -0.00170 114 0.00244 37 0.00355 76 -0.00088 115 0.00865 38 0.00474 77 0.00168 116 0.00092 39 0.00833 78 -0.00891 117 -0.00196 118 0.02521 Table 20 Covariances between the 118 s e c u r i t i e s r i s k premium and the 'unconstrained kernel Z u for the period January 1956 - June 1958 52 i c u r i t y Cov(X. . Z J i t ut Security Cov(X. .Z ) i t ut Security Cov(X.. , Z i t ut 1 0.00011 40 0.00568 79 -0.01577 2 -0.00104 41 -0.00089 80 -0.00084 3 -0.00421 42 -0.01162 81 -0.00058 4 -0.00050 43 -0.00782 82 0.02526 5 -0.01470 44 -0.00322 83 -0.01943 6 0.00416 45 -0.01523 84 -0.01339 7 -0.01436 46 -0.01749 85 -0.00880 8 -0.00925 47 -0.02938 86 0.00262 9 -0.00615 48 -0.00847 87 -0.00133 10 0.01517 49 -0.00401 88 -0.00675 11 -0.01323 50 -0.00191 89 0.00063 12 -0.00803 51 -0.00264 90 0.00632 13 0.00764 52 -0.00059 91 -0.00956 14 -0.01102 53 -0.00576 92 -0.01420 15 0.00036 54 -0.00875 93 -0.00949 16 0.00240 55 -0.01378 94 -0.01944 17 -0.01616 56 -0.00633 95 -0.00083 18 -0.00241 57 0.00100 96 -0.01353 19 0.00307 58 -0.00735 97 0.00016 20 -0.00217 59 -0.01275 98 -0.01021 21 -0.01045 60 -0.03498 99 -0.01201 22 -0.00885 61 -0.01924 100 -0.01482 23 0 .00288 62 -0.00092 101 -0.00248 24 -0.00409 63 -0.00096 102 -0.00478 25 0.00623 64 -0.00403 103 -0.00403 26 -0.00836 65 0.00810 104 -0.00494 27 0.01043 66 -0.00263 105 -0.01331 28 0.00145 67 -0.00485 106 -0.00971 29 0.02211 68 -0.01828 107 -0.00219 30 -0.00491 69 -0.00302 108 0.00158 31 -0.00106 70 0.01102 109 0.01132 32 -0.01346 71 -0.00821 110 -0.01307 33 -0.00601 72 -0.00691 111 0.00425 34 -0.00016 73 -0.02099 112 0.00733 35 -0.00324 74 0.00307 113 0.00383 36 -0.00566 75 0.01208 114 0.00830 37 -0.00778 76 0.00156 115 -0.02053 38 0.00237 77 0.00487 116 -0.00471 39 -0.00063 78 -0.02986 117 118 -0.00525 -0.02116 Table 21 Covariances between the 118 s e c u r i t i e s r i s k premium and the unconstrained kernel Z u f o r the period J u l y 1958 - December 1960 53 i c u r i t y Cov(X.. , Z J i t ut Security Cov(X. . Z ) i t ut Security Cov(X.. ,Z , i t ut 1 -0.00453 40 -0.00395 79 -0.01290 2 -0.00199 41 -0.00449 80 -0.00736 3 -0.01418 42 -0.00563 81 -0.00778 4 -0.00899 43 -0.01058 82 -0.01623 5 -0.01209 44 -0.01277 83 -0.00477 6 -0.00696 45 -0.00739 84 -0.01054 7 -0.00965 46 -0.02316 85 -0.01118 8 -0.00594 47 -0.01856 86 -0.00238 9 0.00837 48 -0.00929 87 -0.00411 10 -0.06562 49 -0.00663 88 -0.01035 11 -0.00920 50 -0.00686 89 -0.02765 12 -0.01596 51 -0.00492 90 -0.01121 13 -0.00758 52 -0.00416 91 -0.00777 14 -0.00504 53 -0.00763 92 -0.01974 15 -0.00481 54 -0.00561 93 -0.01291 16 -0.00133 55 -0.01073 94 -0.01554 17 -0.02317 56 -0.01829 95 0.00002 18 -0.01172 57 -0.01174 96 -0.00880 19 -0.02180 58 -0.00911 97 -0.00570 20 -0.01087 59 -0.00204 98 -0.00437 21 -0.01339 60 -0.00154 99 -0.00495 22 -0.00742 61 -0.01579 100 -0.00512 23 -0.01809 62 -0.01545 101 -0.00913 24 -0.01179 63 -0.01228 102 -0.00799 25 -0.01984 64 -0.00383 103 -0.00569 26 -0.01533 65 -0.01110 104 -0.01164 27 -0.00971 66 -0.00584 105 -0.00382 28 -0.00737 67 -0.00792 106 -0.01007 29 -0.01904 68 -0.01671 107 -0.00891 30 -0.00650 69 -0.00669 108 -0.00825 31 -0.01956 70 -0.01066 109 -0.00318 32 -0.01166 71 -0.00948 110 -0.00375 33 0.00102 72 -0.00672 111 -0.00092 34 0.00255 73 -0.01573 112 0.01133 35 0.00051 74 -0.00384 113 -0.00575 36 0.00709 75 -0.01044 114 0.00641 37 -0.00088 76 -0.00759 115 -0.01177 38 -0.00554 77 -0.01437 116 -0.00222 39 -0.01049 78 -0.01266 117 -0.01344 118 -0.01478 Table 22 Covariances between the 118 s e c u r i t i e s r i s k premium and the unconstrained kernel Z u for the period January 1961 - June 1968 54 Security Cov.(X. .Z ) i t ut 1 -0.00107 2 -0.00448 3 -0.01258 4 -0.00559 5 -0.01640 6 -0.01262 7 -0.01053 8 -0.00749 9 -0.02677 10 -0.12394 11 -0.00912 12 -0.00812 13 -0.00699 14 -0.02486 15 -0.00444 16 0.00158 17 -0.02686 18 -0.01187 19 -0.03126 20 -0.01387 21 -0.00935 22 -0.00194 23 0.00410 24 -0.00626 25 -0.02505 26 -0.02197 27 -0.01195 28 -0.00491 29 0.00063 30 -0.00532 31 -0.01017 32 -0.00598 33 -0.00445 34 -0.01057 35 0.00196 36 0.00228 37 0.00212 38 -0.01920 39 -0.00720 Security Cov(X. ,Z ) 40 -0.00031 41 0.00006 42 -0.00568 43 0.01267 44 -0.00683 45 -0.00742 46 -0.01033 47 -0.00596 48 -0.00489 49 -0.00554 50 -0.01167 51 -0.00255 52 -0.00103 53 ^0.01017 54 -0.00092 55 0.00160 56 -0.00290 57 -0.00512 58 -0.01082 59 -0.00413 60 -0.01055 61 -0.00300 62 -0.01030 63 0.00244 64 0.00456 65 -0.01742 66 -0.00136 67 -0.00660 68 -0.00933 69 -0.00410 70 0.00445 71 -0.01867 72 -0.00294 73 -0.01466 74 -0.00374 75 0.00009 76 -0.00124 77 -0.00669 78 0.00694 Table 23 Security Cov(X ,Z ) 79 -0.00979 80 -0.00739 81 -0.00342 82 0.00143 83 -0.00934 84 -0.00422 85 -0.00771 86 -0.00167 87 -0.00067 88 -0.00179 89 -0.01761 90 -0.00511 91 -0.00050 92 0.00002 93 -0.01308 94 -0.01748 95 0.01651 96 -0.00577 97 0.00027 98 -0.00320 99 0.00700 100 0.01017 101 0.00059 102 0.00545 103 -0.00952 104 -0.01567 105 -0.01443 106 -0.00363 107 -0.00422 108 -0.01056 109 -0.00122 110 0.00044 111 0.00066 112 0.04619 113 0.00173 114 0.01386 115 0.01234 116 -0.00804 117 -0.00514 118 0.00537 Covariances between the 118 s e c u r i t i e s r i s k premium and the unconstrained kernel Z u f o r the period January 1961 - September 1964 55 i c u r i t y Cov(X. .Z J i t ut Security Cov(X..,Z ) i t ut Security Cov(X..,Z i t ut 1 -0.00907 39 -0.01703 79 -0.01441 2 -0.00342 40 -0.01136 80 -0.00952 3 -0.01737 41 -0.01089 81 -0.01352 4 -0.01463 42 -0.00656 82 -0.03205 5 -0.00885 43 -0.03407 83 -0.00367 6 -0.00447 44 -0.01894 84 -0.01673 7 -0.01013 45 -0.00728 85 -0.01433 8 -0.00717 46 -0.03951 86 -0.00490 9 0.00382 47 -0.03000 87 -0.00906 10 -0.02157 48 -0.01358 88 -0.01603 11 -0.01051 49 -0.00913 89 -0.03756 12 -0.02655 50 -0.00221 90 -0.01687 13 -0.00892 51 -0.00878 91 -0.01628 14 0.01657 52 -0.00784 92 -0.03375 15 -0.00567 53 -0.00440 93 -0.01346 16 -0.00327 54 -0.01122 94 -0.01058 17 -0.01704 55 -0.02359 95 -0.01502 18 -0.01321 56 -0.03315 96 -0.01392 19 -0.01127 57 -0.01969 97 -0.01256 20 -0.00632 58 -0.00971 98 -0.00754 21 -0.01628 59 -0.00607 99 -0.01713 22 -0.00952 60 0.00409 100 -0.02015 23 -0.04088 61 -0.03373 101 -0.01843 24 -0.01605 62 -0.01780 102 -0.02135 25 -0.01361 63 -0.02738 103 -0.00373 26 -0.01196 64 -0.01731 104 -0.01020 27 -0.00890 65 -0.00659 105 0.00370 28 -0.00990 66 -0.01273 106 -0.02009 29 -0.03847 67 -0.01280 107 -0.01649 30 -0.00798 68 -0.03036 108 -0.00712 31 -0.02837 69 -0.01155 109 -0.00791 32 -0.01616 70 -0.02497 110 -0.00849 33 0.00320 71 -0.00310 111 -0.00465 34 0.01325 72 -0.01547 112 -0.01816 35 -0.00650 73 -0.02404 113 -0.01599 36 0.00908 74 -0.00655 114 -0.00341 37 -0.00784 75 -0.02650 115 -0.03528 38 0.00488 76 -0.01751 116 0.00002 77 -0.02333 117 -0.02650 78 -0.02972 118 -0.03668 Table 24 Covariances between the 118 s e c u r i t i e s r i s k premium and the unconstrained kernel Z u for the period October 1964 - June 1968 56 Dependent Variable Period July 1958 Jan. 1961 Oct. 1964 Jan. 1956 Jan. 1961 to to to to to Dec. 1960 Sept. 1964 June 1968 Dec. 1960 Dec. 1968 Jan. 1956 to June 1958 R t 0.011445 0.00014 0.12551 -0.4245 0.0822 -3.2232 -0.032302 0.00083 -0.31114 0.37681 0.27246 6.5910 -0.12282 0.02114 -1.5827 Si <ti •H U > c c a c J u l y 1958 to Dec. 1960 Jan. 1961 to Sept. 1964 Oct. 1964 to June 1968 R' t R t R t -0.096361 0.00409 -0.68990 0.043286 0.00144 0.40966 -0.11464 0.02303 -1.6535 0.51453 0.49014 10.560 -0.055318 0.01287 -1.2299 -0.045112 0.00489 -0.75474 -0.019484 0.00051 -0.24407 0.38869 0.46414 10.024 0.43713 0.33507 7.6455 Jan. 1956 to Dec. 1960 R t -0.10136 0.00750 -0.93643 Table 25 Regression c o e f f i c i e n t , R and t - S t a t i s t i c s of the regression between covariances o f the 118 s e c u r i t i e s and Zu over d i f f e r e n t periods of time 57 Period Dependent Variable July 1958 Jan. 1961 Oct. 1964 to to to Dec. 1960 Sept. 1964 June 1968 Jan. 1956 to Dec. .1960 Jan. 1961 to Dec. 1968 Jan. 1956 to June 1958 R' t -0.057382 0.00254 -0.53864 0.003544 0.00001 0.033581 -0.11688 0.00803 -0.96072 0.35780 0.19513 5.2571 -0.044827' 0.00357 -0.63929 Jul y 1958 to Dec. 1960 R t 0.12440 0.01581 1.3531 0.061628 0.0029 0.57543 0.51172 0.51769 11.062 0.090297 0.01880 1.4780 a 0) xs a a T3 C H Jan. 1961 to Sept. 1964 Oct. 1964 to June 1968 t R' t -0.32729 0.07997 -3.1478 0.0594 0.00683 0.88537 -0.065233 0.01103 -1.1277 0.31982 0.23091 5.8504 0.39519 0.47226 10.100 Jan. 1956 to Dec. 1960 R" t -0.039025 0.00178 -0.45040 Table 26 Regression c o e f f i c i e n t , R2 and t - s t a t i s t i c s of the regression between covariances of the 116 s e c u r i t i e s and Zu over d i f f e r e n t periods of time 58 B i b l i o g r a p h y [1] B i c k e l , P . J . and K.A. Doksum. M a t h e m a t i c a l S t a t i s t i c s : B a s i c I d e a s a n d S e l e c t e d T o p i c s . H o l d e n - D a y , I n c . , 1 s t e d . , 1977. [2] B r e n n a n , M.J. and R. Thompson. The No A r b i t r a g e P r o p e r t y  o f S e c u r i t y M a r k e t s . U n i v e r s i t y o f B r i t i s h C o l u m b i a , u n p u b l i s h e d p a p e r , November 1983. [3] B l u m e , M. B e t a s a n d t h e i r R e g r e s s i o n T e n d e n c i e s . J o u r n a l o f F i n a n c e , X, No. 3 ( J u n e 1 9 7 5 ) , p p . 78 5 - 7 9 5 . [4] C e n t r e f o r R e s e a r c h i n S e c u r i t y P r i c e s . CRSP M a s t e r F i l e : M o n t h l y a n d D a i l y D a t a [ m a c h i n e - r e a d a b l e d a t a f i l e ] . C o n d u c t e d b y G r a d u a t e S c h o o l o f B u s i n e s s , U n i v e r s i t y o f C h i c a g o . CRSP e d . , M a r c h , 1 9 8 3 . 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C a p i t a l A s s e t P r i c e s : A T h e o r y o f M a r k e t E q u i l i b r i u m U n d e r C o n d i t i o n s o f R i s k . J o u r n a l o f  F i n a n c e ( S e p t . 1 9 6 4 ) , p p . 4 2 5 - 4 4 2 . [9] Wood Gundy. Wood Gundy E q u i t y a n d B o n d D a t a b a s e s [ m a c h i n e r e a d a b l e d a t a f i l e ] . C o n d u c t e d b y Wood Gundy. Wood Gundy e d . 1 9 8 1 . Wood Gundy [ p r o d u c e r and d i s t r i b u t o r ] , 1 9 8 1 . 1 d a t a f i l e (9181 l o g i c a l r e c o r d s ) a n d a c c o m p a n y i n g c o d e b o o k . 

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