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Journal of Financial Economics 12 (1983) 311-342. North-Holland
A C O M P A R I S O N O F F U T U R E S A N D F O R W A R D P R I C E S *
K e n n e t h R . F R E N C HUniversity o f Chicago, Chicago, IL 60637, USA
Received September 1982, final version received May 1983This paper uses the pricing models of Cox, Ingersoll and Ross (1981), Richard and Sundaresan(1981), and French (1982) to examine the relation between futures and forward prices for copperand silver. There are significant differences between these prices. The average differences aregenerally consistent with the predictions of the futures and forward price models. However, thesemodels are not helpful in describing intra-sample variations in the futures-forward pricedifferences. This failure is apparently caused by measurement errors in both the price differencesand in the explanatory variables.
1 . I n t r o d u c t i o nF u t u r e s a n d f o r w a r d c o n t r a c t s a r e v e r y s im i la r; b o t h c o n t r a c t s r e p r e s e n t
a n a g r e e m e n t t o t r a d e a n a s s e t a t a s p e ci fi c t i m e i n t h e f u t u re . B e c a u s e o f t h iss i m i l a r i t y , t h e s e c o n t r a c t s a r e o f t e n t r e a t e d a s t h o u g h t h e y a r e i d e n t i c a l .H o w e v e r , f u t u r e s a n d f o r w a r d c o n t r a c t s a r e n o t i d e n ti c a l; th e d a i l y g a i n o rl o s s f r o m h o l d i n g a f u t u r e s c o n t r a c t i s t r a n s f e r r e d b e t w e e n t h e t r a d e r s a t t h ee n d o f e a c h d a y , w h i le t h e p r o f i ts o r l os s es f r o m h o l d i n g a f o r w a r d c o n t r a c ta c c u m u l a t e u n t il t h e c o n t r a c t m a t u r e s . A n u m b e r o f r e c e n t p a p e r s h a v ee x a m i n e d t h e t h e o r e t i c a l i m p l i c a t i o n s o f t h is d i f f e r en c e f o r t h e r e l a t i o nb e t w e e n f u t u r e s a n d f o r w a r d p r i c e s . F o r e x a m p l e , M a r g r a b e ( 1 9 7 6 ) , C o x ,I n g e r s o l l a n d R o s s ( 1 9 7 7 ) , a n d J a r r o w a n d O l d f i e l d ( 1 9 8 1 ) d e m o n s t r a t e t h a tt h e s e p r i c e s w i l l n o t b e e q u a l u n l e s s i n t e r e s t r a t e s a r e n o n - s t o c h a s t i c . C o x ,I n g e r s o l l a n d R o s s ( 1 9 8 1 ) , R i c h a r d a n d S u n d a r e s a n ( 1 9 8 1 ) , a n d F r e n c h( 1 98 2 ) b u i l d m o d e l s o f f u t u r e s a n d f o r w a r d p r i c e s w h i c h a l l o w t h e m t o m a k em o r e s p e ci fi c p r e d i c t i o n s a b o u t t h e r e l a t i o n b e t w e e n t h e s e p ri c e s. T h i s p a p e re x a m i n e s t h e a c c u r a c y o f m a n y o f t h es e p r e d i c t io n s f o r s i lv e r a n d c o p p e rc o n t r a c t s .
*This paper is based on my doctoral thesis at the University of Rochester. I am very gratefulto my dissertation committee, G. William Schwert (Chairman), Michael Jensen, John Long, andCharles Plosser, for their guidance. I have also received helpful comments from Robert Jarrow,Scott Richard, Richard Roll, Richard Ruback, Dennis Sheehan, Ren6 Stulz, Lee Wakeman,Jerold Zimmerman and the referee, Douglas Breeden. Financial support was generouslyprovided by the Managerial Economics Research Center, Graduate School of Management,University of Rochester; the Center for the Study of Futures Markets, Graduate School ofBusiness, Columbia University; the Foundation for Research in Economics and Education; andthe Richard D. Irwin Foundation. I would also like to thank Christopher Snyder of DataResources, Inc., and Richard Brealey for providing data. This work was completed while I was apost-doctoral fellow at the University of California, Los Angeles.0304-405x/83/$3.00 1983, Elsevier Science Publishers B.V. (North-Holland)
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312 K.R . French, Futu res and ]brward pr ices
A f o r w a r d c o n t r a c t i s s i m p l y a s a l e s a g r e e m e n t i n w h i c h d e l i v e r y a n dp a y m e n t a r e d e fe r r e d . A ll o f t h e t e rm s o f t h e s a le , s u c h a s t h e a s s e t t o b ed e l i v e r e d , t h e t i m e o f d e l i v e ry , a n d t h e p u r c h a s e p r i c e ( c a ll e d t h e f o r w a r dp r ic e ) , a r e s p e c if i ed w h e n t h e c o n t r a c t is w r i t t e n . N o p a y m e n t s a r e m a d eu n t i l t h e c o n t r a c t m a t u r e s . A t t h a t t i m e t h e s e l l e r d e l i v e r s t h e a s s e t a n d t h eb u y e r p a y s t h e f o r w a r d p r i c e . S i n c e t h e a s s e t c a n b e p u r c h a s e d j u s t b e f o r ed e l i v e r y a n d r e - s o l d i m m e d i a t e l y a f t e r w a r d , a f o r w a r d c o n t r a c t c a n b e v i e w e da s a b e t a b o u t t h e m a t u r i t y p r i c e o f t h e c o m m o d i t y . T h e p a y o f f o n t h i s b e t ise q u a l t o t h e d i f f e r e n c e b e t w e e n t h e f o r w a r d p r i c e a n d t h e m a t u r i t y s p o t p r i c e .
A f u t u r e s c o n t r a c t c a n a l s o b e v i e w e d a s a b e t a b o u t t h e m a t u r i t y p r i c e o ft h e a s s e t , b u t t h e p a r t i e s t o t h i s b e t s e t t l e -u p d a i l y . 1 A t t h e e n d o f e a c h d a y ' st r a d i n g , t h e c u r r e n t f u t u r e s p r i c e i s c o m p a r e d w i t h t h e c l o s i n g p r i c e f r o m t h ep re v i o u s d a y . I f t h e fu t u re s p r i c e h a s f a l le n , t h e i n v e s t o r w h o is l o n g i n t h ec o n t r a c t ( c o m m i t t in g h i m s e l f t o p u r c h a s e t h e c o m m o d i t y ) m u s t p a y t h e s h o r ti n v e s t o r t h e a m o u n t o f t h e d e c r e a s e . I f t h e f u t u r e s p r ic e h a s r is e n , t h e l o n gi n v e s t o r r e c e iv e s t h e a m o u n t o f t h e i n c r e a s e f r o m t h e s h o r t i n v e s to r . W h e nt h e c o n t r a c t m a t u r e s , t h e l o n g i n v e s t o r p u r c h a s e s t h e c o m m o d i t y a t t h ep r e v i o u s d a y ' s c l o s i n g , o r s e t t l e m e n t , p r i c e . S i n c e t h e c o m m o d i t y c a n b e r e -s o ld i m m e d i a t e l y a t t h e p r e v a i l i n g s p o t p r i c e , t h e s u m o f t h e c a sh f l o w sb e t w e e n t h e t w o f u t u r e s t r a d e r s is e q u a l t o t h e d i f f e r e n ce b e t w e e n t h eo r i g i n a l f u t u r e s p r i c e a n d t h e m a t u r i t y s p o t p r ic e . T h i s i s v e r y s im i l a r t o t h ec a s h f l o w f r o m t h e f o r w a r d c o n t r a c t ' b e t ' . H o w e v e r , w i t h a f o r w a r d c o n t r a c tt h e p r o f i t s a r e t r a n s f e r r e d a t m a t u r i t y , w h i l e t h e p r o f i t s f r o m a f u t u r e sc o n t r a c t a r e t r a n s f e r r e d a t t h e e n d o f e a c h d a y . 2
D e s p i t e t h e d if f er e n ce s b e t w e e n t h es e c o n t r a c t s, m o s t c o m m o d i t y t r a d e r st r e a t f u t u r e s c o n t r a c t s a s t h o u g h t h e y a r e f o r w a r d c o n t r a c t s . F o r e x a m p l e , b yi m p l i c i t l y m o d e l l i n g f u t u r e s c o n t r a c t s a s f o r w a r d c o n t r a c t s , f u t u r e s t r a d e r sh a v e d e v e l o p e d s e v e r a l ' a r b i t r a g e ' c o n d i t i o n s r e l a t i n g f u t u r e s p r i c e s , s p o tp r i c e s , a n d s t o r a g e c o s t s . M a n y e c o n o m i s t s s e e m t o a g r e e t h a t t h e d a i l ys e t t l i n g -u p h a s a n e g l i g i b l e e f f e ct o n fu t u re s p r ic e s . I n d e v e l o p i n g t h e i rm o d e l s o f f u t u r e s p ri c e s, D u s a k ( 19 7 3) , G r a u e r ( 1 97 7 ), a n d G r a u e r a n dL i t z e n b e r g e r ( 1 97 9 ) a b s t r a c t f r o m t h e s e t t l in g - u p p r o v i s i o n s o f t h e s e c o n t r a c t se n t i r e l y . O t h e r a u t h o r s , s u c h a s B l a c k (1 9 7 6 ) , e x p l i c i t l y r e c o g n i z e t h e d a i l ys e t t l i n g -u p , b u t t h e y s t i l l c o n c l u d e t h a t f u t u re s a n d fo rwa rd p r i c e s w i l l b e t h es a m e .I n li g h t o f t h is c o n s e n s u s a b o u t t h e s i m i l a r it y b e t w e e n f u t u r e s a n d f o r w a r dp r i c e s , t h e e m p i r i c a l e v i d e n c e i n t h i s p a p e r m a y b e s u r p r i s i n g . T h e r e a r es i g n if i c an t d i f fe r e n c e s b e t w e e n t h e s e p r i c e s fo r b o t h c o p p e r a n d s il v erc o n t r a c t s . F o r e x a m p l e , d u r i n g t h e 1 9 7 4 - 1 9 8 0 s a m p l e p e r i o d , t h e a v e r a g e
o / ofu t u re s fo r w a rd p r i c e d i f f e r e n c e s a r e a b o u t 0 . 1 o , 0 . 4 0 , a n d 0 .8 "4 ; fo r 3, 6 , a n d12 m o n t h s i lv e r c o n t r a c t s , r e s p e c t i v e ly .W h i s s e t t l in g - u p i s o f te n c a l l ed " m a r k i n g - t o - m a r k e t ' .e S t ri c tl y s p e a k i n g , e a c h f u t u r e s c o n t r a c t i s a b e t a b o u t t h e n e x t d a y ' s f u t u r e s p r i ce .
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K.R . French, Futures and forward prices 313T h e n e x t s e c t io n o f t h i s p a p e r o u t l i n e s t h e p r i c i n g m o d e l s o f C o x , I n g e r s o l l
a n d R o s s (1 9 81 ), R i c h a r d a n d S u n d a r e s a n ( 19 8 1) , a n d F r e n c h ( 1 98 2) . T h e s em o d e l s i m p l y t h a t f u t u r e s a n d f o r w a r d p r i ce s w ill d if fe r i n p r e d i c t a b l e w a y s .F o r e x a m p l e , t h e m o d e l s p r e d i c t t h a t t h e p r i c e f o r a f o r w a r d c o n t r a c t i sr e l a t e d t o t h e i n t e r e s t r a t e o n a l o n g - t e r m b o n d t h a t m a t u r e s a t t h e s a m et i m e a s t h e c o n t r a c t , w h i l e t h e f u t u r e s p r i c e i s r e l a t e d t o t h e r e t u r n f r o mr o l l in g o v e r o n e - d a y b o n d s u n t il t h e c o n t r a c t m a t u r e s . T h e s e p r i c e s w ill b ei d e n t ic a l ly e q u a l o n l y i f i n t e r es t r a t e s a r e n o n - s t o c h a s t i c .
S e c t io n 3 c o m p a r e s f u t u r e s a n d f o r w a r d p r ic e s f o r c o p p e r a n d s il v er a n dt e st s m a n y o f t h e p r e d i c t i o n s o f t h e p r i c i n g m o d e l s o u t l i n e d i n s e c ti o n 2. Ing e n e r a l , t h e a v e r a g e p r i c e d i f f e r e n c e s a r e c o n s i s t e n t w i t h t h e p r e d i c t i o n s .H o w e v e r , t h e m o d e l s a r e n o t h e l p f u l i n d e s c r i b i n g i n t r a - s a m p l e v a r i a t i o n s i nt h e f u t u r e s - f o r w a r d p r i c e d i f f e r e n c e s . T h i s f a i l u r e i s a p p a r e n t l y c a u s e d b ym e a s u r e m e n t e r r o r s i n b o t h t h e p r i c e d i f f e r e n c e s a n d i n t h e e x p l a n a t o r yv a r i a b l e s .
S e c t i o n 4 c o n t a i n s a b r i e f s u m m a r y a n d s o m e c o n c lu s i o n s .
2. M odels of futures and forward pricesT h i s s e c t i o n u s e s m o d e l s d e v e l o p e d b y C o x , I n g e r s o l l a n d R o s s ( 1 9 8 1 ) ,
R i c h a r d a n d S u n d a r e s a n ( 1 9 8 1 ) , a n d F r e n c h ( 1 9 8 2 ) t o e x a m i n e t h et h e o r e t i c a l d i f f er e n ce s b e t w e e n f u t u r e s a n d f o r w a r d p r ic e s. A ll o f t h e se m o d e l sa s s u m e t h a t t h e r e a r e n o t a x e s o r t r a n s a c t i o n c o s t s a n d t h a t i n d i v i d u a l s c a nb o r r o w a n d l e n d a t t h e s a m e n o m i n a l i n t e r e s t r a t e ?2.1. Arbitrage models"
F u t u r e s a n d f o r w a r d c o n t r a c t s p r o v i d e a w i d e v a r ie t y o f i n t e r t e m p o r a le x c h a n g e o p p o r t u n i t i e s . F o r e x a m p l e , b y i n it ia t in g a lo n g f o r w a r d c o n t r a c ta n d p u r c h a s i n g r i s k - f r e e b o n d s , a n i n d i v i d u a l c a n b u y a n a s s e t t o d a y t h a tw i ll b e d e l i v e r e d i n t h e f u t u r e . T o s ee th e m e c h a n i c s o f t h i s t r a n s a c t i o n ,d e f in e f ( t , T ) a s t h e f o r w a r d p r i ce o n d a y t f o r a c o n t r a c t t h a t m a t u r e s o nd a y T a n d d e f i n e R(t , T) a s t h e y i e l d t o m a t u r i t y o n a r i s k l e s s d i s c o u n t b o n dt h a t p a y s o n e d o l l a r o n d a y T. T h e c u r r e n t p r i c e o f t h is b o n d i s
U(t, T ) = e x p [ - ( T - t ) R ( t , T ) ] . ( 1 )A t r a d e r c a n m a k e a d e l ay e d p u r c h a s e b y in i ti a ti n g o n e f o r w a r d c o n t r a c ta n d i n v e s t i n g f ( t , T ) B ( t , T ) d o l l a r s i n r i s k l e s s b o n d s . W h e n t h e c o n t r a c tm a t u r e s , h e r e c e i v e s f ( t , T ) d o l l a r s f r o m th e b o n d s a n d e x c h a n g e s th i s f o ro n e u n i t o f t h e c o m m o d i t y . I n e ff ec t, a n i n v e s t m e n t o f f ( t , T ) B ( t , T ) t o d a y
3The mod els also assum e that investors will not default on any contract. T hi s implies thatthere is a finite upper bound on the daily price changes and on the daily interest rates.
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314 K.R . French. Future s and Jorw ard pr ices
y i e ld s o n e u n i t o f t h e c o m m o d i t y o n d a y Z A l t e r n a ti v e l y , t h e t r a d e r c o u l dr e v e r s e t h e s t r a t e g y a n d o b t a i n f ( t , T )B(t , T) d o l l a r s t o d a y i n e x c h a n g e f o rt h e a s s e t a t t i m e T .B e c a u s e t h i s i n t e r t e m p o r a l e x c h a n g e i s a v a i l a b l e , i n v e s t o r s m u s t b em a r g i n a l l y i n d i f f e r e n t b e t w e e n f ( t , T ) B ( t , T ) d o l l a r s t o d a y a n d o n e u n i t o ft h e c o m m o d i t y o n d a y T. B y de f in i n g P ( T ) a s th e ( u n k n o w n ) p r ic e o f t h ec o m m o d i t y a t t i m e T , t h is i n d i f f er e n c e c a n b e e x p r e s s e d in p u r e l y d o l l a rt e rm s ; f(t , T)B(t, T) m u s t b e t h e v a l u e o n d a y t o f P ( T ) d o l l a r s o n d a y T .E q u i v a l e n t l y , t h e f o r w a r d p r i c e m u s t e q u a l t h e p r e s e n t v a l u e o f t h e m a t u r i t ys p o t p r i c e t i m e s t h e g r o s s r e t u r n f r o m a l o n g - t e r m b o n d ,
f ( t , T)=exp[(T- t )R( t , T)]PV, .7 .[P(T)] . (2 )In eq. (2), PV,.T ( ') d e n o t e s t h e p r e s e n t v a l u e a t t i m e t o f a p a y m e n t r e c e i v e da t t ime T.
F o r e x a m p l e , c o n s i d e r a f o r w a r d c o n t r a c t o n a s t o c k t h a t p a y s n od i v i d e n d s. S i n c e t h e c u r r e n t s t o c k p r i c e m u s t b e t h e p r e s e n t v a l u e o f t h efu t u re s l o c k p r i c e , t h e fo rwa rd p r i c e i s e q u a l t o
f ( t , T)=exp[(T- t )R( t , T ) ]P( t ) . ( 3 )T h i s r e s u l t i s i n t u i t i v e l y a p p e a l i n g . U s i n g a f o r w a r d c o n t r a c t t o p u r c h a s e t h es t o c k o n d a y T i s e q u i v a l e n t t o p u r c h a s i n g t h e s t o c k t o d a y , e x c e p t t h ef o r w a r d c o n t r a c t a l l o w s t h e p a y m e n t t o b e d e f e r r e d . T h e r e f o r e , t h e f o r w a r dp r i c e is e q u a l t o t h e d e f e r r e d v a l u e o f t h e c u r r e n t s t o c k p r i ce .
T h e p r e s e n t v a l u e o f t h e m a t u r i t y s p o t p r ic e is n o t o b s e r v a b l e f o r m o s tc o m m o d i t i e s ; a m o r e c o m p l e t e m o d e l m u s t b e i n t r o d u c e d t o e v a l u a t e t h ep a y m e n t i n (2 ) a n d t o d e t e rm i n e t h e fo rwa rd p r i c e . Ho we v e r , e q . ( 2 ) i s u s e fu lf o r h i g h l i g h ti n g t h e d i f fe r e n c e s b e t w e e n f u t u r e s a n d f o r w a r d p r ic e s.
C o x , I n g e r s o l l a n d R o s s ( 1 9 8 1 ) a n d F r e n c h ( 1 9 8 2 ) d e v e l o p a s i m i l a re x p r e s s i o n f o r f u t u r e s p r i c e s . T h e y d e m o n s t r a t e t h a t t h e f u t u r e s p r i c e m u s te q u a l t h e p r e s e n t v a l u e o f t h e p r o d u c t o f t h e m a t u r i t y s p o t p r i ce a n d t h eg r o s s r e t u r n f r o m r o l l i n g o v e r o n e - d a y b o n d s ,
F ( t , T ) = P V , r e x p / ~ ( ~ - . r + l ) P ( T ) .I
( 4 )I n t h i s e q u a t i o n , F(t ,T) i s t h e f u t u r e s p r i c e o n d a y t f o r a c o n t r a c t t h a tm a t u r e s o n d a y T a n d R ( ~ , ~ + 1) is th e c o n t i n u o u s l y c o m p o u n d e d i n te r e str a t e o n a o n e - d a y b o n d f r o m d a y T t o d a y + 1 .
E q s . (2 ) a n d ( 4) i n d i c a t e t h a t t h e d i f fe r e n c e b e t w e e n f o r w a r d a n d f u t u r e sp r ic e s is r e l a t e d t o t h e d i f f e re n c e b e t w e e n h o l d i n g a l o n g - t e r m b o n d a n dr o l li n g o v e r a s e r ie s o f o n e - d a y b o n d s . T h e o n l y c a s h f lo w t h a t is r e le v a n t t o
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K .R . F r e n c h ,F u t u r e sa n df o r w a r dp r i c e s 315t h e f o r w a r d t r a d e r is a g r e e d o n t o d a y a n d p a i d o n t h e m a t u r i t y d a te .T h e r e f o r e , t h e r e l e v a n t i n t e r e s t r a t e i n d e t e r m i n i n g t h e f o r w a r d p r i c e i s t h ek n o w n y i e l d o n a m u l t i - p e r i o d b o n d . O n t h e o t h e r h a n d , w h i l e t h e f u t u r e st r a d e r k n o w s t h e t o t a l p a y m e n t s h e w ill h a v e t o m a k e , t h e t i m i n g o f t h es ec a s h f l o w s is o n l y d e t e r m i n e d a s th e c o n t r a c t m a t u r e s . B e c a u s e o f t h isu n c e r t a i n t y , t h e f u tu r e s p r ic e is a fu n c t i o n o f t h e u n k n o w n o n e - d a y i n te r e s tr a t e s t h a t a r e e x p e c t e d t o a r i s e o v e r t h e l ife o f c o n t r a c t . I n g e n e r a l , t h ef u t u r e s p r i c e w i l l n o t e q u a l t h e f o r w a r d p r i c e u n l e s s t h e s e i n t e r e s t r a t e s a r en o n - s t o c h a s t i c .
C o x , I n g e r s o l l a n d R o s s u s e t h e a r b i t r a g e m o d e l s i n e q s . ( 2 ) a n d ( 4 ) t od e v e l o p s e v e ra l p r o p o s i t i o n s a b o u t t h e r e l a ti o n b e t w e e n f o r w a r d a n d f u t u r e sp r i c e s . I n a c o n t i n u o u s - t i m e , c o n t i n u o u s - s t a t e e c o n o m y , t h e y f i n d t h a t t h ed i f f e r e n c e b e t w e e n t h e s e p r i c e s i s e q u a l t o
F ( t , T ) - f ( t , T ) = - P V ~ . T { i F ( w , T ) c o v [ F ( w , T ) ,B ( w , T ) ] d w } /B ( t , T ) . ( 5 )I n t h i s e q u a t i o n , c o v [ F ( w , T ) , B ( w , T ) ] is d e f i n e d a s t h e l o c a l c o v a r i a n c e a tt i m e w b e t w e e n t h e p e r c e n t a g e c h a n g e i n th e f u t u re s p r i ce a n d t h e p e r c e n t a g ec h a n g e i n t h e b o n d p r i c e . T h i s r e s u l t h a s s e v e r a l i m p l i c a t i o n s . F o r e x a m p l e ,t h e l o c a l c o v a r i a n c e b e t w e e n t h e f u t u r e s a n d b o n d p r i c e s i s a l m o s t c e r t a i n l yp o s i t i v e f o r f i n a n c i a l a s s e t s , s u c h a s t r e a s u r y b i l l s . T h e r e f o r e , t h e f o r w a r dp r i ce s h o u l d b e a b o v e t h e f u t u r e s p r ic e f o r t h e se a s s et s. O n t h e o t h e r h a n d ,o n e w o u l d e x p e c t t h e f u t u r e s p r i c e t o b e a b o v e t h e f o r w a r d p r i c e f o r m o s tr e a l c o m m o d i t i e s . U n e x p e c t e d i n f l a t i o n a n d c h a n g e s i n e x p e c t e d i n f l a t i o np r o b a b l y p l a y a m a j o r r o l e i n d e t e r m i n i n g t h e c o v a r i a n c e b e t w e e n b o n dp r i c e s a n d c o m m o d i t y p r i c e s . S i n c e u n e x p e c t e d i n f l a t i o n m o v e s b o n d p r i c e sa n d c o m m o d i t y p r ic e s i n o p p o s i t e d ir e ct io n s , t h e c o v a r i a n c e s h o u l d b en e g a t i v e a n d t h e f u t u r e s - f o r w a r d p r i c e d i ff e re n c e s h o u l d b e p o s i t iv e . 4
E q . ( 5 ) a l s o i m p l i e s t h a t t h e d i f f e r e n c e b e t w e e n t h e f u t u r e s a n d f o r w a r dp r i c e s w i ll b e r e l a t e d t o t h e v a r i a n c e o f b o t h t h e f u t u r es p r i c e s a n d t h e b o n dp r i c e s. S p e c if i ca l ly , i f t h e c o r r e l a t i o n b e t w e e n t h e f u t u r e s a n d b o n d p r i c e s isc o n s t a n t , t h e a b s o l u t e v a l u e o f t h e p r i c e d if f e r e n c e w i ll b e a n i n c r e a s i n gf u n c t io n o f t h e m a r k e t ' s e x p e c t a ti o n o f b o t h t h e f u tu r e s a n d b o n d p r ic ev a r i a n c e s .
C o x , I n g e r s o l l a n d R o s s a l s o u s e t h e a r b i t r a g e m o d e l s t o s h o w t h a t t h e
4The local covariance between the futures price and the bo nd price may be positive for somecommodities. For example, if a commodity is used in the production of a durable good, anincrease in the exp ected real interest rate will reduce the dem and for the comm odity. Therefore,changes in the expected rea l interest rate w ill ten d to make the comm odity and bond pricesmov e together. How ever, since the inflation rate is mu ch more volatile than the exp ected realinterest rate, the covariance is still expected to be negative for most comm odities.
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316 K.R . French, F utures and /orwa rd prices
d i f fe r e n c e b e t w e e n t h e f u t u r e s a n d f o r w a r d p r i ce s c a n b e e x p r e s s e d a s
F ( t , T ) f ( t , T ) = P V t , r e x p i { w ) d w [ P { w ) / ~ ( w , T ) ]
{v a r [ /~ (w , T ) ] - c o v [P (w} , /~ (w , T ) ]} d w l , ) {6}
i f t h e c o m m o d i t y i s s t o r e d c o s t l e s s ly o v e r t h e c o n t r a c t p e r i o d . I n t h i se q u a t i o n , f ( w ) is t h e i n s t a n t a n e o u s i n t e r e st r a t e a t t i m e w , v a r [ B ( w , T ) ] is t h el o ca l v a r i a n c e o f th e p e r c e n t a g e c h a n g e in t h e b o n d p r ic e , a n dc o y [ P ( w ) , / ~(w , T ) ] is t h e l o c a l c o v a r i a n c e b e t w e e n t h e p e r c e n t a g e c h a n g e int h e s p o t p r i c e a n d t h e p e r c e n t a g e c h a n g e i n t h e b o n d p r i c e . T h i s r e s u l ti m p l i e s t h a t t h e fu t u re s fo r w a rd p r i c e d if f e r e n c e w ill b e a d e c re a s i n g fu n c t i o no f th e m a r k e t ' s e x p e c t a t i o n o f t h e s p o t p r i c e v a r i a n c e i f t h e p r i c e d i f fe r e n c e isn e g a t i v e a n d i f t h e lo c a l c o r r e l a t i o n b e t w e e n t h e sp o t a n d b o n d p r i c e s isc o n s t a n t .
2.2. Uti l ity b ase d models"T h e r e l a t i o n b e t w e e n f u t u r e s a n d f o r w a r d p r i c e s c a n b e e x p l o r e d f u r t h e r b ya s s u m i n g t h a t m a r k e t s a r e c o m p l e t e a n d t h a t t h e r e i s s o m e r a t i o n a l
i n d i v i d u a l w h o a c t s t o m a x i m i z e a t i m e - a d d i t i v e e x p e c t e d u t i l i t y f u n c t i o n o ft h e f o r m
J = E , { ~ , e x p [ - t , t r - - t ) ] U [ 8 ( ~ ) {7)
In e q . ( 7) , ~ '(~) is a v e c t o r i n d i c a t i n g t h e (n o n -n e g a t i v e ) q u a n t i t y o f e a c hg o o d c o n s u m e d o n d a y r , a n d p i s a u t i l i t y d i s c o u n t f a c t o r . F u r t h e r , U ( . ) i s as i n g l e p e r i o d , v o n N e u m a n n - M o r g e n s t e r n u t i l i t y f u n c t i o n t h a t i s i n c r e a s i n g ,s t r i c t l y q u a s i - c o n c a v e , a n d d i f f e r e n t i a b l e . F i n a l l y , d e f i n e 2 ( r ) a s t h e m a rg i n a lv a l u e o f a d o l l a r t h a t i s r e c e i v ed a t t im e r . F o r n o t a t i o n a l c o n v e n i e n c e , th i sm a rg i n a l v a l u e i s d i s c o u n t e d b a c k t o d a y t . I n o t h e r wo rd s , 7 , ( r ) i s t h ed i s c o u n t e d m a r g i n a l u t i li t y o f m o n e y o n d a y -c,
2,(r) = e x p [ - p ( r t )]i f( i ,z) /P(i ,r) . {8)
In t h i s e x p re s s i o n , P(i, r ) is t h e p r i c e o f a n y c o m m o d i t y t h a t is c o n s u m e d o nd a y r a n d i f( i, r) is th e m a r g i n a l u t il i ty o f t h is c o m m o d i t y o n d a y r . N o t i c etha t . u n l ike 2 (r}, if (i , r ) is no t d i sc ou n te d bac k to da y t .T h e d i s c u s s i o n i n t h e p r e v i o u s s e c t i o n s h o w s t h a t i n v e s t o r s m u s t b e m a r -
g i n a l l y i n d i f f e r e n t b e t w e e n f ( t , T ) d o l l a r s t o d a y a n d e x p [ ( T - t ) R ( t , T ) ] P ( T )
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K.R . French, F utures and forw ard pr ices 317d o l l a r s o n d a y T . F o r a n i n d i v i d u a l w i t h t h e t i m e - a d d i t i v e u t i l i t y f u n c t i o nin (7), t h i s indi f fe renc e ca n b e exp ressed as
f ( t , T ) 2 ( 0 = e xp [ ( T - t ) R ( t , T ) ] E t [ P ( T ) ~ (T ) ] ; (9 )t h e m a r g i n a l u t i li ty o f f ( t , T ) d o l la r s t o d a y m u s t e q u a l t h e e x p e c te d m a r g i n a lu t i l i t y o f e x p [ ( T - t ) R ( t , T ) ] P ( T ) d o l l a rs a t ti m e T . E q u i v a l e n t ly , t h e fo r w a r dp r i c e m u s t e q u a l
f ( t , T ) = e x p [ ( T - t ) R ( t , T ) ] E t [ P ( T ) ~ ( T ) /2 ( t ) ] . (1o)I f t h i s c on d i t i o n w e r e no t sa t is f ie d , the i n d i v i du a l c ou l d i nc r e a se h i s l if e - ti mee x p e c t e d u t i li ty b y u s i n g a p o r t fo l i o o f f o r w a r d c o n t r a c t s a n d b o n d s t ot ra n s f e r m o n e y b e t w e e n d a y t a n d d a y T. 5
Th e t i me - a dd i t i ve u t i l it y f unc t i on c a n a l so be use d t o c ha r a c t e r i z e f u t u r e sp r ic e s . E q . (4 ) i mpl i e s t ha t , i n e qu i l i b r i um, 6 inve s t o r s a r e i nd i f f e r e n t be t w e e nF ( t , T ) d o l la r s t o d a y a n d e x p [ ~ T , ~ / ~ ( z ,~ + 1)] P ( T ) d o l la r s a t t i m e T. T h i si nd i f f e re nc e c a n be e xpr e sse d a s
o r
F(t,T),~(t)=E, exp ~(~,~+1) P(T)Tt(T), (11)
{ I ] tF ( t , T ) = E t e x p ~ / ~ ( z , ~ + l ) P ( T ) ~ ( T ) / X ( t ) . (12)t = tEqs . (10 ) a n d ( 12) r e e m pha s i z e t h a t t he f o r w a r d p r ic e i s a f unc t i on o f t heg r o s s r e t u r n f r o m h o l d i n g a l o n g - t e r m b o n d w h i l e t h e f u t u r e s p r i c e i s af u n c t i o n o f t h e g r o s s r e t u r n f r o m r o l li n g o v e r o n e - d a y b o n d s . 7
3 . A com pa r ison of futures and forward pr icesAs t he d i sc u ss i on i n se c t i on 1 i nd i c a t e s , the s i m i l a r i t y be t w e e n f u t u r e s a n df o r wa r d p r i c e s o f t e n l e a ds pe op l e t o v i e w t he m a s i de n t i c a l c on t r a c t s . Them o d e l s o f f u t u r e s a n d f o r w a r d p r ic e s d e s c r ib e d i n t h e p r e v i o u s s e c t io nh i gh l i gh t t he t he or e t i c a l d i f f e r e nc e s be t we e n t he se c on t r a c t s . Th i s se c t i one xa mi ne s t he e mpi r i c a l e f f e c t s o f t he se d i f f e r e nc e s a nd t e s t s se ve r a l
p r e d i c t i o n s o f t h e m o d e l s .5Eq. (10) and eq . (12) be low ac tua l ly fo l low di rec t ly f rom eqs . (2) and (4) us ing the re la t ionP Vt, r(Y ) - E, [ } '~(T )/2( t)].61n e qu i l i b r i um, no i nve s t o r wa n t s t o ma ke a dd i t i ona l t r a de s a t t he e x i s t i ng p r i c e s .7The mode l s i n e qs . (10 ) a nd (12 ) a re de ve l ope d by R i c ha rd a nd S unda re sa n (1981) a ndF re nc h (1982) . In a dd i t i on , t he fo rwa rd p r i c e mode l i n (10 ) i s s i mi l a r t o t he fu t u re s p r i c e mode l si n G r a u e r ( 1 9 7 7 ) a n d G r a u e r a n d L i t z e n b e r g e r ( 1 9 7 9 ) .
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318 K.R. French, Futures and Jbrward priees3 .1 . M a t c h i n g f u t u r e s a n d f o r w a r d p r ic e s
T h e t e st s in t h i s s e c ti o n c o m p a r e f u t u re s a n d f o r w a r d p r i c e s f o r s i lv e r a n dc o p p e r f r o m 1 96 8 t h r o u g h 1 98 0. M o s t o f t h e o r g a n i z e d f o r w a r d a c t i v i t yd u r i n g t hi s p e r i o d o c c u r r e d o n t h e L o n d o n M e t a l E x c h a n g e ( L M E ) . 8M e m b e r s o f t h i s e x c h a n g e t r a d e s p o t a n d f o r w a r d c o n t r a c t s o n s i lv e r, c o p p e r ,a n d s e v e ra l o t h e r m e t a l s . A l t h o u g h o t h e r c o n t r a c t s a r e a v a i l a b l e , t hes t a n d a r d s i lv e r c o n t r a c t s h a v e m a t u r i t i e s o f 3 , 6 , a n d 12 m o n t h s a n d t h es t a n d a r d c o p p e r c o n t r a c t h a s a 3 - m o n t h m a t u r i t y . A n e w s et o f c o n t r a c t s i sw r i t t e n e a c h d a y . F o r e x a m p l e , th e 3 - m o n t h s i lv e r c o n t r a c t i n i t i a t e d o nF e b r u a r y 10, 19 77 , m a t u r e d o n M a y 10, w h i l e t h e c o n t r a c t w r i t t e n o nF e b r u a r y 11 m a t u r e d o n M a y 11.
T h e f u t u r e s p r i c e s r ef le c t t r a d i n g o n t w o e x c h a n g e s i n t h e U n i t e d S t a t e s - -t he C o m m o d i t y E x c h a n g e ( C o m e x ) in N e w Y o r k a n d t h e C h i c a g o B o a r d o fT r a d e ( C B T ) . 9 T h e c o n t r a c t m a t u r i t i e s u s e d in t h e s e m a r k e t s f o l l o w ad i ff e r en t c o n v e n t i o n t h a n t h a t u s e d i n f o r w a r d m a r k e t s . W h i l e f o r w a r dt r a d e r s w r i t e c o n t r a c t s w i t h a s p e c if i c t i m e u n t i l m a t u r i t y , s u c h a s 3 m o n t h s ,f u t u r e s t r a d e r s w r i t e c o n t r a c t s f o r s p e ci fi c m a t u r i t y m o n t h s . F u r t h e r , f u t u r e sc o n t r a c t s a r e t r a d e d c o n t i n u a l l y f o r u p t o 2 y e a r s. F o r e x a m p l e , a n i n v e s t o rc o u l d i n i t ia t e a D e c e m b e r 1 97 8 s i lv e r c o n t r a c t a n y t i m e f ro m J a n u a r y 1 97 7u n t il i t m a t u r e d t w o y e a r s l a te r .
I n c o m p a r i n g f u t u r es a n d f o r w a r d p r ic e s o n e w o u l d l i k e t o s i m u l t a n e o u s l yo b s e r v e f u t u r e s a n d f o r w a r d c o n t r a c t s f o r t h e s a m e c o m m o d i t y a n d t h e s a m em a t u r i t y . T h e d i f f e re n t m a t u r i t y s t r u c t u r e s u s e d i n t h e f u t u r e s a n d f o r w a r dm a r k e t s m a k e t h is d if f i c ul t. T h e d i s c u s s i o n in t h e fi rs t t w o s e c t i o n s a s s u m e st h a t f u t u r e s a n d f o r w a r d c o n t r a c t s h a v e p r e c i s e m a t u r i t y d a t e s . W h i l e t h is ist r u e fo r f o r w a r d c o n t r a c t s , i t is n o t t r u e f o r m o s t f u t u r e s c o n t r a c t s , i n c l u d i n gt h o s e e x a m i n e d h e re . I ns t e a d , a s h o r t f u t u re s t r a d e r m a y c h o o s e t o m a k ed e l i v e ry a n y t i m e d u r i n g t h e m a t u r i t y m o n t h . T h e e x c h a n g e ' s c l e a r i n g h o u s et h e n a s s i g n s t h e s h i p m e n t t o th e l o n g t r a d e r w i t h t h e ' o l d e s t ' c o n t r a c t . T h et e s t s r e p o r t e d i n t h i s s e c t i o n a s s u m e t h a t f u t u r e s c o n t r a c t s m a t u r e i n t h e f i r s tw e e k o f t h e d e l i v e r y m o n t h . 1
T h e f u tu r e s a n d f o r w a r d p r ic e s h a ve s e v e r a l o th e r c h a r a c t e r i s t i c s t h a t m a k ec o m p a r i s o n s b e t w e e n t h e m d i ff ic u lt . F o r e x a m p l e , b o t h C o m e x a n d t he C B Ti m p o s e l i m i t s o n t h e d a i l y p ri c e m o v e m e n t o f a n y f u t u r e s c o n t r a c t t h a t is n o t
SSilver contra cts began trad ing on the LM E in F ebru ary 1968. The London Bullion Ma rketprices for silver spot and forward contr acts are used before 1973."The commodity price data are obtained from several sources: the Commodity Services, Inc.,data bank, provided by the Center for the Study of Futures Markets at Columbia University;the D ata Resources. Inc ., commodities da ta bank: the W all Street Journal: and the Journal ofCommerce.~'Specifically, the tests use prices observed on Fridays. The relevant Fr ida y for each futurescontra ct is chosen so that the m atching forward con tract m atures during the first we ek of thefutures delivery month. All of the tests have been replicated using four other maturity periods:the second, third, and fourth w ee k of the m aturity m onth and the first business day o f thatmonth. The results are not substa ntially different for the different matu rity assumptions.
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K .R . French, Futures and forw ard prices 319
i n i t s de l i ve r y mont h . The se l i m i t s c ons t r a i n t he f u t u r e s p r i c e t o l i e wi t h i n ar a n g e d e t e r m i n e d b y t h e p r e v i o u s d a y ' s s e t t l e m e n t p r ic e . I f a l i m i t is r e a c h e d ,t h e d a y ' s t r a d i n g i s e ff e ct iv e ly s t o p p e d u n l e ss t h e e q u i l i b r i u m p r ic e m o v e sb a c k w i t h i n t h e l i m i t s . I n o t h e r w o r d s , w h e n a l i m i t m o v e o c c u r s t h er e p o r t e d p r i c e m a y b e s i g n if i ca n t ly d i f fe r e n t f r o m t h e u n o b s e r v e d m a r k e t -c l e a r i ng p r ic e . To r e duc e t he e f fe c t o f t h i s m e a s ur e m e n t e r r o r , a n y f u t u r e sp r i c e t ha t r e f l e c t s a l i m i t move i s no t i nc l ude d i n t he t e s t s .P e r h a p s t h e b i g g e s t p r o b l e m s e n c o u n t e r e d i n t r y i n g t o m a t c h f u t u r e s a n df o r w a r d p r i c e s a r i s e b e c a u s e t h e f o r w a r d c o n t r a c t s a r e t r a d e d i n G r e a tBr i t a i n w hi l e t he f u t u r e s c o n t r a c t s a r e t r a d e d i n t he U ni t e d S t a te s . 11 I de a l l y ,t h e f u t u r e s a n d f o r w a r d p r i c e s s h o u l d b e o b s e r v e d s i m u l t a n e o u s l y . I n f a c t ,p r i c e s f r o m t h e L o n d o n e x c h a n g e s a r e r e c o r d e d s e v e r a l h o u r s b e f o r e t h eA m e r i c a n p r ic e s a r e o b s e r v e d . T h i s d if fe r en c e i n t r o d u c e s m e a s u r e m e n t e r r o rb e t w e e n t h e f u t u r e s a n d f o r w a r d p r i c e s .
T w o o t h e r c o m p l i c a t i o n s a re p o t e n t i a l ly m o r e s e r io u s . F i rs t , A m e r i c a nf u t u r e s p r i c e s a r e d e n o m i n a t e d i n d o l l a r s w h i l e L o n d o n f o r w a r d c o n t r a c t sa r e d e n o m i n a t e d i n p o u n d s s t e r l i n g . B e f o r e t h e s e p r i c e s c a n b e c o m p a r e d ,t h e y m u s t b e c o n v e r t e d i n t o t h e s a m e c u r r e n c y . S e c o n d , s i l v e r o r c o p p e r i nL o n d o n i s n o t e x a c t l y t h e s a m e c o m m o d i t y a s i t i s i n N e w Y o r k o r C h i c a g ob e c a u s e o f t r a n s p o r t a t i o n c o s t s a n d i n t e r n a t i o n a l t r a d e r e s t r ic t io n s .O n e w a y t o d e a l w i t h t h e s e p r o b l e m s i s t o a s s u m e t h a t t h e e x p e c t e dd i ff e re n c e b e t w e e n t h e e x c h a n g e - a d j u s t e d s p o t p r i ce s is a c o n s t a n t f r a c t i o n o ft he e xpe c t e d spo t p r i c e s ,
E , [ P ( T , $ ) ] = ( 1 +b)E,[P(T,)X(T,T) ] . (13)I n t h i s e xpr e ss i on , P ( T ,$ ) i s t he spo t p r i c e a t t i me T de nomi na t e d i n do l l a r s ,P ( T , ) i s t h e s p o t p r i c e i n p o u n d s , a n d ) ~ ( T , T ) i s t h e s p o t e x c h a n g e r a t eb e t w e e n d o l l a r s a n d p o u n d s a t t i m e T . A l t h o u g h t h e e x p e c te d b a s isd i ff e r en t ia l , b , is a s s u m e d t o b e c o n s t a n t t h r o u g h t im e , i t m a y v a r y a c r o s sc o m m o d i t i e s . F o r e x a m p l e , si n ce t h e c o s t o f t r a n s p o r t i n g s il ve r b e t w e e nL o n d o n a n d t h e U n i t e d S t a te s is s m a l l r e la t iv e t o t h e v a l u e o f t h ec o m m o d i t y , o n e e x p e c t s t h a t t h e a b s o l u t e v a l u e o f t h e b a s i s d i f fe r e n ti a l f o r
l i T h e p r i c e c o n t r o l s i m p o s e d i n t h e U n i t e d S t a t e s f r o m A u g u s t 1 5 , 1 9 7 1 , t h r o u g h A p r i l 3 0 ,1 9 7 4 , c o u l d c a u s e m o r e p r o b l e m s . B o t h t h e s p o t p r i c e s a n d t h e f u t u r e s p r i c e s f o r c o p p e r a n ds i lv e r w e r e s u b je c t t o t h e s e c o n t r o l s . F o r e x a m p l e , d u r i n g t h e f i r s t 9 0 d a y s o f th e c o n t r o l p e r i o dn o n e o f t h e f u t u r es c o n t r a c t s w a s a l l o w e d t o t r a d e a b o v e i ts M a y 2 5, 1 9 70 , p ri c e o r t h e a v e r a g eo f t he p r i c e s fo r t he 30 da ys p re c e d i n g Augu s t 15, 1971 , wh i c he ve r wa s h i ghe r . B e c a use o f t hep o t e n t i a l d i s t o r t i o n s c a u s e d b y t h e p r i c e c o n t r o l s , a l l o f t h e t e s t s h a v e b e e n d u p l i c a t e d u s i n gt h r e e s e p a r a t e s u b p e r i o d s : a p r e - c o n t r o l p e r i o d , fr o m J a n u a r y 1, 19 68 , t h r o u g h A u g u s t 1 4, 1 97 1;a p r i c e c on t ro l pe r i od f rom A ugu s t 15 , 1971 , t h ro ug h A pr i l 30 , 1974 ; a nd a pos t - c o n t ro l pe r i o df rom Ma y 1 , 1974 , t h rough De c e mbe r 31 , 1980 . S u rp r i s i ng l y , t he p r i c e c on t ro l s do no t a f f e c t t her e s u l ts o f t h e s e t es t s. F o r a m o r e c o m p l e t e d e s c r i p t io n o f th e p r i c e c o n t r o l s i m p o s e d o n t h ec o p p e r a n d s i lv e r m a r k e t s a n d s o m e t e s ts o f t h e e f fe c t o f th e s e c o n t r o l s , s ee L e v i c h a n d W h i t e(1981).LF.E. B
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3 2 0 K.R . French, Futures and Jbrward pricess i lv e r is sm a l l. O n t h e o t h e r h a n d , t h e b a s is d i f f e r e n ti a l f o r c o p p e r m a y b eh i g h e r b e c a u s e t h e r e l a t i v e t r a n s p o r t a t i o n c o s t s a r e h i g h e r .
U n d e r t h is c o n s t a n t - e x p e c t e d - b a s i s - d i f f e r e n t i a l m o d e l , t h e A m e r i c a nf o r w a r d p r ic e c a n b e e s t i m a t e d b y t h e p r o d u c t o f t h e L o n d o n f o r w a r d p r ic e ,t h e f o r w a r d e x c h a n g e r a t e , a n d t h e b a s is a d j u s t m e n t , ~2
f ( t , T , $ ) = ( 1 +b)f(t, T,)X(t, T). (14)T a b l e 1 p r e s e n t s e s t i m a t e s o f t h e b a s i s d i f fe r e n t ia l fo r s il v e r a n d c o p p e r a s
a p e rc e n t a g e o f t h e s p o t p r i c e . 13 T h e B r i t is h s p o t p r i c e s u s e d i n t h e s ee s t i m a t e s a r e m e a s u r e d b y p r i c e s f r o m t h e L o n d o n M e t a l E x c h a n g e . E x p l i c i ts p o t p r i c e s f o r c o p p e r a n d s il v e r a r e n o t a v a i l a b l e f r o m t h e A m e r i c a n f u t u r e se x c h a n g e s . I n s t e a d , t h e p r i c e f o r t h e d e l i v e r a b l e f u t u r e s c o n t r a c t - - c a l l e d t h ec a s h p r i c e - - i s u s e d . F o r e x a m p l e , t h e D e c e m b e r s i l v e r f u t u r e s p r i c e i s u s e dt o e s ti m a t e t h e A m e r i c a n s p o t p r i c e d u r i n g D e c e m b e r .
T h e e s t i m a t e s o f th e b a s i s d i f f e r e n t i a l f o r s i l v e r i n t a b l e 1 a r e v e ry s m a l l.F o r e x a m p l e , t h e a v e ra g e d i f f e r e n t ia l o v e r t h e fu ll 1 9 68 1 9 80 s a m p l e p e r i o d i sless tha n 0 .1~ ,, wi th a t - s ta t i s t i c o f 0 .53.14 Th is i s co ns i s ten t w i th th er e l a t i v e l y s m a l l t r a n s p o r t a t i o n c o s t s f o r s i l v e r . T h e e s t i m a t e s f o r c o p p e r a r em u c h l a rg e r ; t h e a v e ra g e b a s i s d i f f e r e n t i a l f o r t h e fu l l s a m p l e p e r i o d i s-0 . 3 4 . ~ , a n d t h e a v e ra g e fo r t h e f i r s t s u b p e r i o d , f ro m 1 9 6 8 t h ro u g h 1 9 7 3 , i s
1.87J~,.T h e f o r w a r d p r i c e s t h a t a r e c o n v e r t e d f r o m p o u n d s t o d o l l a r s i n t h e t e s t sb e l o w a l l h a v e m a t u r i t i e s o f 3 m o n t h s o r m o r e . T h e r e f o r e , it is o n l y n e c e s s a r yt o a s s u m e t h a t t h e e x p e c t e d b a s i s d i f f e r e n t i a l i s c o n s t a n t f o r f o r e c a s t h o r i z o n so f a t l e a s t 3 m o n t h s . T h i s is we a k e r t h a n a s s u m i n g t h a t t h e b a s is d i f f e re n t i a la l w a y s e q u a l s b ,
P ( t , $ ) = ( 1 +b)P(t ,)X(t , t) . (15)1 2 T hi s c o n v e r s i o n a l s o a s s u m e s t h a t, c o n d i t i o n a l o n t h e i n f o r m a t i o n a v a i l a b le a t t i m e t, t h ec o v a r i t m c e o f t h e m a r g i n a l u t i l it y o f a d o l l a r w i t h t h e d i ff e r en c e b e t w e e n t h e b a s i s - a d j u s t e d
m a tu r i ty spo t p r i c e s i s z e r o ; c ov t [ ~ - ( T , $ ) , ( I + b ) P ( T , ) .~ ( T ,T ) P[ T, $)] = (/.F r e n c h ( 1 9 8 2 ) d e m o n s t r a t e s t h a t e q . ( 1 4 ) h o l d s u n d e r t h e a s s u m p t i o n t h a t t h i s c o v a r i a n c e i sz e r o a n d t h a t t h e e x p e c t e d b a s i s d i f f e r e n t i a l , h , i s z e r o . E x t e n d i n g t h i s r e s u l t t o t h e g e n e r a l c a s eo f a n y c o n s t a n t e x p e c t e d b a s i s d if f e re n t i al is s t r a i g h t f o r w a r d .~ 3 A l t h o u g h t h e m o d e l i n e q . ( 1 3 ) i s s p e c i f i e d i n t e r m s o f t h e l e v e l s o f t h e s p o t p r i c e s , t h e s ee s t i m a t e s a n d a ll o f t h e t e s t s b e lo w u s e t h e l o g a r i t h m s o f t h e p r ic e s. T h i s e l i m i n a t e s s o m eh e t e r o s k e d a s t i c i t y .T h e e x c h a n g e r a te s u s e d i n t a b l e 1 a n d i n t h e t e s t s b e l o w a r e o b t a i n e d f r o m t h e Bank ~/"England Quarterly Bulletin, t h e F e d e r a l R e s e rv e B a n k o f N e w Y o r k , a n d D a t a R e s o u r c es , I n c.A l t h o u g h d a i l y d a t a a r e a v a il a b l e d u r i n g m o s t o f t h e p e ri o d , o n l y t h e e x c h a n g e r a te s f o r F r i d a ya r e a v a i l a b l e f r o m 1 9 6 8 t h r o u g h 1 97 0. T h i s d o e s n o t c a u s e p r o b l e m s f o r m o s t o f t h e t e s tsb e c a u s e t h e y o n l y u s e f u t u re s a n d f o r w a r d p r i c e s t h a t a r e o b s e r v e d o n F r i d a y . H o w e v e r , f o r t es t si n v o l v i n g d a il y d a t a , t h e F r i d a y e x c h a n g e r a t e i s u se d f o r t h e n e x t f o u r d a y s d u r i n g t h i s p e ri o d .~ * T h e t - s t a t i s ti c s i n t a b l e 1 a r e a d j u s t e d f o r a u t o c o r r e l a t i o n a t l a g s o n e a n d t w o s i n c e t h e s ea r e a l l o w e d u n d e r t h e c o n s t a n t - e x p e c t e d - b a s is - d i f f er e n t i a l m o d e l .
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Table1
Estimatesofthebasisdifferentialsandtestsoftheconstant-expected-basis-differentialmodel,a
NumberMeanStd.dev.t-stat.S(r3)
rI
r2
r~
r4
r5
ro
Silver
1/6812/80146
0.0801.863
0.5260.086
0.025-0.044
0.011
0.249
0.053
0.018
1/68-12/73
66
-0.0151.533
0.0560.160
0.4460.179-0.0020.182
0.269
0.259
1/74-12/80
80
0.1582.104
1.0000.120
0.1490.143-0.0190.283
0.011
0.065
e~
Copper
1/68-12/80112
-0.3493.5580.6050.178
0.7950.5200.5760.4280.376
0.400
1/68-12/73
37
-1.8683.955
2.2360.395
0.8970.1460.6820.4090.366
0.716
1/74-12/80
75
0.4003.1070.5780.218
0.8390.665
0.5800.411
0.288
0.219
~Thepercentagebasisdifferentialisestimatedbylog[P(t,$)/P(t,)X(t,t)]*100.Undertheconstant-expected-basis-differential
model,theautocorrelationsafterlag2shouldnotbesignificantlydifferentfromzero.Theautocorrelationatlagiisdenotedby
r~andS(r3)isthestandarderrorfortheautocorrelationatlag3,estimatedusingBartlett's(1946)approximation.Copperhas
fewerobservationsthansilverduringthefirstsampleperiodbecausefewercoppercontractsweretradedduringthisperiod.
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322 K.R. French, Futures and Jorward pricesThe expectational model does not rule out differences between the basis-adjusted spot prices. However, since the conditional expected value of thesedifferences is zero, differences that are at least 3 months apart must beindependent. The autocorrelat ions reported in table l provide a test of thisimplication.
The estimates for silver support the constant-expected-basis-differentialmodel, Since these estimates use monthly observations, the autocorrelationsshould be approxima tely zero after the second lag. In fact, almost all of theautocorrelations for the silver price differences are close to zero. Forexample, only the autocorrelation at lag 4 is significant during the 1968-1980period. 15 The results for copper are less consistent with the model. Forexample, all of the estimates for the 1968-1980 sample period aresignificantly positive. Because of these large autocorre lat ions, the futures andforward price comparisons involving converted copper prices should beinterpreted cautiously. t 6
3 . 2 . A p r e l i m i n a r y l o o k a t t h e d a t aBefore comparing individual futures and forward prices, it is helpful to
examine the general properties of the cash, spot, futures, and forward pricesthat are used in the tests below. Tables 2 and 3 summarize the dailypercentage changes in the prices for silver and copper contracts. Thesechanges are equal to the daily logarithmic price relatives. For example, thedaily percentage change in the spot price for London silver is equal tolog [P(t, ) / P ( t -- 1, )] 100.
Most of the tests in this paper use copper and silver prices from 1968through 1980. However, metal prices were unusually volatile during the lasttwo years of this sample period. For example, the cash price for silver rosefrom $5.98 per ounce on Janua ry 2, 1979, to $52.25 on January 21, 1980. Bythe end of 1980, the cash price had fallen back down to $16.58. Because ofthis unusual behavior during 1979 and 1980, tables 2 and 3 summarize thedaily price changes when these two years are included in the sample andwhen they are not.
Compa riso ns of the behavior of futures and forward prices are complicatedby the price limits in the futures markets. Although limit moves are notincluded in the futures price series (nor in any of the tests below), the price
~SThe standard errors for the autocorrelations beyond the second lag are estimated usingBartlett's (1946) approximation. Under the hypothesis that the differences are uncorrelated afterlag 2, these standard errors are equal toS(r 3)-IN 111+2r12+2r22]}~.16A11 of the tests below were also performed using a second conversion technique.This technique assumes that the market uses the current basis differential as its forecast ofthe future differential, so the American forward price is estimated as f ( t , T ,$)=f(t ,T, z3Xt t , T)~P(t ,$)/P(t ,)X(t , t )) . The results using this conversion technique are verysimilar to the results reported below.
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K .R . French, Futures and forw ard prices 323
l im i t s re d u c e t h e a p p a r e n t v o l a t i li t y o f t h e f u t u r e s p ri ce s b e c a u s e t h e y m a k ei t i mp oss i b l e t o ob se r ve l a r ge p r ic e c ha ng e s . Be c a use o f t h i s p r ob l e m , t a b l e s 2a n d 3 r e p o r t e s t im a t e s f o r t w o d i f fe r e n t s e ts o f c o n v e r t e d f o r w a r d p r ic es . T h ef ir s t s e t i nc l ude s a ll o f t he f o r w a r d p r i c e s t ha t a r e a v a i l a b l e du r i n g t he sa m pl epe r i od . T he se c on d se t is mo r e r e s tr ic t ive ; it on l y i nc l ude s a f o r w a r d p r i c e if ama t c h i ng f u t u r e s p r i c e i s a l so a va i l a b l e . I n o t he r wor ds , a f o r wa r d p r i c e i sn o t i n c l u d e d i f t h e m a t c h i n g f u t u r e s c o n t r a c t w a s n o t t r a d e d o r i f i ts p r ic ere f l ec t s a l imi t move .The s t a nda r d de v i a t i ons f o r s i l ve r i n t a b l e 2 i nd i c a t e t ha t t he p r i c e l i m i t sdo ha ve a no t i c e a b l e e f f e c t . Fo r e xa mpl e , , t he e s t i ma t e d s t a nda r d de v i a t i onf o r t he da i l y c ha nge i n t he 3 - mont h f u t u r e s p r i c e s ove r t he f u l l 1968- 1980sa m pl e p e r i od i s 1 .5~ o , wh i le the L o nd o n f o r w a r d p r i c e s a n d t he f ir s t s e t o fc o n v e r t e d f o r w a r d p r i ce s h a v e s t a n d a r d d e v i a t i o n s o f a p p r o x i m a t e l y 2 .7 ~o . I ft h e c o n v e r t e d f o r w a r d p r i c e s e ri es i s r e s t ri c te d t o d a y s w h e n n o n - l i m i t f u t u r e spr i ces a re a va i l able , the e s t im ate fa i ls to 1 .6~o. H ow eve r , the evid ence s t i lls u g g e s t s t h a t u n c o n s t r a i n e d A m e r i c a n p r i c e s w o u l d b e l e s s v o l a t i l e t h a n t h eL o n d o n o r c o n v e r t e d p r i c e s . F i r s t , t h e s t a n d a r d d e v i a t i o n s f o r t h e r e s t r i c t e df o r w a r d p r i c e s r e m a i n h i g h e r t h a n t h e s t a n d a r d d e v i a t i o n s f o r t h e f u t u r e spr i ces . W hi le th i s d i f fe rence m ay s t il l be cau sed by the pr i ce l im i t s , 17 these c on d p i e c e o f e v i de nc e i s no t a f f e c te d by t h i s b i a s . S i nce s i lve r ( a nd c oppe r )f u t u r e s p r i c e s a r e n o t c o n s t r a i n e d b y p r i c e l i m i t s d u r i n g a c o n t r a c t ' s d e l i v e r ym o n t h , t h e s t a n d a r d d e v i a t i o n s f o r t h e c a s h p r i c e s a r e n o t a r t i f i c i a l l yr e d u c e d . T h e r e f o r e , d i r e c t c o m p a r i s o n s b e t w e e n t h e c a s h a n d s p o t p r i c e s a r ea p p r o p r i a t e . T h e s e c o m p a r i s o n s i n d i c a t e t h a t t h e s t a n d a r d d e v i a t i o n s f o r t h eA m e r i c a n c a s h p r ic e s a r e c o n s i s te n t l y l o w e r t h a n t h e s t a n d a r d d e v i a t i o n s f o rt h e L o n d o n o r c o n v e r t e d s p o t p r i c e s .Be c a u se o f t he p r i c e l im i t s , t he s t a nd a r d de v i a t i ons f o r t he f u t u r e s p r i c e sa r e f a ir l y c o n s t a n t t h r o u g h t im e . H o w e v e r , t h e e s t im a t e s f o r t h e c a sh , s p o ta nd f o r wa r d p r i c e s i nd i c a t e t ha t t he s i l ve r p r i c e vo l a t i l i t y i nc r e a se dd r a m a t i c a l l y o v e r t h e s a m p l e p e r i o d . F o r e x a m p l e , e v e n i f t h e v e r y t u r b u l e n t1 9 7 9 - 1 9 8 0 p e r i o d i s e x c l u d e d , t h e s t a n d a r d d e v i a t i o n s f o r t h e s p o t a n df o r w a r d p r i c es i n c r e a s e d b y m o r e t h a n 5 0 ~ o f r o m t h e f ir s t t o t h e s e c o n ds u b p e r i o d . I t i s a l s o i n t e r e s t i n g t h a t t h e e s t i m a t e d s t a n d a r d d e v i a t i o n s f o r t h ef o r w a r d p r ic e s d o n o t a p p e a r t o b e r e la t e d t o t h e m a t u r i t y o f t h e c o n t r a c ts .T h e s u m m a r y s t a t i s t i c s f o r t h e d a i l y c h a n g e s i n t h e c o p p e r p r i c e s a r ep r e se n t e d i n t a b l e 3 . The se r e su l t s a r e s l i gh t l y d i f f e r e n t f r om t he r e su l t s i n
~TSelection bias can still occur because the futures and forward prices are not perfectlycorrelated. A large change in the forward price can be included in the sample if it is associatedwith a smaller change in the futures price. However, large changes in the futures price are neverincluded.It may appear that the difference between the closing times of the Lo ndon and Americanmarkets can also contribute to the selection bias problem. For example, if a limit move is causedby information that arrives after the London market closes on day t, this information will leadto a large change in the forward price on day t + I. However, a limit move on day t eliminatesthe price changes for both day t and day t+ 1.
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Means,standarddeviations,andt-statisticsofthe
Table2
dailypercentagechangesinthefuturesandforwardpricesforsil,,'er:
AmericanLondon
prices
prices
Restricted
Restricted
Convertedconverted
AmericanLondonConvertedconverted
prices
prices
prices
prices
prices
prices
CashorspotMean
0.085
0.060
contracts
Std.dev.
2.372
2.849
t-star.
1.805
I.190
Number
2563
3145
3-month
Mean
0.060
0.059
contracts
Std.dev.
1.479
2.719
t-stat.
1.941
1.225
Number
2266
3143
168-12/80
1/6812'78
0.054
0.(134
0.059
0.047
0.037
0.034
2.880
2.923
1.831
2.223
2.223
2.332
1.046
0.574
1.502
1.092
0855
0.657
3145
2441
2166
2662
2662
2064
0.058
0.017
0.053
0.047
0.044
0.017
2.772
1.623
1.451
2.019
2.047
1.518
1.164
0.480
1.662
I.199
1.101
0.491
3070
2167
2044
2662
2599
1955
e~ "n e~
CashorspotMean
0.022
0.045
contracts
Std.dev.
1.549
1.586
t-stat.
0.488
1.092
Number
1166
1456
3-month
Mean
0.040
0.046
contracts
Std.dev.
1.431
1.586
t-stat.
0.909
1.119
Number
1456
1/6812/73
0.042 1.6
100.983 145
60.041 1.6
300.934 1405
0.025 1.6
450.510
1124
0.036 1.4
660.7951026
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1/74-12/80
1/74-12/78
CashorspotMean
0.137
0.074
0.064
0.042
0.102
0.050
0.032
0.044
contracts
Std.dev.
2.883
3.599
3.635
3.679
2.113
2.180
2.809
2.952
t-stat.
1.771
0.839
0.726
0.410
1.530
0.614
0.391
0.459
Number
1397
1689
1689
1317
1000
1206
1206
940
3-month
Mean
0.079
0.017
0.073-0.001
0.069
0.047
0.048-0.005
contracts
Std.dev.
1.522
3.406
3.454
1.753
1.472
2.442
2.449
1.573
t-stat.
1.792
0.851
0.864
0.018
1.449
0.674
0.683
0.091
Number
1187
1687
1665
1141
965
1206
1194
929
6-month
Mean
0.020
0.086
0.089-0.003
0.014
0.052
0.056-0.021
contracts
Std.dev.
1.470
3.852
3.887
2.150
1.463
3.189
3.183
2.154
t-stat.
0.460
0.876
0.899
0.037
0.288
0.567
0.608
0.288
Number
1159
1551
1525
1024
951
1191
1175
886
12-month
Mean
0.076
0.074
0.074
0.007
0.081
0.049
0.049
0.083
contracts
Std.dev.
1.623
3.487
3.538
2.219
1.635
2.653
2.672
2.274
t-stat.
1.590
0.839
0.817
0.104
1.531
0.635
0.623
0.109
Number
1167
1545
1515
1021
957
1191
1171
881
"Thepercentagepricechangeisdefinedaslog(Pt/Pt-1)*100.TheAmericanpricesarecashandfuturesprices,TheLondonprices
arespotandforwardprices,denominatedinpoundssterling.TheconvertedpricesareLondonspotandforwardpricesconvertedto
dollars.Theestimatesfortheunrestrictedseriesusealloftheavailableconvertedprices,whiletherestrictedestimatesonlyuseprices
ifthematchingAmericanpricesarealsoavailable.
e~ 3"
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Table3
Means,standarddeviations,andt-statisticsofthedailypercentagechangesinthefuturesandforwardpricesforcopper?
Restricted
Restricted
AmericanLondonConvertedconverted
AmericanLondonConvertedconverted
prices
prices
prices
prices
pricesprices
prices
prices
1/6812/80
CashorspotMean
-0.011
0.010
0.006
Contracts
Std.dev.
1.793
1.904
1.926
t-stat.
-0.272
0.303
0.164
Number
1793
3216
3216
3-month
Mean
-0.013
0.009
0.005
contracts
Std.dev.
1.540
1.879
1.924
t-stat.
-0.322
0.273
0.138
Number
1542
3215
3136
i~812/73
CashorspotMean
0.032
0.038
0.036
contracts
Std.dev.
1.500
1.863
1.869
t-stat.
0.484
0.781
0.733
Number
519
1481
1481
3-month
Mean
-0.013
0.035
0.029
contracts
Std.dev.
1.298
1.271
1.312
t-stat.
0.266
1.051
0.823
Number
745
1482
1427
1/7412/80
CashorspotMean
-0.029
0.013-0.020
contracts
Std.dev.
1.899
1.939
1.974
t-stat.
0.548
0.288
0.423
Number
1274
1735
1735
3-month
Mean
0.013-0.013
0.015
contracts
Std.dev.
1.736
2.274
2.314
t-stat.
0.205
0.236
0.271
Number
797
1733
1709
1/6812/78
-0.02I
0.0230.012
0.004-0.037
1.895
1.5531.717
1.726
1.466
0.473
0.5740.372
O.112
0.939
1751
1440
2728
2728
1413
--0.419
0.0180.011
0.003
0.042
1.371
1.4451.343
1.373
1.267
1.176
-0.457
0.427
0.117
1.187
1483
1333
2727
2660
1283
0.038 1.5
940.542
512
-0.022 1.1
800.501
713
1/7412/78
--0.014
0.055-0.018-0.034-0.036
2.006
1.582
1.525
1.539
1.389
0.254
1.049
0.419
0.748
0.773
1239
921
1247
1247
901
-0.060
0.025
0.017-0.026
0.067
1.527
1.634
1.424
1.441
1.369
1.093
0.375
0.428
0.643
1.165
770
588
1245
1233
570
~'Thepercentagepricechangeisdefinedaslog(P,/P,1)*100.TheAmericanpricesarecashandfuturesprices.TheLondonprices
arespotandforwardprices,denominatedinpoundssterling.TheconvertedpricesareLondonspotandforwardpricesconvertedto
dollars.Theestimatesfortheunrestrictedseriesusealloftheavailableconvertedprices,whiletherestrictedestimatesonlyusepricesif
thematchingAmericanpricesarealsoavailable.
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K.R. French, Futures and forward prices 327t a b l e 2 . F o r e x a m p l e , t h e s t a n d a r d d e v i a t i o n s f o r t h e f u t u r e s p r i c e s a r ea p p r o x i m a t e l y e q u a l t o t h e s t a n d a r d d e v i a t i o n s f o r b o t h t h e r e s t r i c t e d a n dt h e u n r e s t r i c t e d f o r w a r d p r i c e s e r i e s d u r i n g b o t h t h e 1 9 6 8 - 1 9 7 3 s u b p e r i o da n d t h e 1 9 7 4 - 1 9 7 8 s u b p e r i o d . T h e e f f ec t o f t h e f u t u r e p r i c e l i m i ts i s o n l yn o t i c e a b l e w h e n t h e m o r e v o l a t i l e 1 9 7 9 a n d 1 9 8 0 p r i c e s a r e i n c l u d e d i n t h es a m p l e . i S
T h e b e h a v i o r o f t h e c o p p e r p r i c e v a r i a n c e s t h r o u g h t i m e is a l s o s li g h tl yd i f fe r e n t f r o m t h e b e h a v i o r f o r s il ve r. T h e s t a n d a r d d e v i a t i o n s f o r t h e c o p p e ra n d s il v e r p r i c e c h a n g e s a r e a l l a b o u t 1 . 5~ o d u r i n g t h e f i rs t s u b p e r i o d .H o w e v e r , t h e e s t i m a t e s f o r c o p p e r a r e m u c h l o w e r t h a n t h e e s t i m a t e s f o rs i l v e r f ro m 1 9 7 4 t h ro u g h 1 9 8 0 ; wh i l e s i l v e r ' s v a r i a n c e i n c re a s e s f ro m t h e f i r s tt o t h e s e c o n d s u b p e r i o d , c o p p e r ' s v a r i a n c e r e m a i n s f a i rl y c o n s t a n t . I n f a ct ,t h e s t a n d a r d d e v i a t i o n s f o r c o p p e r d o n o t i n c r e a s e u n le s s 1 9 79 a n d 1 98 0 a r ea d d e d t o t h e s a m p l e p e r i o d .
A u t o c o r r e l a t i o n s f o r t h e d a i l y c o p p e r a n d s i l v e r p r i c e c h a n g e s a r ep r e s e n t e d i n t a b l e 4 . T h e a u t o c o r r e l a t i o n s f o r t h e s p o t , c a s h , a n d 3 - m o n t hp r i c e s e r i e s a r e e s t i m a t e d u s i n g d a t a f ro m 1 9 6 8 t h ro u g h 1 9 8 0 , wh i l e t h ea u t o c o r r e l a t i o n s f o r t h e 6 - a n d 1 2 - m o n t h s e r i e s a r e e s t i m a t e d f r o m 1 9 7 4t h r o u g h 1 98 0. T h e m o s t s t r i k i n g r e s u l t in t a b l e 4 is t h a t a l m o s t a l l o f t h ef i r s t - o r d e r a u t o c o r r e l a t i o n s a r e n e g a t i v e a n d r e l a t i v e l y l a r g e . O n l y f i v e o ft h e s e a u t o c o r r e l a t i o n s a r e w i t h in f o u r s t a n d a r d e r r o r s o f z e r o . T h i s s e r ia lc o r r e l a t i o n w o u l d s e e m t o s u g g e s t a p r o f i t a b l e t r a d i n g o p p o r t u n i t y ; b u y o nt h e d a y a f t e r a p r i c e d ro p a n d s e l l a f t e r a r i s e . Ho we v e r , i t i s m o re l i k e l y t h a tt h e c o r r e l a t i o n i s c a u s e d b y m e a s u r e m e n t e r r o r t h a n b y m a r k e t i n e f f i c i e n c y .F o r e x a m p l e , s u p p o s e t h e m e a s u r e m e n t e r r o r i n t o d a y ' s r e p o r t e d s p o t p r i c efo r s i l v e r i s p o s i t i v e . Th i s i n t ro d u c e s a p o s i t i v e b i a s i n t o d a y ' s p r i c e c h a n g ea n d a n e g a t i v e b i a s i n t o m o r r o w ' s p r i c e c h a n g e . T h i s p a t t e r n w o u l d c a u s e an e g a t i v e f ir s t - o r d e r a u t o c o r r e l a t i o n i n t h e o b s e r v e d p r i c e c h a n g e s . I f t h em e a s u r e m e n t e r r o r i s n e g a t i v e , t h e p a t t e r n i s r e v e r s e d b u t t h e f i n a l r e s u l t i st h e s a m e . I n v i e w o f t h is m e a s u r e m e n t e r r o r h y p o t h e s i s , it is in t e r e s t i n g t on o t e t h a t f o u r o f th e f i ve s m a l l e st f i r s t - o r d e r a u t o c o r r e l a t i o n s a r e f o r c a s ha n d f u t u r e s p r i c e s .
T o s u m m a r i z e t h e r e s u l t s f r o m t a b l e s 2 t h r o u g h 4 , t h e r e a r e s o m en o t i c e a b l e d i f fe r e n c e s b e t w e e n t h e b e h a v i o r o f th e f u t u r e s p r i c e s a n d t h eb e h a v i o r o f t h e f o r w a r d p r i c es . F o r e x a m p l e , t h e e v i d e n c e i n ta b l e s 2 a n d 3i n d i c a t e s t h a t , b e c a u s e o f th e f u t u r e s p r i c e l im i t s, t h e v a r i a n c e o f th e o b s e r v e df u t u r e s p r ic e s is g e n e r a l l y l o w e r t h a n t h e v a r i a n c e o f th e f o r w a r d p r ic e s . I na d d i t i o n , t h e f i r s t - o r d e r a u t o c o r r e l a t i o n s i n t a b l e 4 s u g g e s t t h a t t h e L o n d o ns p o t a n d f o r w a r d p r i c e s c o n t a i n m o r e m e a s u r e m e n t e r r o r t h a n t h e A m e r i c a nc a s h a n d f u t u r e s p ri c e s. T h e d a t a a l s o i n d ic a t e t h a t t h e v a r i a b i l it y o f t h ed a i l y p r i c e c h a n g e s i n c re a s e s f ro m 1 9 6 8 t o 1 9 8 0 . Th e s i l v e r p r i c e v o l a t i l i t y
l S l n fa c t, o n l y 3 ~ o f t h e 3 - m o n t h c o p p e r f u t u r e s p r i c e s o b s e r v e d f r o m 1 9 68 t h r o u g h 1 97 8r e fl e c t l im i t m o ve s , wh i l e 9~ o o f t he p r i c e s f o r 1979 a nd 1980 a r e l im i t m ov e s .
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328 K .R . French, Futures and f i)rward prices
T a b l e 4A u t o c o r r e l a t i o n s o f t h e d a il y p e r c e n t c h a n g e s in t h e c a s h , s p o t, f u t u r e s a n d f o r w a r d p r i c e s f o r
c oppe r a nd s i l ve r . "N u m b e r S(r ~) r~ r 2 r 3 r~ r~ r~,
SilverCa sh pr ice s 2364 0 .021 0 .050 0 .008 -0 .0 2 6 0 .018 0 .057 0 .036L o n d o n s p o t p r i c e s 3 0 54 0 .0 18 - 0 . 1 7 2 - 0 . 0 2 5 0 .0 5 2 0 .0 2 0 - 0 . 0 5 4 0 .0 35C o n v e r t e d s p o t p r i ce s 3 0 54 0 .0 18 O . 1 7 0 - 0 . 0 2 9 0 .0 48 0 .0 16 - 0 . 0 5 2 - 0 . 0 3 4Re st r ic te d sp ot pr ice s 2213 0 .021 - 0 .216 0 .006 0 .025 0 .037 0 .015 0 .0133 - m o n t h c o n t r a c t sFu tur es pr ice s 2081 0 .022 -0 .1 11 0 .010 0 .044 0 .081 0 .083 0 .034L o n d o n f o r w a r d p r i c es 3 05 1 0 .0 18 - 0 . 1 3 1 - 0 . 0 2 3 0 .0 45 0 . 02 6 - 0 . 0 5 1 - 0 . 0 4 5C o nv e r t e d fo rw a rd p r i c e s 2947 0 .018 -0 .13 3 -0 .0 28 0 .041 0 .022 0 .048 0 .037R e s t r i c t e d fo r w a rd p r i c e s 1954 0 .023 -0 . 20 3 -0 .0 21 0 .012 0 .020 -0 .0 12 I ). 0366 - m o n t h c o n t r a c t sF u t u re s p r i c e s 1037 0 .031 -0 .1 48 0 .020 -0 .0 15 0 .028 -0 .0 62 0 .005Lo nd on fo rw a rd p r i c e s 1505 0 .026 -0 .1 95 -0 .0 45 0 .038 0 .018 0 .054 0 .043C o nv e r t e d fo r w a rd p r i c e s 1467 0 .026 -0 . 19 5 -0 .0 42 0 .026 0 .026 0 .059 0 .045R e s t r i c t e d fo r wa rd p r i c e s 905 0 .033 -0 .2 44 0 .007 0 .038 0 .078 0 .028 0 .019
1 2 -m o n t h c o n t r a c t sFu tur es pr ices 1057 0 ,03l -0 .0 18 0 .038 0 .015 0 .017 - --0 .043 0 .019L o n d o n f o r w a r d p r i ce s 1 49 6 0 .0 2 6 - 0 . 1 5 0 - 0 . 0 2 8 0 . ( } 3 8 0 .0 3 4 0 .0 7 5 - 0 . 0 6 4C o n v e r t e d f o r w a r d p r i c e s 1 45 2 0 .0 2 6 - 0 . 1 5 0 - 0 . 0 2 8 ( I . 0 2 2 0 .0 3 9 - ~0 .0 79 - ( I. 0 5 8R e s t r i c t e d fo rw a rd p r i c e s 906 0 .033 -0 .14 1 -0 .03 5 0 .038 0 .035 0 .017 0 .018Copper
C a sh p r i c e s 1636 0 .025 0 .023 -0 .0 39 0 .042 0 .027 0 .069 -0 .0 60L on do n sp ot pr ice s 3160 0 .018 0 .106 -0 .0 5 0 0 .058 0 .015 0 .015 0 .003C o nv e r t e d spo t p r i c e s 3160 0 .018 -0 .0 93 -0 .04 8 0 .052 0 .015 0 .016 0 .008Re st r ic te d sp ot pr ice s 1585 0 .025 0 .137 -0 .0 6 2 0 .054 0 .036 0 .056 0 .0033 - m o n t h c o n t r a c t sFu tur es pr ice s 1407 0 .027 0 .064 -0 .0 3 9 ( I.007 0 .001 0 .08l 0 .058Lo nd on fo rw a rd p r i c e s 3158 0 .018 0 .192 -0 .0 20 -0 .0 13 0 .064 0 .005 0 .018C o n v e r t e d f o r w a r d p r i c e s 3 04 3 0 .0 1 8 - 0 . 1 8 8 - 0 . 0 2 4 - 0 . 0 2 0 0 .0 5 6 0 .0 0 9 0 .0 1 9R e s t r i c t e d fo rw a rd p r i c e s 1329 0 .027 -0 .0 53 -0 .0 57 0 .048 0 .064 0 .061 0 .056" r e i s t h e a u t o c o r r e l a t i o n a t l a g r ; S(r 0 i s t h e s t a n d a r d e r r o r o f t h e f i r s t - o r d e r a u t o c o r r e l a t i o n .T h e c a s h , s p o t , a n d 3 - m o n t h a u t o c o r r e l a t i o n s a r e e s t i m a t e d f r o m 1 /6 8 to 1 2/ 80 . T h e 6 - a n d 1 2-m o n t h a u t o c o r r e l a t i o n s a r e e s ti m a t e d f r o m 1 /7 4 t o 1 2/ 80 . T h e c o n v e r t e d p r i ce s a r e L o n d o n s p o ta n d f o r w a r d p r i c e s c o n v e r t e d t o d o l l a r s . T h e e s t i m a t e s f o r t h e u n r e s t r i c t e d s e r i e s u s e a l l o f t h ea v a i l a b l e c o n v e r t e d p r i c e s , w h i l e t h e r e s t r i c t e d e s t i m a t e s o n l y u s e t h e c o n v e r t e d p r i c e s i f t h em a t c h i n g A m e r i c a n p r i c e s a r e a l s o a v a i l a b l e .
a p p e a r s t o g r o w o v e r t h e w h o l e s a m p l e p e r i o d . A l t h o u g h t h e v a r i a n c e o f t h ec o p p e r p r i c e s i s f a i r ly c o n s t a n t f r o m 1 9 68 t h r o u g h 1 9 78 , t h e v a r i a n c e s f o rb o t h c o m m o d i t i e s i n c r e a s e s i g n i f i c a n t l y d u r i n g t h e 1 9 7 9 - 1 9 8 0 p e r i o d .3 .3 . T h e d ( f f e r e n c e s b e t w e e n f u t u r e s a n d J o r w a r d p r i c e s a n d t e sts ' o 1 t h e C o x -
I n g e r s o l l - R o s s p r o p o s i t i o n sT h e s i m p l e s t w a y t o e x a m i n e w h e t h e r t h e r e i s a n e m p i r i c a l l y r e l e v a n t
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K.R. French, Futures and forward prices 3 2 9d i f f e r e n c e b e t w e e n f u t u r e s a n d f o r w a r d p r i c e s i s t o c o m p a r e t h e mi n d i v i d u a l l y . T a b l e 5 s u m m a r i z e s t h e p e r c e n t a g e d i f f e r e n c e s b e t w e e nm a t c h i n g f u t u r e s a n d f o r w a r d p ric es ,, d e f in e d a s l o g [ F ( t , T ) / f ( t , T ) ] * 1 0 0, f o rb o t h c o p p e r a n d s il ve r. T h e s e d i f fe r e nc e s a r e m e a s u r e d a t 3 m o n t h s t om a t u r i t y f o r t h e c o p p e r c o n t r a c t s a n d a t 3 , 6 , a n d 1 2 m o n t h s t o m a t u r i t y f o rt h e s i lv e r c o n t r a c t s . S i n c e t h e f u t u r es c o n t r a c t s m a t u r e a t m o n t h l y i n te r v a ls ,t h is p r o c e s s g e n e r a t e s m o n t h l y o b s e r v a t i o n s . 19 F o r e x a m p l e , t h e J a n u a r yp r i c e d i f f e r e n c e f o r t h e 3 - m o n t h s i l v e r s e r i e s r e f l e c t s f u t u r e s a n d f o r w a r dc o n t r a c t s t h a t m a t u r e i n A p r il , w h i l e t h e F e b r u a r y d i ff e re n c e i n v o l v e sc o n t r a c t s t h a t m a t u r e i n M a y . T h e f o r w a r d p r i c e s in t a b le 5 ( a n d i n t he t e st sb e l o w ) a r e c o n v e r t e d f r o m p o u n d s s t e r l i n g t o d o l l a r s u s i n g t h r e e d i f f e r e n te s t i m a t e s o f t h e b a s i s d i ff e re n t ia l f o r e a c h c o m m o d i t y T h e f o r w a r d p r i ce s int h e 1 9 6 8 - 1 9 8 0 c o m p a r i s o n s a r e c o n v e r t e d u s i n g t h e f u l l - p e r i o d e s t i m a t e s o ft h e b a s i s d i f f e r e n t i a l . T h e c o n v e r s i o n s f o r t h e s u b p e r i o d t e s t s u s e t h e b a s i sd i f f e r e n t i a l e s t i m a t e d o v e r t h e m a t c h i n g s u b p e r i o d .
T h e f u t u r e s - f o r w a r d p r i c e d if f e r en c e s f o r si l v e r i n t a b l e 5 i n d i c a t e t h a t , o na v e r a g e , th e f u t u r e s p r i c e s a r e l a r g e r t h a n t h e f o r w a r d p r i c es . F o u r o f t h e f iv ee s t i m a t e s a r e s i g n i f i c a n t l y p o s i t i v e a t t h e 5 % l e v e l . M o r e o v e r , t h e d i f f e r e n c eb e t w e e n t h e f u t u r e s a n d f o r w a r d p r i ce s in c r e a s e s w i t h t h e m a t u r i t y o f t h e
o /c o n t r a c t ; t h e a v e r a g e d i ff e re n c e s f r o m 1 97 4 t h r o u g h 1 98 0 a r e a b o u t 0 . 1 /o ,0 .4 % , a n d 0 .8 ~ o f o r th e 3 -, 6 - a n d 1 2 - m o n t h c o n t r a c t s , r e s p e c t iv e l y .
T h e r e l a t i o n b e t w e e n t h e f u t u r e s a n d f o r w a r d p r i c e s fo r c o p p e r i s l es sc l e a r c u t . I t a p p e a r s t h a t t h e f u t u r e s p r i c e s a r e l o w e r t h a n t h e f o r w a r d p r i c e sd u r i n g t h e f i r s t s u b p e r i o d a n d h i g h e r t h a n t h e f o r w a r d p r i c e s d u r i n g t h es e c o n d s u b p e r i o d ; t h e a v e r a g e p r i c e d i f f er e n c e s a r e - 0 . 9 ~ a n d 0 1 ..... /o,r e s p e c t i v e l y . H o w e v e r , t h e t - s t a t i s t i c s f o r t h e s e e s t i m a t e s a r e o n l y 1 . 5 5 a n d0.21. 20
T a b l e 5 a l so c o n t a i n s s o m e e v id e n c e a b o u t t w o o f t h e p r o p o s i t i o n sd e v e l o p e d b y C o x , I n g e r s o l l a n d R o s s ( 1 9 8 1 ) . T h e y h y p o t h e s i z e t h a t t h ef u t u r e s - f o r w a r d p r i c e d i f f e r e n c e i s e q u a l t o
r ( t , T ) - f ( t , T ) = - P V t , T { i F ( w , T ) c o v [ P ( w , T ) , B ( w , T ) ] d w ) / B ( t , T ) , (16)w h e r e c o v [ F ( w , T ) , B ( w , T ) ] i s t h e l o c a l c o v a r i a n c e a t t i m e w b e t w e e n t h ep e r c e n t a g e c h a n g e i n t h e f u t u r e s p r i c e a n d t h e p e r c e n t a g e c h a n g e i n t h e b o n dp r i c e . T h i s i n s t a n t a n e o u s c o v a r i a n c e m a y b e c h a n g i n g s t o c h a s t i c a l l y a s t h ef u t u r e s c o n t r a c t a n d t h e b o n d c o n t r a c t m o v e t o w a r d m a t u r i t y a t t i m e T .
~gFutures contracts do not mature ev ery m on th so s om e m onth s will not be represented inthese series.2It is interesting that, altho ug h the av era ge pric e difference for the second subperiod ispositive, the t-statistic for the averag e difference of the full period , - 1.68, is mo re ne gative thanthe t-statistic for the first subp eriod, - 1.55. Th is happ ens be cause different estimate s of the b asisdifferential are used for the full sample period and for eac h subperiod.
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330 K.R. French, Futures and forward pricesH o w e v e r , i f t h e c o v a r i a n c e is a l w a y s p o s i t i v e d u r i n g t h e c o n t r a c t p e r i o d , t hei n t e g r a l i n ( 16 ) w i ll b e p o s i t i v e a n d t h e f o r w a r d p r i c e s h o u l d b e h i g h e r t h a nt h e f u t u r e s p r ic e . O n t h e o t h e r h a n d , i f t h e l o c a l c o v a r i a n c e i s a l w a y sn e g a t i v e , t h e f u t u re s p r i c e s h o u l d b e a b o v e t h e f o r w a r d p r ic e .
C o x , I n g e r s o l l a n d R o s s a l s o s h o w t h a t t h e f u t u r e s - f o r w a r d p r i c e d i f fe r e n c ec a n b e e x p r e s s e d a s
F ( t, T ) - - f ( t , T ) = P V t, T e x p ~ ( w ) d w ( [ P ( w ) / / ~ (w , T ) ]
x { v a r [ /~ ( w , T ) ] - - c o v [ P ( w ) , / ~ ( w , T ) ] } d w } , ( 1 7)
i f t h e c o m m o d i t y is s t o r e d c o s t l e s s l y o v e r t h e c o n t r a c t p e r i o d . T h i s l e a d s t ot h e p r e d i c t i o n t h a t , i f t h e l o c a l v a r i a n c e o f t h e b o n d p r i c e is a l w a y s l a r g e rt h a n t h e lo c a l c o v a r i a n c e b e t w e e n t h e s p o t p r ic e a n d t h e b o n d p r ic e , t h ef u t u r e s p r i c e w i ll b e a b o v e t h e f o r w a r d p r ic e . I f t h e v a r i a n c e is s m a l l e r t h a nt h e c o v a r i a n c e , t h e f u tu r e s p r i c e s h o u l d b e b e l o w t h e f o r w a r d p r i c e .
T h e s i m p l e s t w a y t o t e s t t h e s e h y p o t h e s e s is t o a s s u m e t h a t t h e l oc a lv a r i a n c e s a n d c o v a r i a n c e s a r e c o n s t a n t a n d t h e s a m e f o r a ll c o n t r a c t s . 2~U n d e r t h is a s s u m p t i o n t h e l o c a l c o v a r i a n c e i n e q. ( 16 ) is m e a s u r e d i n t w os t ep s . F i r s t , t h e c o v a r i a n c e b e t w e e n t h e d a i l y p e r c e n t a g e c h a n g e in th ef u t u r e s p r ic e a n d t h e d a i l y p e r c e n t a g e c h a n g e i n th e b o n d p r i c e is e s t i m a t e df o r e a c h p a i r o f f u tu r e s a n d f o r w a r d c o n t r a c t s . T h e s e c o v a r i a n c e s a r e t h e na v e r a g e d a c r o s s c o n t r a c t s . T h e l o c a l v a r i a n c e s a n d c o v a r i a n c e s i n e q . ( 17 ) a r ee s t i m a t e d i n t h e s a m e w a y . T h e r e s u l ts o f t h is p r o c e s s a r e r e p o r t e d i n t h es e c o n d a n d t h i r d c o l u m n s o f t a b l e 5. z2
U n d e r t h e f i rs t C I R p r o p o s i t i o n , i f t h e c o v a r i a n c e b e t w e e n t h e b o n d p r ic e sa n d t h e f u t u r e s p r i c e s is p o s it i v e t h e f u tu r e s f o r w a r d p r i c e d i ff e r e nc e s h o u l db e n e g a t iv e . T h e r e s u l t s in t a b l e 5 p r o v i d e s o m e s u p p o r t f o r t h is h y p o t h e s i s .
2~This assumption can not be strictly true since the local variances and c ovariances mustconverge to zero as the contracts approach maturity and the bond prices converge to one.However, estimates o f the variances and covariances m easured over the full con tract period canbe used to predict w hether th e relevant integrals ar e positive or negative. F or exam ple if thefutures price is roughly constant, the average covariance between the futures price and the bondprice is appro xim ately pro por tion al to the integral in eq. (16).22A11 of the variances an d c ovariances in tabl e 5 have been multiplied by 106. The bon d pricesused to estimate these variables are measured by the 3-, 6- and 12-month treasury bill pricesprovided by the Federal Reserve Bank of New York and Data Resources, Inc. For example,when a futures contract has 12, 11, or 10 months to maturity, its daily price changes arecompared with the price changes for treasury bills that will mature in approximately 12 months.When the futures contract has between 9 and 5 months to maturity, the 6-month treasury billseries is used. The 3-month bills are used during the last four months of the con tract period. A 1-month treasu ry bill series is also available, but its bid/as k spread is very large. Fo r example, theaverage d aily sprea d from 1973 throug h 1980 is 0.32~;,. The average daily bid /ask sp read for the3-month series over this period is 0.025~.
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K .R . French, Futures and forw ard prices 33 1T a b l e 5
F u t u r e s - f o r w a r d p r ic e d i ff e re n c es a n d t e s ts o f t h e C o x I n g e r s o l l - R o s s p r o p o s i t i o n s ?
log ( f u r / f o r ) * 100 c oy ( F , B ) va r ( B ) - c ov ( P , B )1 9 6 8 - 1 9 8 0
3 - m o n t h M e a n 0 .2 9 7 0 .2 3 1 0 . 80 3silv er Std . dev . 1.325 1.917 4.203t- s ta t . 2 .372 0 .889 1 .319N u m b e r 1 12 1 40 1 543 - m o n t h M e a n - 0 . 7 0 1 0 .3 71 0 .1 5 8co pp er S td . dev . 3 .744 1 .442 0 .843t-st a t . 1.685 2.556 1.506N u m b e r 8 1 1 27 1 54
1 9 6 8 - 1 9 7 33 - m o n t h M e a n 0 .4 8 5 0 .1 3 9 0 . 1 1 0si lve r S td . dev . 1 .332 0 .530 0 .795t-s ta t . 2.651 1.193 0.653N u m b e r 5 3 6 8 7 03 - m o n t h M e a n - 0 . 8 6 1 0 .1 8 9 0 . 0 08c op pe r S td . de v . 3 . 464 0 . 734 0 . 310t-s ta t . 1.552 1.195 0.15 0N u m b e r 3 9 5 6 7 0
1 9 7 4 - 1 9 8 03 - m o n t h M e a n 0 . 13 6 0 .3 1 8 1 .3 8 0si lve r S td . dev . 1 .325 2 .630 5 .595t- s ta t . 0 .788 0 .648 1 .267N u m b e r 5 9 7 2 8 46 - m o n t h M e a n 0 . 44 4 0 .0 1 9 1 .5 43si lve r S td . dev . 1 .489 2 .338 4 .985t-s ta t . 2.251 0.041 1.369N u m b e r 5 7 6 3 8 412 - m on th M e a n 0 . 846 - 1 .945 2 . 187si lve r S td . dev . 1 .678 5 .240 4 .038t- s ta t . 3 .670 1 .533 2 .294N u m b e r 5 3 7 2 7 93 - m o n t h M e a n 0 .1 0 9 0 . 51 5 0. 2 83co pp er S td . dev . 3 .325 1 .808 1 .094t- s ta t . 0 .212 2 .347 1 .556N u m b e r 4 2 71 8 4
a U n d e r t h e f i rs t C I R p r o p o s i t i o n , i f t h e c o v a r i a n c e b e t w e e n t h e d a i l y p e r c e n t a g ec h a n g e i n t h e f u t u r e s p r i c e a n d t h e d a i l y p e r c e n t a g e c h a n g e i n t h e b o n d p r i c e i sp o s i t i v e , t h e f u t u r e s - f o r w a r d p r i c e d i f f e r e n c e s h o u l d b e n e g a t i v e . I f t h e c o v a r i a n c e i sn e g a t i v e , t h e p r i c e d i f f e r e n c e d i f f e r e n c e ~ i l o u l d b e p o s i t i v e . U n d e r t h e s e c o n d C I Rp r o p o s i t i o n , i f t h e b o n d p r i c e v a r i a n c e is l a r g e r t h a n i t s c o v a r i a n c e w i t h t h e s p o tp r ic e , t h e f u t u r e s - f o r w a r d p r i c e d i ff e r en c e s h o u l d b e p o s i t iv e . I f t h e v a r i a n c e -c o v a r i a n c e d i f f e r e n c e i s n e g a t i v e , t h e p r i c e d i f f e r e n c e s h o u l d a l s o b e n e g a t i v e . T h et - s t a t i s t i c s f o r t h e c o v a r i a n c e s a n d t h e v a r i a n c e - c o v a r i a n c e d i f f e r e n c e s a r e a d j u s t e d f o rse r i a l c o r r e l a t ion .
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332 K .R. French, F utures and j i~rward pricesF o r e x a m p l e , t h e a v e r a g e c o v a r i a n c e f o r c o p p e r f r o m 1 96 8 t h r o u g h 1 98 0 is0 . 3 7, w i t h a t - s t a t i s t i c o f 2 .5 6. 23 U s i n g t h e C I R m o d e l , t h i s i m p l i e s t h a t t h ef u t u r e s p r i c e s w i l l b e b e l o w t h e f o r w a r d p r i c e s . T h e a v e r a g e p r i c e d i f f e r e n c eof -0 .71~,~, i s c o n s i s t e n t w i t h t h i s p r e d i c t i o n . T h e r e s u l t s f o r 1 2 - m o n t h s i l v e rc o n t r a c t s a l so s u p p o r t t h e C I R h y p o t h e s is : th e a v e r a g e c o v a r i a n c e f r o m 1974t h r o u g h 1 9 80 i s - 1 .9 a n d , a s t h e m o d e l p r e d i c t s , t h c a v e r a g e f u t u r e s f o r w a r dp r i c e d i f f e r e n c e i s s i g n i f i c a n t l y p o s i t i v e . U n f o r t u n a t e l y , t h e o t h e r c o m p a r i s o n sd o n o t p r o v i d e m u c h e v i d e nc e . A ll b u t o n e o f t h e o t h e r a v e r a g e e o v a r i a n c e sa r e a p p r o x i m a t e l y z er o . T h e e s t i m a t e d c o v a r i a n c e f o r 3 - m o n t h c o p p e rc o n t r a c t s f r o m 1 9 74 t h r o u g h 1 98 0 is s i g n i f i c a n t ly p o s i ti v e . H o w e v e r , s i n ce th et - s t a t i s t i c f o r t h e a v e r a g e f u t u r e s f o r w a r d p r i c e d i f f e r e n c e f o r t h e s e c o n t r a c t si s o n l y 0 . 2 1 , o n e c a n n o t r e j e c t t h e h y p o t h e s i s t h a t t h e t r u e d i f f e r e n c e i sn e g a t i v e , a s t h e m o d e l p r e d i c t