French_a comparison of futures and forward prices

8/7/2019 French_a comparison of futures and forward prices

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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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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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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.


13/32

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.


14/32

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


15/32

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"


16/32

Table3

Means,standarddeviations,andt-statisticsofthedailypercentagechangesinthefuturesandforwardpricesforcopper?

Restricted

Restricted



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.


17/32

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 .


18/32

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

French_a comparison of futures and forward prices

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