Beta and Return Uk

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N O R T H - H ( X l A N D

A n E x a m i n a t io n o f th e C r o s s -S e c t io n a lR e l a t io n s h i p o f B e t a a n d R e tu r n :U K E v id e n ceJ o n a t h a n F l e t c h e r

T h i s p a p e r e x a m i n e s t h e c o n d i ti o n a l r e la t i o n sh i p b e t w e e n b e t a a n d r e t u rn i n U Ks to c k r e tu r n s . T h e r e i s n o e v id e n c e o f a s i g n i f i c a n t r i s k p r e m iu m o n b e t a w h e n th eu n c o n d i t i o n a l r e l a t i o n s h i p b e t w e e n b e t a a n d r e t u r n i s c o n s i d e r e d . W h e n t h es a m p l e i s s p l i t i n to p e r io d s a c c o r d in g t o w h e th e r t h e e x c e s s m a r k e t r e tu r n i sp o s i t i v e o r n e g a t iv e , t h e r e i s a s i g n i f i c a n t r e l a t i o n s h ip b e tw e e n b e t a a n d r e tu r n .H o w e v e r , t h e r e l a t io n s h i p i s s t r o n g e r i n m o n t h s w h e n t h e e x c e ss m a r k e t r e t u r n isn e g a t iv e t h a n w h e n i t is p o s i ti v e . S u b s id ia r y r e s u lt s o f t h e p a p e r a l s o i n d i c a t e t h ea b s e n c e o f t h e s iz e e f f e c t i n U K s to c k r e tu r n s . 1 9 97 T e m p l e U n iv e r s i t yKeywords: C A P M ; B e t a ; C o n d i t i o n a l r e l a t i o n s h i pJEL classif ication: G 1 2

I . I n t r o d u c t i o nT h e C a p i t a l A s s e t P ri c in g M o d e l ( C A P M ) d e v e l o p e d b y S h a r p e ( 19 64 ), L i n t n e r( 1 9 65 ) a n d M o s s in ( 1 96 6 ) h a s b e e n o n e o f t h e p r e m ie r m o d e l s 1 i n fi n a n c e . I t h a sb e e n w i d e l y u s e d i n c o st o f c a p i ta l e s t i m a t i o n a n d t h e p e r f o r m a n c e m e a s u r e m e n to f m a n a g e d f u n ds . T h e r e h a s b e e n r e n e w e d i n t e r e st i n t h e C A P M f o ll o w in g t h er e c e n t s t u d y o f F a m a a n d F r e n c h ( 19 9 2) . F a m a a n d F r e n c h f o u n d t h a t , u s in g n e a rl y5 0 y e a r s o f U S s to c k r e tu r n d a t a , t h e r e w a s a fi a t r e l a t io n s h ip b e tw e e n r e tu r n a n d

D e p a r t m e n t o f F i n a n c e a n d A c c o u n t i n g , G l a s g o w C a l e d o n i a n U n i v e r s it y , G l a s go w , U n i t e d K i n g d o m(JF) .A d d r e s s c o r r e s p o n d e n c e t o : D r . J o n a t h a n F l e t c h e r , D e p a r t m e n t o f F i n a n c e a n d A c c o u n t i n g ,G l a s g o w C a l e d o n i a n U n i v e rs i ty , C o w c a d d e n s R o a d , G l a sg o w , G 4 0 B A , U K .1 O the r p opu l a r m od e l s o f r isk a nd r e tu r n i nc l ude t he A r b i t r a ge P r i c ing T h e o r y [ R oss (1976 )] , t hem u l t i - b e t a C A P M [ M e r t o n ( 19 73 )] a n d t h e c o n s u m p t i o n C A P M [ B r e e d e n (1 97 9)] .

Journa l of Econom ics and Business 1997; 49:211 221 0148-619 5/97/$17.00 1997 Tem ple Univ ersity PII S0148-6195(97XI0006-4


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212 J. Fletcherbeta. This was widely reported in the financial press as the death knell o f beta. 2Similar findings of an insignificant relationship between beta and return have beenobserved in UK stock returns by Strong and Xu (1994) for the period 1973-1992.

Recent studies have tended to counter the findings of Fama and French (1992).Chan and Lakonishok (1993), Jagannathan and Wang (1996), Kothari et al. (1995a)and Kim (1995) suggested some support of a positive relationship between returnand beta. The differences seem to be due to the time period examined [Chan andLakonishok (1993)], return interval over which beta is estimated [Kothari et al.(1995a)], 3 the form in which the CAPM is tested [Jagannathan and Wang (1996)]and statistical issues [Kim (1995)].An alternative explanation of the fiat relationship between return and beta wasproposed by Pettengill et al. (1995). They argued that the statistical methodologyused to evaluate the relationship between beta and return requires adjustment totake account of the fact that realized returns and not ex ante returns have beenused in the tests. They developed a conditional relationship between return andbeta which depends on whether the excess return on the market index is positive ornegative. In periods when the excess market return is positive (up market), thereshould be a positive relationship between beta and return. In periods when theexcess market return is negative (down market), there should be a negativerelationship between beta and return. This is because high beta stocks are moresensitive to the negative market excess return and will have a lower return than lowbeta stocks. The evidence in Pettengill et al. (1995) shows that for the period1936-1990, there is strong support for beta when the sample period is split into upmarket and down market months.This paper examines the cross-sectional relationship between beta and return inUK stock returns between January 1975 and December 1994. The main objectiveof the paper is to examine the conditional relationship between beta and returnproposed by Pettengill et al. (1995) to UK stock returns. The paper also considersthe role of size in UK stock returns.

Consistent with Strong and Xu (1994) and Fama and French (1992), there is noevidence of a significant risk premium on beta when the unconditional relationshipbetween beta and return is examined. However in periods when the market returnexceeds the risk-free return, there is a significant positive relationship betweenbeta and return. When the market return is less than the risk-free return, there is asignificant negative relationship between beta and return. The surprising finding ofthis study is that the relationship is stronger in down market months than upmarket months. Subsidiary results of the paper provide little support for the sizeeffect in UK stock returns for this time period. This appears to be due to a possiblenonlinear relationship between portfolio average return and the proxy for portfo-lio size.

The paper is outlined as follows. Section II contains the regression methodologyused in the tests. Section III reports the data and portfolio grouping strategy.Section IV includes the main empirical results. The final section contains conclud-ing comments.

2 See, for example, Peltz (1992).3See, also, the subsequent papers by Fama and French"(1996) and Kothari et al. (1995b).


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Beta and Return: UK Evidence 213

I I. M e t h o d o l o g yT h e z e r o - b e t a C A P M o f B l a ck (1 97 2) p r e d i c ts t h a t :

E ( R i ) = 3'0 + 3 ' 1 /9 / fo r a l l i = 1 . . . . . N (1 )w h e r e E ( R ) i s t h e e x p e c t e d r e t u r n o n a s s e t i ; /3i i s t h e b e t a o f a s s e t i w h e r e~ i = c v ( R i , Rm)/var (Rm); To is t h e e x p e c t e d r e t u r n o n t h e p o r t f o l io , w h i c h h a s az e r o c o v a r i a n c e w i t h t h e m a r k e t p o r t f o li o 4 ; 71 i s t h e e x p e c t e d r is k p r e m i u m o f th em a r k e t p o r t f o l i o . M o s t s t u d i e s h a v e a d o p t e d a tw o - p a s s r e g r e s s io n m e t h o d o l o g y t oe x a m i n e w h e t h e r t h e r e i s a s ig n i f ic a n t p o s i ti v e ri sk p r e m i u m o n b e t a . T h e t w o - p a ssm e t h o d o l o g y s t e m s m a i n ly f r o m F a m a a n d M a c B e t h ( 19 73 ). S h a n k e n ( 1 99 2)p r e s e n t e d a u s e f u l re v i e w o f th e s t a ti s ti c a l f r a m e w o r k o f t h e t w o - p a s s m e t h o d o l o g ya n d s o m e o f t h e e c o n o m e t r i c i s s u e s i n v o l v ed , in c l u d i n g t h e r o l e o f s i ze . 5

I n t h e f i r s t s t e p , / 3 ~ i s e s t i m a t e d f r o m t h e r e g r e s s i o n m o d e l :R i t = ~ i + ~ i R m t + c 'it (2 )

w h e r e R i t is t h e r e t u r n o n a s s e t i i n p e r i o d t ; Rmt i s t h e r e t u r n o n t h e m a r k e tp r o x y p o r t f o l i o i n p e r i o d t ; c-it i s a r a n d o m e r r o r t e r m ; / 3 i i s t h e e s t i m a t e d b e t a o fa s s e t i . I t i s a s s u m e d t h a t t h e e r r o r t e r m s i n e q u a t i o n ( 2 ) a r e i n d e p e n d e n t l y a n di d e n t ic a l l y d i s t r i b u t e d w i t h m e a n z e r o a n d s t a t i o n a r y c o v a r i a n c e m a t r ix , a n d R ,~ t isd r a w n f r o m a s t a t i o n a r y d i s t r i b u t i o n .

I n t h e s e c o n d s t a g e , a c r o s s - s e c t i o n a l r e g r e s s i o n e q u a t i o n i s e s t i m a t e d e a c hm o n t h a s :

R i t = T o t - k- T i t ~ i q - u i t (3 )w h e r e /3 is e s t i m a t e d f r o m e q u a t i o n ( 2) , a n d uit is a r a n d o m e r r o r t e r m . E q u a t i o n( 3) i s e s t i m a t e d b y O r d i n a r y L e a s t S q u a r e s ( O L S ) , 6 w h i c h g i v es e s t im a t e s 3 '0 t a n d~/1~ f o r e a c h m o n t h i n t h e s a m p l e p e r i o d . T h e a v e r a g e v a l u e s o f t h e m o n t h l yc o e f f i c i e n t s ( 7 0 , 7 1 ) a r e c a l c u l a t e d , a n d t h e a v e r a g e v a l u e c a n b e t e s t e d t o s e e i f i tis si g n if i ca n t ly d i f f e r e n t f r o m z e r o u s i n g t h e t t e s t o f F a m a a n d M a c B e t h ( 19 7 3) . 7T h e t t e s t c a n b e m o d i f ie d t o t a k e a c c o u n t o f t h e e f f e c t o f m e a s u r e m e n t e r r o r i nb e t a , a s a n e s t i m a t e o f b e t a is u s e d i n e q u a t i o n ( 3 ) [s ee S h a n k e n ( 19 92 )]. E q u a t i o n( 3 ) c a n a l s o b e e x t e n d e d t o i n c l u d e a d d i t i o n a l v a r i a b l e s s u c h a s s i z e . T h e s i z e e f f e c tc a n b e e x a m i n e d b y t es t in g w h e t h e r t h e a v e r ag e v a l u e o f t h e m o n t h l y c o e ff i ci e n tso n t h e s i ze v a r i a b l e i s s ig n i f i c a n tl y d i f f e r e n t f r o m z e r o .

4 When a risk-free asset exists, 2'0 will be the risk-free return, and this is the traditional form of theCAPM of Sharpe (1964).5 Banz (1981) found that the market value of the firm was inversely related to re turn after the role ofbeta had been adjusted for. Berk (1995) has pointed out that a market value variable is a usefulspecification test o f an asset pricing model.6 Shanken (1992) has po inted out that equation (3) can also be estimated by Generalized LeastSquares (GLS) and Weighted Least Squares (WLS).7 This is simply the average value o f the coefficients divided by the standard error of the coefficients.The standard e rror of the coefficients equals the standard deviation of the monthly coefficients dividedby the square root o f the numbe r of observations.


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214 J. Fletche rT h e m a i n o b j e c t i v e o f t h i s s t u d y i s t o e x a m i n e t h e c o n d i t i o n a l r e l a t i o n s h i pb e t w e e n b e t a a n d r e t u r n . P e t t e n g i l l e t a l. ( 19 9 5) a r g u e d t h a t s t u d i e s f o c u s in g o n

t h e r e l a ti o n s h i p b e t w e e n r e t u r n a n d b e t a n e e d t o t a k e a c c o u n t o f t h e f a ct t h a t e xp o s t r e t u r n s h a v e b e e n u s e d i n t h e t e s t s a n d n o t e x a n t e r e t u r n s . W h e n r e a l i z e dr e t u r n s a r e u s e d , a c o n d i t i o n a l r e l a t i o n s h i p b e t w e e n b e t a a n d r e t u r n s h o u l d e x is t.T h i s o c c u r s a s th e r e m u s t b e s o m e p r o b a b i l it y w h e r e i n v e s t o r s e x p e ct t h a t t h er e a l i z e d r e t u r n o n a l o w b e t a p o r t f o l i o w i ll b e g r e a t e r t h a n t h e r e t u r n o n a h i g hb e t a p o r t f o l i o . T h i s is b e c a u s e n o i n v e s t o r w o u l d h o l d t h e l o w b e t a p o r t f o l i o i f t h isw e r e n o t t h e c a s e. P e t t e n g i l l e t al . ( 19 9 5) a s s u m e d t h a t t h i s o c c u r s w h e n t h e m a r k e tr e t u r n i s l o w e r t h a n t h e r i s k - f r e e r e t u r n , w h i c h t h e y s u g g e s t e d i s i m p l i e d b y t h ee x c es s r e tu r n s m a r k e t m o d e l . T h e i m p l i c a t i o n o f t h i s i s t h a t t h e r e s h o u l d b e ap o s i t i v e r e l a t i o n s h i p b e t w e e n b e t a a n d r e t u r n w h e n t h e e x c e s s m a r k e t r e t u r n i sp o s i ti v e , a n d a n e g a t i v e r e l a t io n s h i p w h e n t h e e x c es s m a r k e t r e t u r n i s n e g a ti v e .

T o t e s t t h e c o n d i t i o n a l r e l a ti o n s h i p , t h e s a m p l e p e r i o d w a s d i v i d e d i n t o u pm a r k e t m o n t h s a n d d o w n m a r k e t m o n t h s . T h i s i s e q u i v a l e n t t o s p l i t t i n g t h et i m e - s e r i es c o e f f ic i e n ts , 3 ' 1t , i n t o t w o s u b s a m p l e s . F o r a ll t h e m o n t h s w h e n t h em a r k e t r e t u r n e x c e e d e d t h e r i s k - f r e e r e t u r n , t h e c o r r e s p o n d i n g 3 ' 1 t w e r e g r o u p e di n t o th a t s u b s a m p l e . F o r t h e m o n t h s w h e n t h e m a r k e t r e t u r n w a s le ss th a n t h er i s k - f r e e r e t u r n , t h e c o r r e s p o n d i n g 3 ' 1, w a s p u t i n t o t h a t s u b s a m p l e . ~ '2 t i s d e -f i n ed as t h e m o n t h l y r is k p r e m i u m e s t im a t e s i n u p m a r k e t m o n t h s a n d ~3t as t h er i s k p r e m i u m e s t i m a t e s i n d o w n m a r k e t m o n t h s . T h e h y p o t h e s e s , p r e d i c t e d b yPe t t e ng iU e t a l . , ( 1995) a re :

H o : 7 2 = 0Ha: 72 > OHo: 73 = 0H a ' 7 3 * ( 0

w h e r e 7 2

(4 )

a n d 73 are t h e a v e r a g e v a l u e s o f t h e c o e f f i c i e n ts ~2t a n d 3 '3 t. T h e s e c a nb e t e s t e d b y t h e s t a n d a r d t t e s ts o f F a m a a n d M a c B e t h ( 19 73 ).P e t t e n g i l l e t a l . ( 1 9 9 5 ) p o i n t e d o u t t h a t t h e a b o v e c o n d i t i o n a l r e l a t i o n s h i p d o e s

n o t g u a r a n t e e a p o s i t i v e r i s k a n d r e t u r n t r a d e o f f . T h e y s t a t e d t h a t t w o c o n d i t i o n sa r e n e c e s s a r y f o r a p o si ti v e t r a d e o f f b e t w e e n r is k a n d r e t u r n . T h e s e a r e t h a t t h ee x c e s s m a r k e t r e t u r n s h o u l d b e p o s i t i v e o n a v e r a g e a n d t h e r i s k p r e m i u m i n u pm a r k e t s a n d d o w n m a r k e t s s h o u l d b e s y m m e t ri c a l. T h e s y m m e t ri c a l r e la t io n s h i pc a n b e t e s t e d b y t h e f o l l o w i n g h y p o t h e s i s :

H 0 : 7 2 - 7 3 = 0 . ( 5 )T h i s c a n b e t e s t e d b y a t w o - p o p u l a t i o n t t e st , b u t t h e s ig n o f t h e "Y3t c o e f f i c i e n t sn e e d s t o b e r e v e r s e d a n d t h e a v e r a g e v a l u e r e c a l c u l a t e d .

I I I . D a t aR e t u r n s w e r e c o l l e c t e d f o r s e c u r i t i e s i n c l u d e d i n t h e L o n d o n B u s i n e s s S c h o o lS h a r e P r i c e D a t a b a s e ( L B S ) b e t w e e n J a n u a r y 1 9 75 a n d D e c e m b e r 1 99 4. T h er e s u l t s o v e r t w o t e n - y e a r s u b - p e r i o d s w e r e a l s o c o n s i d e r e d . W e b e g a n t h e s a m p l ep e r i o d h e r e b e c a u s e i t i s o n l y s i nc e 1 9 75 t h a t t h e L B S h a s i n c l u d e d r e t u r n d a t a o n


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Beta and Return: UK Evidence 215all securities which have traded on the London Stock Exchange and UnlistedSecurities Market. The monthly return on a 30-day UK Treasury Bill was used asthe risk-free return obtained from LBS. The regression approach described in theprevious section used portfolios of securities formed from the LBS. Securities weregrouped into 100 size-beta portfolios following Fama and French (1992). Securitieswere included in the portfolios each year if they existed for at least three yearsprior to the start of that year and were in existence at the start of the year.

Beginning with January 1975, all securities with non-zero market values wereranked on the basis of market value (as recorded on LBS) and grouped into 10portfolios in ascending order. Within each size decile, the beta of the security wasestimated from the regression of the security return on a constant and the returnon the Financial Times All Share Index (FFA) 8 as the market proxy using pastreturn data over the previous 36 to 60 months. This can be viewed as thepre-ranking beta estimate. The securities within each size decile were then groupedinto 10 beta portfolios on the basis of their pre-ranking betas. Equally-weightedreturns were calculated each month on the 100 portfolios over the subsequent year.This procedure was repeated for each year and gave 240 monthly return observa-tions between January 1975 and December 1994, for each portfolio. The number ofsecurities included in each portfolio varied f rom year to year. For 1975 to 1977, thenumber of securities within each por tfolio was between 13 to 14, and for the lateryears ranged between 17 to 19, on average.

Table 1 presents the average monthly return of the 100 size-beta portfolios.The re is quite a wide dispersion in the monthly average returns of the 100 size-betaportfolios, which ranged be tween 0.33% to 2.22%. Each of the 10 portfolios in thesmallest market value decile had a higher average return than the correspondingportfolios in the largest marke t value decile. There does not appear to be a strongrelationship between average return and beta within a given size decile. In amajority of the size deciles, the low beta portfolio had a higher mean return thanthe high beta portfolio. Also, the relationship between average return and sizeportfolios within a given beta decile is not monotonic. The average returns tend to

8 This is a value-weighted index of the largest 750 compan ies on the Lond on Stock Exchange.

Table 1. Average Return of 100 Size-Beta Portfolios a

MV 1 1.95 1.83 1.79 1.66 1.87 2.14 1.45 2.06 2.22 1.65MV 2 1.4 1.41 1.26 1.68 1.65 1.65 1.48 1.22 1.44 0.33MV3 1.39 1.47 1.49 1.61 1.69 1.27 1.01 1.17 1.54 0.38MV4 1.31 1.15 1.38 1.38 1.72 1.55 1.25 0.82 1.6 1.05MV5 0.91 1.04 1.44 1.38 1.1 1.49 0.73 1.2 1.0l 1.21MV6 1.61 1.33 1.39 1.35 1.46 1.39 1.51 1.12 1.0 0.87MV 7 1.14 1.36 1.29 1.49 1.21 1.15 1.4 1.6 1.23 0.88MV8 1.42 1.14 1.44 1.32 1.62 1.47 1.38 1.45 1.6 0.78MV9 1.24 1.37 1.53 1.56 1.42 1.42 1.57 1.44 1.54 1.11MV10 1.27 1.59 1.32 1.55 1.24 1.48 1.25 1.77 1.56 1.28

a The table reports the monthly average return (%) of 100 size-beta portfolios for the period January 1975to December 1994. The individual securities are first grouped into 10 size (rows) deciles for each year, and thenwithin each decile they are further grouped into 10 beta portfolios (columns).


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216 J. Fletc her

T a b l e 2 . P o r t f o l i o B e t a o f 1 0 0 S i z e - B e t a P o r t f o l i o s a31 3z 33 34 35 136 37 /38 39 310

MV 1 0.59 0.63 0.78 0.72 0.92 1.04 1.02 1.19 1.23 1.37MV2 0.64 0.78 0.74 0.95 0.9 1.04 1.08 1.15 1.17 1.34MV3 0.64 0.75 0.79 0.92 0.98 1.00 1.05 1.08 1.36 1.22MV4 0.69 0.74 0.82 0.91 1.02 1.04 1.07 1.14 1.29 1.47MV5 0.71 0.91 0.88 1.03 1.01 1.11 1.12 1.22 1.25 1.42MV6 0.68 0.91 0.95 1.01 1.05 1.12 1.11 1.21 1.26 1.45MV 7 0.74 0.91 1.01 0.99 1.02 1.13 1.18 1.09 1.3 1.46MV8 0.82 0.98 1.07 1.07 1.1 1.08 1.24 1.21 1.32 1.45MV9 0.87 1.03 1.08 1.1 1.21 1.12 1.19 1.26 1.27 1.36MVI0 0.88 0,98 0.92 0.99 1.08 1.12 1.11 1.13 1.16 1.25

a The table repor ts were the portfolio beta of 100 size-beta portfolios for the period January 1975 toDecember 1994. The portfolio betas were computed relative to the equally-weighted index (EWl) as themarket proxy. The individual securities are first grouped into 10 size (rows) deciles for each year, and thenwithin each decile they are further grouped into 10 beta portfolios (columns).

d e c l i n e a n d t h e n r is e a g a in a s w e m o v e d o w n t h e c o l u m n s i n T a b l e 1 i n a n u m b e ro f c a s e s .

T a b l e 2 r e p o r t s t h e p o s t - r a n k i n g b e t a e s t i m a t e s o f t h e 1 0 0 s i z e - b e t a p o r t f o l i o s .T h e b e t a s w e r e e s t i m a t e d f r o m t h e r e g r e s s i o n o f t h e p o r t f o l i o r e t u r n s o n ac o n s ta n t a n d t h e r e t u r n s o f th e m a r k e t p r o x y b e t w e e n J a n u a r y 1 97 5 a n d D e c e m b e r1 99 4. T h e F T A a n d a n e q u a l l y - w e i g h t e d in d e x ( E W I ) w e r e u s e d a s t h e m a r k e tp r o x i e s . 9 T h e E W I p r o x y i s a n e q u a l l y - w e i g h t e d p o r t f o l i o o f a ll s e c u r i t ie s w i thr e t u r n o b s e r v a t i o n s f r o m L B S in a g i ve n m o n t h . O n l y t h e b e t a s e s t i m a t e d u s in g t h eE W I p r o x y a r e r e p o r t e d i n T a b l e 2 . T a b l e 2 s h o w s t h a t t h e r e is a c lo s e r e la t i o n s h i pb e t w e e n t h e p o s t - r a n k i n g b e t a s a n d t h e p r e - r a n k i n g b e t a s .

T a b l e 3 g iv e s t h e l o g o f t h e t i m e - s e r i e s a v e r a g e o f t h e m a r k e t v a l u e s o f th e 1 0 0s i z e -b e t a p o r t f o l io s b e t w e e n J a n u a r y 1 97 5 an d D e c e m b e r 1 99 4. T h e s iz e o f t h e

9 The results of this study should no t be viewed as strict tests of the CAPM because of Roll's (1977)critique about the unobservability of the market portfolio. Given this, the results of this paper can beviewed as tests of a single factor model in UK stock returns.

T a b le 3 . A ve r a ge M a r ke t V a lue (Log) o f 100 S i z e -Be t a Po r tf o l io safll 32 33 34 35 36 37 38 39 310

MV 1 0.39 0.36 0.42 0.39 0.44 0.44 0.47 0.43 0.42 0.44MV2 1.07 1.08 1.08 1.13 1.11 1.12 1.11 1.12 1.12 1.11MV3 1.62 1.62 1.62 1.64 1.66 1.62 1.64 1.66 1.64 1.65MV4 2.08 2.08 2.09 2.11 2.10 2.09 2.11 2.11 2.10 2.11MV5 2.51 2.51 2.53 2.52 2.51 2.53 2.54 2.54 2.54 2.53MV6 2.99 2.98 2.99 3.02 3.01 3.01 3.03 3.01 3.01 3.00MV 7 3.52 3.52 3.54 3.54 3.57 3.54 3.56 3.55 3.56 3.54MVs 4.15 4.17 4.18 4.19 4.2 4.22 4.24 4.24 4.23 4.20MV9 5.06 5.08 5.09 5.08 5.09 5.16 5.16 5.21 5.15 5.14MV10 7.26 7.45 7.33 7.10 7.07 7.02 7.10 7.09 7.02 6.81

'~ The table repor ts the log of the time-series average market value of 100 size-beta portfolios for the periodJanuary 1975 to December 1994. The size of the portfolios was calculated for each year as the average marketvalue of the star t of the year of the securities in the portfolio. The individual securities are first grouped into 10size (rows) deciles for each year, and then within each decile they are further grouped into 10 beta portfolios(columns).


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Beta and Return: UK Evidence 217p o r t f o l i o s w a s c a l c u l a t e d f o r e a c h y e a r a s th e a v e r a g e m a r k e t v a l u e a t t h eb e g i n n i n g o f t h e y e a r ( o b t a i n e d f r o m L B S ) o f t h e s e c u r it i es i n t h e p o r t f o l i o .T h e v a l u e s r a n g e d b e t w e e n 0 . 3 6 t o 7 . 4 5 ( i n m i l l i o n p o u n d s ) .

I V . T h e R o l e o f B e t a a n d S i z e i n R e t u r n sI n i ti a ll y t h e r e l a t i o n s h i p b e t w e e n p o r t f o l i o b e t a a n d s i z e a n d t h e m o n t h l y r e t u r n so f t h e 1 00 s i ze - b e t a p o r t f o l io s w a s e x a m i n e d w i t h o u t d i s t i n c t io n b e t w e e n u pm a r k e t s a n d d o w n m a r k e ts . T h e b e t a s o f t h e p o r t f o l io s w e r e e s t i m a t e d f o r t h e f u llp e r i o d b e t w e e n J a n u a r y 1 9 7 5 a n d D e c e m b e r 1 9 9 4 , w i t h r e s p e c t t o t h e F T A a n dE W I p r o x i e s. T a b l e 4 p r e s e n t s t h e r e s u l t s o f th e m o n t h l y c r o s s - s ec t i o n a l r eg r e s -s i o ns o f p o r t f o l io r e t u r n s , b e t a a n d s iz e . T h e c o e f f i ci e n t s w e r e e s t i m a t e d b y O L S ,a n d t h e t s t a ti s ti c s o f F a m a a n d M a c B e t h ( 1 9 73 ) h a v e b e e n c o r r e c t e d f o r th em e a s u r e m e n t e r r o r i n b e t a a s s u g g e s t e d b y S h a n k e n ( 1 9 9 2 ) . P a n e l A r e f e r s t o t h eF T A p r o x y a n d p a n e l B t o t h e E W I p r o x y.

T h e e v i d e n c e i n T a b l e 4 s u g g e s t s t h a t t h e r e i s n o s i g n i f i c a n t p o s i t i v e ri s kp r e m i u m o n b e t a i n U K s t o c k r et u r n s. I n d e e d , th e e s t i m a t e d ri s k p r e m i u m o n b e t ai s a c t u a l l y n e g a t i v e , a l t h o u g h i n s i g n i f i c a n t , w h i c h s u g g e s t s a d o w n w a r d s l o p i n gr e l a t i o n s h i p b e t w e e n b e t a a n d r e t u r n . O n a n a n n u a l i z e d b a s i s , t h e e s t i m a t e d r i s kp r e m i u m o n b et a is - 3 . 7 6 % ( F T A ) a n d - 4 . 8 2 % ( E W I ) . W h e n t h e an al ys is w a sr e p e a t e d o v e r t w o te n - y e a r s u b p e r io d s ( J a n u a r y 1 9 7 5 - D e c e m b e r 1 98 4 a n d J a n u a r y1 9 8 5 - D e c e m b e r 1 99 4), t h e e s t i m a t e d r is k p r e m i u m s o n b e t a w e r e 0 .0 0 04 9 ( t =0 .073) a n d - 0 .00762 ( t = - 1 .22 ) fo r the F T A p roxy , a n d 0 .00143 ( t = 0 .254) a n d- 0 . 0 1 1 5 ( t = - 1 .9 4) f o r t h e E W I p r o x y . A g a i n , t h e r e i s n o s u p p o r t o f a s i g n i f i c a n tp o s i t i v e r i s k p r e m i u m o v e r e i t h e r s u b - p e r i o d . T h e f i n d i n g s o f a n i n s i g n i f i c a n t r i s k

Table 4. Cross-Sectional Regressions of Portfo l io R etu rns on B eta and SizeaPanel A b

Inter cept -T0 Beta-~,l Size-~'2Coeff icient 0.0162 - 0.00313t 5.25" - 0.66Coeffi cient 0.0163 - 0.0035 0.000068t 6.35" - 0.703 0.13

Pane l B cInt e rc ept -% Beta-~ i S ize-5'2

Coeff icient 0.01799 - 0.00402t 6.97" - 0.94Coeff icien t 0.0181 - 0.0038 - 0.000099t 6.46* - 0.94 - 0.21

* Significant at 5%.a Monthly cross-sectional regressions were run on portfolio returns of the 100 size-beta portfolios on aconstant and the e stim ated beta for the period Janu ary 1975 to Dece mb er 1994. The regressions were alsoestim ated including a proxy for portfolio size. The coefficients Y0, ~1 and 5'2 are the time-series averages of%t, ~/lt and ~/2t estim ated using Ordina ry Least Squ ares (OLS). The t statistics have been correct ed for themea sur eme nt errors in beta, as suggested by Shanke n (1992), and test whet her the es timate d coefficient issignificantly different fr om zero.b Panel A of the table uses portfolio betas es timat ed from the Financial Times All Share index (FTA).c Panel B uses the eq ually-weighted index (EWI) to esti mate betas.


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2 1 8 J. Fletcherpremium on beta 1 is consistent with Fama and French (1992) and Jagannathanand Wang (1996) for US market proxies, and also with Strong and Xu (1994) in UKstock returns. This may be due to the low power of these tests [see Affleck-Gravesand Bradfield (1993)].

The other interesting observation from Table 4 is the absence of the size effectin UK stock returns, as the coefficient on the size variable is insignificantlydifferent from zero. The coefficient on the size effect continues to be insignificantover the two ten-year sub-periods. This contrasts with most US studies, but isconsistent with Strong and Xu (1994), n who found no evidence of a size effect forthe period 1973 to 1992.Examination of Table 1 suggests a possible explanation of the insignificantrelationship between size and returns. When a proxy for the size variable isincluded in the regressions, it is assumed that the relationship between size of theportfolios and average return is linear. However Table 1 shows that for many of thepre-ranking beta deciles, the relationship between the size of the portfolios andaverage return appears U-shaped. Although the average return of the portfolios ofthe smallest size decile are the highest, there is not a monotonic decline in theaverage returns as we move across the size portfolios. The absence of this linearsize relationship has been checked by adding a squared size variable to thecross-sectional regression. The coefficient on the squared size variable is statisti-cally significant over the whole sample period. This suggests that the absence of thesize effect is caused by a possible non-l inear relationship between portfolio averagereturn and the proxy for portfolio size.

A number of explanations have been put forward for the findings of aninsignificant relationship between beta and return. The one which is considered inthis study is that proposed by Pettengill et al. (1995), who argued that the finding ofthe fiat slope arises when realized returns are used in the tests. This is becausethe relationship between beta and return in this situation will be conditional on theexcess returns on the market index.

The hypotheses of Pettengill et al. (1995) can be evaluated by splitting themonthly risk premium estimates into two subsamples according to whether theexcess market return is positive or negative. The hypotheses were tested forthe whole sample period and for the two ten-year sub-periods. Table 5 contains theresults. Panel A refers to the VIA proxy and panel B to the EWI proxy. The tablereports the mean values of ~ 2 t and ~ / 3 t with corresponding t statistics and thenumber of up market and down market months over the sample period (reportedin parentheses). The table also includes t values of the test that 92 - 93 = 0,which tests whether the relationship between beta and return in up market anddown market months is symmetrical.Table 5 shows that there are a substantial number of down market months overthe sample period, 97 for the VIA proxy and 100 for the EWI proxy. The evidencein Table 5 suggests that in periods of up market months, there is a significant

10 T h e r e g r e s s i o n s w e r e a l s o e s t i m a t e d b y G L S a n d W L S a s g i v e n in S h a n k e n ( 19 9 2) , b u t t h i s h a s n oi m p a c t o n t h e i n f e r e n c e s i n T a b l e 4 .n S t r o n g a n d X u ( 1 9 9 4 ) d i d f i n d a s i g n if i ca n t b o o k - t o - m a r k e t e q u i t y e f f e c t a s in F a m a a n d F r e n c h(1992) .


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B e t a a n d R e t u r n : U K E v i d e n c e

T a b l e 5 . T e s t s o f t h e C o n d i t i o n a l R e l a t i o n s h i p B e t w e e n B e t a a n d R e t u r n a

219

P a n e l A bA l l 1 / 7 5 - 1 2 / 8 4 1 / 8 5 - 1 2 / 9 4

7 2 0 . 0 3 1 7 5 ( 1 4 3 ) 0 . 0 3 5 7 9 ( 7 4 ) 0 . 0 2 6 8 ( 6 9 )t 5 . 8 9 * 4 . 6 7 * 4 . 0 1 "7 3 - 0 . 0 5 4 5 6 ( 9 7 ) - 0 . 0 5 6 2 8 ( 4 6 ) - 0 . 0 5 4 1 9 ( 5 1 )t - 1 0 . 3 4 " - 7 . 4 8 * - 7 . 1 "7 2 - 7 3 = 0 - 3 . 0 2 * - 1 . 9 8 - 2 . 7 *

P a n e l B cA l l 1 / 7 5 - 1 2 / 8 4 1 / 8 5 - 1 2 / 9 4

~ 2 0 . 0 2 7 7 8 ( 1 4 0 ) 0 . 0 2 2 6 6 ( 7 8 ) 0 . 0 3 0 3 ( 6 2 )t 5 . 8 4 * 3 . 3 8 * 5 . 2 8 *7 3 - 0 . 0 4 8 5 4 ( 1 0 0 ) - - 0 . 0 3 7 9 ( 4 2 ) - 0 . 0 5 6 ( 5 8 )t - - 9 . 2 9 * - 5 . 5 * - 8 . 4 3 *7 2 - - ~ 3 = 0 - 2 . 9 4 * - 1 . 5 9 - 2 . 9 5 *

* S i g n i f i c a n t a t 5 % .a T h e m o n t h l y r i s k p r e m i u m c o e f f i c i e n ts o n b e t a , " Y l t h a v e b e e n d i v i d e d i n to t w o s u b s a m p l e s a c c o r d i n g tow h e t h e r t h e e x c e s s m a r k e t r e t u r n w a s p o s i t i v e o r n e g a t i v e . T h e c o e f f i c i e n t s ~ 2 a n d ~ 3 a r e t h e t i m e - s e r i e sa v e r a g e s o f t h e r i s k p r e m i u m e s t i m a t e s i n u p m a r k e t s ( p o s it i v e ex c e s s m a r k e t r e t u r n s ) a n d d o w n m a r k e t s( n e g a t i v e m a r k e t e x c e s s r e t u r n s ) , r e s p ec t i v e ly . T h e t s t at i s ti c s [ c o m p u t e d u s i n g t h e t i m e - s e r i e s a p p r o a c h o fF a m a a n d M a c B e t h ( 1 9 73 ) ] a r e o n e - s i d e d t e s t s o f w h e t h e r ~ 2 i s s i g n i fi c a n t ly p o s i ti v e a n d ~ 3 i s s i g n i fi c a n t lyn e g a t i v e , r e s p e ct i v e l y . T h e l a s t r o w i s a t w o p o p u l a t i o n t t e s t o f ~ 2 - ~ 3 = 0 . T h e r e s u l t s a re r e p o r t e d f o r th ep e r i o d J a n u a r y 1 9 75 t o D e c e m b e r 1 9 94 , a n d o v e r t w o t e n - y e a r s u b - p e r io d s .b P a n e l A o f t h e t a b l e u s e s p o r t f o l i o b e t a s e s t i m a t e d f r o m t h e F i n a n c i a l T i m e s A l l S h a r e i n d e x ( F T A ) .c P a n e l B u s e s t h e e q u a l l y - w e i g h t e d i n d e x ( E W l ) t o e s t i m a t e b e t a s .

positive relationship between beta and return. High beta portfolios exhibitedhigher returns than low beta portfolios. In periods of down-markets, there is asignificant negative relationship between beta and return. High beta portfoliosearned a lower return than low beta portfolios. Similar patterns are observed overthe two sub-periods and with both market proxies. 12 This is consistent with thehypotheses and the evidence given in Pettengill et al. (1995). It is important to takeaccount of the conditional relationship between beta and return in empirical tests.

The hypothesis that the relationship between beta and return in up market anddown market months is symmetrical, is rejected in Table 5. The estimated riskpremiums in down markets are higher than those in up markets. This contradictsone of the necessary conditions of a positive risk-return tradeoff, This is inconsis-tent with Pettengill et al. (1995), who found that the relationship was symmetricalin US stock returns. It is a puzzle why the relationship between beta and returnshould be stronger in down markets.The results in Table 5 were checked to examine if the October 1987 crash or theJanuary effect documented by Rozeff and Kinney (1976), amongst others, had anyimpact on the inferences. The October 1987 observation and the January monthswere excluded from the analysis. Similar inferences were found in both cases toTable 5. Neither the October 1987 crash or the January effect explain why therelationship between beta and return is stronger in down markets.

~2 S i m i l a r r e su l t s w e r e f o u n d w h e n b e t a s w e r e e s t i m a t e d u s i n g a n e x c e s s r e t u r n s r e g r e s s i o n .


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2 20 J . F le tcherT h e e v i d e n c e w i t h in t h e p a p e r d o e s s u g g e s t th a t t h e r e i s a c o n d i t i o n a l r e l a ti o n -

s h i p b e t w e e n b e t a a n d r e t u r n i n U K s t o c k r e t u r n s . I t a l s o i n d i c a t e s t h a t b e t a m a ys ti ll h a v e a u s e f u l r o l e t o p l a y f o r p o r t f o l i o m a n a g e r s . I n v e s t o r s w h o a r e c o n c e r n e da b o u t t h e r is k o f p e r i o d s w h e r e t h e m a r k e t r e t u r n f al ls b e l o w t h e r is k - fr e e r e tu r n ,c o u l d p r o t e c t t h e m s e l v e s b y in v e s t i n g i n lo w b e t a s t o c k s. B e t a s e e m s t o b e a g o o di n d i c a t o r o f h o w s t o c ks r e a c t i n p e r i o d s o f d o w n m a r k e t m o n t h s . R e l a t e d e v i d e n c eis C h a n a n d L a k o n i s h o k ( 19 9 3 ) a n d G r u n d y a n d M a l k i e l (1 9 96 ).

V . C o n c l u s i o n sT h i s p a p e r h a s e x a m i n e d t h e c o n d i t i o n a l re l a t io n s h i p b e t w e e n b e t a a n d r e t u r n int h e U K b e t w e e n J a n u a r y 1 97 5 a n d D e c e m b e r 1 9 9 4. C o n s i s t e n t w it h th e f i nd in g s o fF a m a a n d F r e n c h ( 1 9 92 ) a n d S t r o n g a n d X u ( 1 9 94 ), th e r e w a s n o e v i d e n c e o f as ig n if ic a n t r is k p r e m i u m o n b e t a w h e n t h e u n c o n d i t i o n a l r e l a t io n s h i p b e t w e e n b e t aa n d r e t u r n w a s e x a m i n e d . A l s o , t h e r e w a s n o s i g n i f i c a n t r e l a t i o n s h i p b e t w e e n s i z ea n d r e t u r n s . T h i s a p p e a r s t o b e d u e t o a p o s s ib l e n o n - l i n e a r r e l a t io n s h i p b e t w e e np o r t f o l i o a v e r a g e r e t u r n s a n d t h e p r o x y fo r p o r t f o l i o si ze .

W h e n t h e s a m p l e p e r i o d w a s s p li t i n to p e r i o d s o f w h e t h e r t h e e x c e ss m a r k e tr e t u r n w a s p o s i t i v e o r n o t , t h e r e w a s a s i g n i f i c a n t p o s i t i v e r e l a t i o n s h i p b e t w e e nb e t a a n d r e t u r n i n p e r i o d s o f p o s i t iv e e x c e ss m a r k e t r e t u r n s , a n d a s i g n if ic a n tn e g a t iv e r e l a t io n s h i p b e t w e e n b e t a a n d r e t u r n i n p e r i o d s o f n e g a t iv e e x c es s m a r k e tr e t u r n . T h i s i s c o n s i s t e n t w i t h P e t t e n g i l l e t a l . ( 1 9 9 5 ) , a n d s u g g e s t s t h e n e e d t of o c u s o n t h e c o n d i t i o n a l r e l a ti o n s h i p b e t w e e n b e t a a n d r e t u r n . H o w e v e r , t h ec o n d i t i o n a l re l a t io n s h i p b e t w e e n b e t a a n d r e t u r n i n u p m a r k e t a n d d o w n m a r k e tm o n t h s w a s n o t s y m m e t r i c a l , as p r e d i c t e d b y P e t t e n g i l l e t a l. ( 1 9 95 ) . T h e r e l a t i o n -s h ip w a s s t r o n g e r i n d o w n m a r k e t s . T h i s c o n t r a d i c ts o n e o f th e c o n d i t i o n s o f ap o s i t iv e ri s k a n d r e t u r n t r a d e o f f . T h e r e s u l ts o f t h e p a p e r d o s u g g e st t h a t t h em a r k e t b e t a s ti ll h a s a r o l e t o p l a y f o r p o r t f o l i o m a n a g e r s .

Extremely helpful comm ents w ere received from K. P audyal, the editor (J. A ffleck-Graves)and twoanonym ous referees.

R e f e r e n c e sAff leck-Graves , J . D. and Bradf ie ld , D. J . Feb . 1993 . An exam ina t ion of the pow er ofunivar ia te t es ts of the C APM : A s imula t ion approach. Journal o f Eco nom ics and Business45(1):17-33.Ba nz , R . W . M a r . 198 1 . T h e r e l a ti ons h ip be t we e n r e t u r n a nd m a r ke t va lue o f c om m onstocks. Journal o f Financia l Econom ics 9(1) :3 -18 .Berk , J . Su m me r 199 5. A cr i t ique o f si ze re la ted anomal ies . Review of Financial Studies8(2):275-286.Black , F . Ju ly 1972 . Capi ta l ma rke t equi l ibr ium wi th res t r i c ted bor rowing. Journal ofBusiness 45(3):444-455.Black , F . Fa l l 1 993 . Be ta and re turn . Journal o f Port fol io M anage men t 19(1):8-18.Bree den , D. T . Sept . 1979. An in te r t em pora l asse t pr ic ing mode l wi th s tochas t i c consump -t ion and inves tment oppor tuni t i es . Journal o f Financial Eco nom ics 7(3):265-296.


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Beta and Return Uk