mehra and prescott 1985

8/9/2019 Mehra and Prescott 1985

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Journa l of M one tary Economics 15 (1985) 145-161. N orth-H olland

T H E E Q U I T Y P R E M I U MA P u z z l e *

R a j n i s h M E H R AColumbia University, Ne w York, N Y 10027, USA

E d w a r d C . P R E S C O T TFederal Reserve Ban k of M inneapolis

University of M innesota, M inneapolis, M N 5545.5, USA.Re strict ions that a class of general equil ibrium models place upon the average returns of equityand Trea sury bil ls are found to be strongly violated by the U.S. dat a in the 1889-1978 period. Thisresult is rob ust to model specification and measurement problems. W e conclude tha t , most l ikely ,an e quil ibrium mo del which is not an A rrow -D ebre u economy will be the one th at Simultaneouslyratio nal izes bo th historically observed large average equity return and the sm all average risk-freereturn.

1 . I n t r o d u c t i o nH i s t o r i c a l l y th e a v e r a g e r e t u r n o n e q u i ty h a s f a r e x c e e d e d t h e a v e r a g e r e t u r n

o n s h o r t - t e r m v i r t u a l l y d e f a u l t - f r e e d e b t . O v e r t h e n i n e t y - y e a r p e r i o d 1 8 8 9 - 1 9 7 8t h e a v e r a g e r e a l a n n u a l y i e l d o n t h e S t a n d a r d a n d P o o r 5 0 0 I n d e x w a s s e v e np e r c e n t , w h i l e t h e a v e r a g e y ie l d o n s h o r t - t e r m d e b t w a s l es s t h a n o n e p e r c e n t .T h e q u e s t i o n a d d r e s s e d i n t h i s p a p e r i s w h e t h e r t h is l a r g e d i f f e r e n t i a l i na v e r a g e y i e l d s c a n b e a c c o u n t e d f o r b y m o d e l s t h a t a b s t r a c t f r o m t r a n s a c t i o n sc o s t s , l i q u i d i t y c o n s t r a i n ts a n d o t h e r f r ic t io n s a b s e n t i n t h e A r ~ o w - D e b r e us e t - u p . O u r f i n d i n g i s t h a t i t c a n n o t b e , a t le a s t n o t f o r t h e c la s s o f e c o n o m i e sc o n s i d e r e d . O u r c o n c l u s i o n i s t h a t m o s t l i k e l y s o m e e q u i l i b r iu m m o d e l w i t h a

*This research was initia ted at the U niversity of Chicago where M ehr a was a visiting scholar atthe Graduate School of Business and Prescott a Ford foundation visi t ing professor at theD epa rtm ent o f Economics. Earlier versions of th is paper, entit led 'A Test of the Intertem poralAsset Pricing ModeF, were presented at the University of Minnesota, University of Lausanne,Ha rva rd U niversity , NBER Conference on Intertem poral Puzzles in Macroeconomics, and theAm erica n Finan ce M eetings. W e wish to thank the workshop pa rticipants, George Coustantinides,Eugene Fa m a, M erton Miller , and particula rly an anonymous referee, Fischer Black, StephenLeRoy and Charles Plosser for helpful discussions and constructive criticisms. We gratefullyacknowledge financial support from the Fac ulty Research Fun d of the Grad uate School ofBusiness , C o lum bia Universi ty , the N at ional Sdence F oundat ion and the Fe deral Reserve Bank o fMinneapo l is .0304-3923/85/$3.301985, Elsevier Science Publishers B.V. (North-Holland)


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146 R. Mehra and E .C Prescott, The equitypremiumf r i c t i o n w i ll b e t h e o n e t h a t s u c c e s s f u l ly a c c o u n t s f o r t h e l a rg e a v e r a g e e q u i t yp r e m i u m .

W e s t u d y a c l a s s o f c o m p e t i t i v e p u r e e x c h a n g e e c o n o m i e s f o r w h i c h t h ee q u i l i b r i u m g r o w t h r a te p r o c es s o n c o n s u m p t i o n a n d e q u i l ib r i u m a s s e t re t u r n sa r e s t a t i o n a r y . A t t e n t i o n is r e s tr i c te d t o e c o n o m i e s f o r w h i c h t h e e l a s t i c it y o fs u b s t i t u t i o n f o r t h e c o m p o s i t e c o n s u m p t i o n g o o d b e t w e e n t h e y e a r t a n d y e a rt + 1 is c o n s i s t e n t w i t h fi n d i n g s i n m i c r o , m a c r o a n d i n t e r n a t i o n a l e c o n o m i c s .I n a d d i t i o n , t h e e c o n o m i e s ar e c o n s t r u c t ed t o d i s p l a y e q u i l ib r i u m c o n s u m p t i o ng r o w t h r a t e s w i t h t h e s a m e m e a n , v a r i a n c e a n d s e r i a l c o r r e l a t i o n a s t h o s eo b s e r v e d f o r t h e U . S . e c o n o m y i n t h e 1 8 8 9 -1 9 7 8 p e r io d . W e f i n d th a t f o r s u c he c o n o m i e s , t h e a v e r a g e re a l a n n u a l y i e ld o n e q u i t y is a m a x i m u m o f f o u r - t e n t h so f a p e r c e n t h i g h e r t h a n t h a t o n s h o r t - te r m d e b t , in s h a r p c o n t r a s t t o t h e si xp e r c e n t p r e m i u m o b s e rv e d . O u r r e s u l ts a r e r o b u s t t o n o n - s t a ti o n a r i ti e s i n t h em e a n s a n d v a r i a n c e s o f t h e e c o n o m i e s ' g r o w t h p r o c es se s .

T h e s i m p l e c l a s s o f e c o n o m i e s s t u d i e d , w e t h i n k , i s w e l l s u i t e d f o r t h eq u e s t i o n p o s e d . I t c l e a r l y i s p o o r l y s u i t e d f o r o t h e r i s su e s , i n p a r t i c u l a r i ss u e ss u c h a s t h e v o l a t i l i ty o f a s s e t p r ic e s . 1 W e e m p h a s i z e t h a t o u r a n a l y s i s i s n o t a ne s t i m a t i o n e x e r c i s e , w h i c h i s d e s i g n e d t o o b t a i n b e t t e r e s t i m a t e s o f k e ye c o n o m i c p a r a m e t e r s . R a t h e r i t i s a q u a n t i t a t i v e t h e o r e t i c a l e x e r c i s e d e s i g n e dt o a d d r e s s a v e r y p a r t i c u l a r q u e s t i o n . 2

I n t u i t i v e l y , t h e r e a s o n w h y t h e l o w a v e r a g e r e a l r e t u r n a n d h i g h a v e r a g er e t u r n o n e q u i t y c a n n o t s i m u l t a n e o u s l y b e r a t i o n a li z e d in a p e r f e ct m a r k e tf r a m e w o r k i s a s f o ll o w s : W i t h r ea l p e r c a p i t a c o n s u m p t i o n g r o w i n g a t n e a r l yt w o p e r c e n t p e r y e a r o n a v e r a g e , t h e e l a s t i c i t i e s o f s u b s t i t u t i o n b e t w e e n t h ey e a r t a n d y e a r t + 1 c o n s u m p t i o n g o o d t h a t a r e s u f f ic i e n tl y s m a l l t o y i el d th es i x p e r c e n t a v e r a g e e q u i t y p r e m i u m a l so y i e ld r ea l r a t e s o f r e t u r n f a r i n e x c es so f t h o s e o b s e r v e d . I n t h e c a s e o f a g r o w i n g e c o n o m y , a g e n t s w i t h h i g h r i s ka v e r s i o n e f f e c ti v e l y d i s c o u n t t h e f u t u r e t o a g r e a t e r e x t e n t t h a n a g e n t s w i t h l o wr i s k a v e r s i o n ( r e l a t i v e t o a n o n - g r o w i n g e c o n o m y ) . D u e t o g r o w t h , f u t u r ec o n s u m p t i o n w i ll p r o b a b l y e x c ee d p r e s e n t c o n s u m p t i o n a n d s in c e t h e m a r g i n a lu t i l i t y o f f u t u r e c o n s u m p t i o n i s l e s s t h a n t h a t o f p r e s e n t c o n s u m p t i o n , r e a li n t e r e s t r a t e s w i l l b e h i g h e r o n a v e ra g e .

T h i s p a p e r i s o r g a n i z e d a s fo l lo w s : S e c t i o n 2 s u m m a r i z e s t h e U .S . h i s t o r i c ale x p e r i e n c e f o r t h e n i n e t y - y e a r p e r i o d 1 8 8 9 - 1 9 7 8 . S e c t i o n 3 s p ec if ie s t h e s e t o fe c o n o m i e s s t u d i e d . T h e / r b e h a v i o r w i t h r e sp e c t t o a v e r a g e e q u i t y a n d s h o r t - t e r md e b t y i e l d s , a s w e l l a s a s u m m a r y o f t h e s e n s i t iv i t y o f o u r r e s u l t s to t h es p e c i f i c a t i o n s o f t h e e c o n o m y , a r e r e p o r t e d i n s e c t i o n 4 . S e c t i o n 5 c o n c l u d e st h e p a p e r .

1Th ere are other interesting eatures of time series and procedures for testing them. The variancebound tests of L eR oy and Porter (1981) and Shiller (1980) a r e p a r t i c u l a r l y innovative andconstructive. T he y did indicate that consum ptionrisk was imp ortant [see Grossm an and Shiller(1981) and LeR oy and LaC avita 1981)].

2See Lucas (1980) for a n articulationof this methodology.


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K M ehra and E.C. Prescott, The equitypremiumTable I

147

growth rate of ~ real return on aper capita r e a l relat ivelyisldess real return onconsumption security % risk premium S&P 500Time Standard Standard Standard Standardperiods Mean deviation Mean deviation Mean deviation Mean deviation1.83 3.57 0.80 5.67 6.18 16.67 6.98 16.541889-1978 (Std error (Std error (Std error (Std error0.38) ffi 0.60) = 1.76) = 1.74)

1889-1898 2.30 4.90 5.80 3.23 1.78 11.57 7.58 10.021899-1908 2.55 5.31 2.62 2.59 5.08 16.86 7.71 17.211909-1918 0.44 3.07 - 1.63 9.02 1.49 9.18 - 0.14 12.811919-1928 3.00 3.97 4.30 6.61 14.64 15.94 18.94 16.181929-1938 - 0.25 5.28 2.39 6.50 0.18 31.63 2.56 27.901939-1948 2.19 2.52 - 5.82 4.05 8.89 14.23 3.07 14.671949-1958 1.48 1.00 -0.81 1.89 18.30 13.20 17.49 13.081959-1968 2.37 1.00 1.07 0.64 4.50 10.17 5.58 10.591969-1978 2.41 1.40 -0.72 2.06 0.75 11.64 0.03 13.11

2 . DataThe data used in this study consists of five basic series for the period

1889-1 978. 3 The first four are identical t o those used by Grossman and Shiller(1981) in their study. The series are ind ividual ly described below:( i) S e r i e s P : An nu al average Standard and Poor 's Composi te Stock Price

Ind ex divided by the Co nsu mp tio n Deflator, a plot of which appears inGr os sm an an d Shiller (1981, p. 225, fig. 1).

(ii) S e r i e s D : Real an nua l dividends for the Stan dard and Poor's series.(iii) S e r i e s C : Kuzn e t s - Ken dr ik -US NIA pe r cap it a r ea l consumpt ion onnon -dur able s and services.

(iv) S e r i e s P C : Consumption deflator series, obtained by dividing real con-sump tion in 1972 dollars on non-durab les and services by the nomin alcon sum pti on on non-d urable s and services.

(v) S e r i e s RF: No mi na l yield on relatively riskless short-term securities overthe 18 89-19 78 period; the securities used were ninet y-d ay gove rnme ntTreasury Bills in the 1931-1978 period, Treasury Certificates for the

3We thank Sanford Grossman and Robert Shiller for providing us with the data they used intheir study (1981).


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148 IL M ehra a nd E . C. P rescott, The equity prem ium

- 1 1 -

- . ,~Et 8 8 Q

/ : \ mI 1 I I1 9 6 7 1 9 2 E 1 9 4 2 1 9 6 0 1 9 7 e

F i g . 1. R e a l a n n u a l r e t u r n o n S & P 5 00 , 1 8 8 9 - 1 9 7 8 ( p e rc e n t ).

1 9 2 0 - 1 9 3 0 p e r i o d a n d s i x ty - d a y t o n i n e t y - d a y P ri m e C o m m e r c i a l P a p e rp r i o r t o 1 9 2 0 . 4

T h e s e s e r i e s w e r e u s e d t o g e n e r a t e t h e s e r i e s a c t u a l l y u t i l i z e d i n t h i s p a p e r .S u m m a r y s t a ti s ti c s a r e p r o v i d e d in t a b l e 1 .

S e r ie s P a n d D a b o v e w e r e u s e d t o d e t e r m i n e t h e a v e r a g e a n n u a l r e a l r e t u r no n t h e S t a n d a r d a n d P o o r ' s 5 0 0 C o m p o s i t e I n d e x o v e r th e n i n e ty - y e a r p e r i o do f s t u d y . T h e a n n u a l r e t u r n f o r y e a r t w a s c o m p u t e d a s ( Pt +x + D t - P t ) / P t T h e r e t u r n s a r e p l o t t e d i n fig . 1 . S e ri es C w a s u s e d t o d e t e r m i n e t h e p r o c e s s o nt h e g r o w t h r a t e o f c o n s u m p t i o n o v e r t h e s am e p e r io d . M o d e l p a r a m e t e r s w e r er e s t r i c t e d t o b e c o n s i s t e n t w i t h t h i s p r o c e ss . A p l o t o f t h e p e r c e n t a g e g r o w t h o fr e a l c o n s u m p t i o n a p p e a r s i n fig . 2 . T o d e t e r m i n e t h e r ea l r e t u r n o n a r e la t i v e lyr is k l es s s e c u r i t y w e u s e d t h e s e ri es R F a n d P C . F o r y e a r t t h i s i s c a l c u l a t e d t ob e R F - ( P C , + 1 - P C t ) / P C , .

T h i s s e r i e s i s p l o t t e d i n f i g . 3 . F i n a l l y , t h e R i s k P r e m i u m ( R . P ) i s c a l c u l a t e da s t h e d i f f e re n c e b e t w e e n t h e R e a l R e t u r n o n S t a n d a r d a n d P o o r ' s 50 0 a n d t h eR e a l R e t u r n o n a R i s k l es s s e c u r it y a s d e fi n e d a b o v e.

4 T h e d a t a w a s o b t a i n e d f r o m H o m e r ( 19 63 ) a n d I b b o t s o n a n d S i n gu e fi e ld ( 19 79 ).


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P~ Mehra and E.C. Prescott, T he equity prem ium 1 4 9

l S

1 8 8 9 ] l I 11 9 ~ 7 1 9 2 S 1 9 4 2 1 9 0 9F i g . 2. G r o w t h r a t e o f r e a l p e r c a p i t a c o n s u m p t i o n , 1 8 8 9 - 1 9 7 8 ( p e r c e n t) .

1 9 7 8

1 1

- 1 1 -

- S 3 -

t- S S I 1 I 11 8 8 9 1 9 9 7 1 9 2 5 1 9 4 2 1 . 9 1 ~ 8 I9 7 8F i g . 3 . R e a l a n n u a l r e t u r n o n a r e l a t i v e l y r i s k le s s s e c u r i t y , 1 8 8 9 - 1 9 7 8 ( p e r c e n t ) .


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150 R. M ehra and E .C. P rescott, The equity prem ium3 . T h e e c o n o m y , a s s e t p r i c es a n d r e tu r n s

In this paper, we employ a variation of Lucas' (1978) pure exchange model.Since per capita consumption has grown over time, we assume that the growthrate of the endowment follows a Markov process. This is in contrast to theassumption in Lucas' model that the endowment leoel follows a Markovprocess. Our assumption, which requires an extension of competitive equi-librium theory, enables us to capture the non-stationarity in the consumptionseries associated with the large increase in per capita consumption thatoccurred in the 1889-1978 period.

The economy we consider was judiciously selected so that the joint processgoverning the growth rates in aggregate per capita consumption and assetprices would be stationary and easily determined. The economy has a singlerepresentative 'stand-in' household. This unit orders its preferences over ran-dom consumption paths by

, /.o , O)where c, is per capita consumption, /~ is the subjective time discount factor,E0{. } is the expectation operator conditional upon information available attime zero (which denotes the present time) and U: R+--* R is the increasingconcave utility function. To insure that the equilibrium return process isstationary, the utility function is further restricted to be of the constant relativerisk aversion class,

c 1-a - 1U(c,a)= 1 -a ' O


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R . M e h r a a n d E . C . P r e sc o tt , T h e e q u i t y p r e m i u m 151

w h e r e x t + 1 E (h 1 . . . . hn} i s the g row th ra te , andPr{ x t+ 1 = h i ; x , = h i} = ~ / j . (4 )

I t i s a l so a s su m ed tha t the M arkov cha in i s e rgod ic . The h i a re al l pos i t ive andYo > 0 . T he ran do m va r iab le Y t i s obse rved a t the beg inn ing o f the pe r iod , a twhich t ime d iv idend payments a re made . A l l s ecur i t i e s a re t r aded ex-d iv idend .W e a l s o a s s u m e t h a t t h e m a t r i x A w i t h e l e m e n t s a i y = [ ~ d P i j h ~ ' a for i , j =1 . . . . n is s table ; tha t i s, li ra A m as m ~ co i s z e ro . In M ehra and P re sco t t(1984) i t i s shown tha t th is is necessary and suff ic ient for expec ted ut i l i ty toe x i st i f th e s t a n d - i n h o u s e h o l d c o n s u m e s Y t eve ry pe r iod . T hey a l so de f ine andes tab l i sh th e ex i s tence o f a De breu (1954) com pe t i t ive equ i l ib r ium wi th a p r ices y s t e m h a v i n g a d o t p r o d u c t r e p r e se n t a ti o n u n d e r t h is c o n d i ti o n .

N ex t w e fo rmula te expre s s ions fo r the equ i l ib r ium t im e t p r ic e o f the equ i tysha re and the r i sk - f ree b i l l . We fo l low the conven t ion o f p r ic ing secur i t i e sex-d iv idend o r ex - in te re s t payments a t t ime t , in t e rms o f the t ime t consump-t ion good . F o r any secur i ty w i th p roces s { d , } on paym ents , i t s p r ic e in pe r iodt is

P t = E t { ~ , f l ' - t U ' ( y , ) d J U ' ( Y t ) } ,s - - t + l ( 5 )a s e q u i l i b r i u m c o n s u m p t i o n i s t h e p r o c e s s ( y ~ ) a n d t h e e q u i l i b r i u m p r i c es y s t e m h a s a d o t p r o d u c t r e p re s e n ta t io n .Th e d iv id end pay m ent p roces s fo r the equ i ty sha re in th is e con om y is { Ys }-C o n s e q u e n t l y , u s i n g t h e f a c t th a t U ' ( c ) = c - a ,

e , e = P e ( x , , y , )oo y , }

= E ~ a s - t t x ., - . -~ , r , , t , Y t (6 )s - - t + l Y s

V a r i a b l e s x t a n d Y t a r e suff ic ient re la t ive to the ent i re h is tory of shocks upto , and inc lud ing , t ime t fo r p red ic t ing f l a e sub sequ en t evo lu t ion o f theeconomy. They thus cons t i tu te l eg i t ima te s ta te va r iab le s fo r the mode l . S inceY s = Y t " x t + t . . . . . x s , t h e p r i ce o f t h e e q u i t y s e c u ri t y i s h o m o g e n e o u s o f d e g r e eo n e i n Y t , w h i c h i s t h e c u r r e n t e n d o w m e n t o f t h e c o n s u m p t i o n g o o d . A s t h eequ i l ib r ium va lue s o f the econom ies be ing s tud ied a re t ime inva r ian t f tme t ionso f t h e s t a t e ( x t, Y t ) , t h e subsc r ip t t c an be d ropped . Th is i s a ccompl i shed byrede f in ing the s ta te to be the pa i r ( c , i ) , i f y t = c a n d x t = h ~. W i t h t h i s


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1 5 2 K M ehra and E.C. Prescot t, The equity prem ium

c o n v e n t i o n , t h e p r i c e o f th e e q u i t y s h a r e f r o m ( 6) sa t is fi e s/I

- a C ~ j ] Ca .e ( c , i ) f f l E k ij ( A , ) [ p ( h j c , j ) + (7 )j - 1

U s i n g t h e r e s u l t t h a t p e ( c , i ) i s h o m o g e n e o u s o f d e g r ee o n e i n c , w er e p r e s e n t t h i s f u n c t i o n a s

p O ( c , i ) = w , c , ( 8 )

w h e r e w i s a c o n s t a n t . M a k i n g t h i s s u b s t i t u t i o n i n ( 7 ) a n d d i v i d i n g b y c y i e l d s

w i = f l ~ e p i jh S l -a ) (w + 1 ) f o r i = 1 . . . . . n . ( 9 )j - - 1

T h i s i s a s y s t e m o f n li n ea r e q u a t io n s i n n u n k n o w n s . T h e a s s u m p t i o n t h a tg u a r a n t e e d e x i s te n c e o f e q u i l ib r i u m g u a r a n t e e s t h e e x is t en c e o f a u n i q u ep o s i t i v e s o l u t i o n t o t h i s s y s t e m .

T h e p e r i o d r e t u r n i f t h e c u r r e n t s t a t e i s ( c , i ) a n d n e x t p e r i o d s t a t e (h ~ c , j ) i s

r ,~ = P e ( X j c ' j ) + > ~ jc p e ( c , i )p e ( c , i )_ X j ( w j + l )

w,. 1 , ( 1 0 )

u s i n g ( 8 ) .T h e e q u i t y ' s e x p e c t e d p e r i o d r e t u r n i f t h e c u r r e n t s t a t e i s i i s

R = F . , % , ; ; . . ( n )j - 1

C a p i t a l l e t t e rs a r e u s e d t o d e n o t e e x p e c t e d r e tu r n . W i t h t h e s u b s c r i p t i, i t i s t h ee x p e c t e d r e t u r n c o n d i t i o n a l u p o n t h e c u rr e n t s t a t e b e i n g ( c , i ). W i t h o u t t hi ss u b s c r i p t i t i s t h e e x p e c t e d r e t u r n w i t h r e s p e c t t o t h e s t a t i o n a r y d i s t r i b u t i o n .T h e s u p e r s c r i p t i n d i c a t e s t h e t y p e o f s e c u r it y .

T h e o t h e r s e c u r i t y c o n s i d e r e d i s t h e o n e - p e r i o d r e a l b i l l o r r i s k l e s s a s s e t ,w h i c h p a y s o n e u n i t o f t h e c o n s u m p t i o n g o o d n e x t p e r i o d w i t h c er ta i n ty .


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R. M ehra and E.C. Prescott, The equity prem ium

F r o m ( 6 ) ,p : = p ' ( c , )

= , , j v , ( x : ) / u ' ( c )j - 1

= f l e P u X ~ .j - -1

153

( 1 2 )

T h e c e r t a i n r e t u r n o n t h is r is k le s s s e c u r i t y i sR [ = 1 / p : - 1 , (13 )

w h e n t h e c u r r e n t s t a t e i s ( c , i ) .A s m e n t i o n e d e a r l i e r , t h e s t a t i s t i c s t h a t a r e p r o b a b l y m o s t r o b u s t t o t h e

m o d e l l i n g s p e c i f i c a t io n a re t h e m e a n s o v e r t i m e . L e t ~r ~ R n b e t h e v e c t o r o fs t a t i o n a r y p r o b a b i l i t i e s o n i. T h i s e x i st s b e c a u s e t h e c h a i n o n i h a s b e e na s s u m e d t o b e e r g o d i c . T h e v e c t o r ~r i s t h e s o l u t i o n t o t h e s y s t e m o f e q u a t i o n s

~ r = ~ r r r ,

w i t h

~ r i = l and ~ r = { ~ j , } .i - - 1

T h e e x p e c t e d r e t u r n s o n t h e e q u i t y a n d t h e r is k - f re e s e c u r i ty a r e , r e s p e c ti v e ly ,nR e= E ~riR: and R f= ~ ~'iR[. (14 )

i - 1 i - 1T i m e s a m p l e a v e r a g e s w i l l c o n v e r g e i n p r o b a b i l i t y t o t h e s e v a l u e s g i v e n t h ee r g o d i c i t y o f t h e M a r k o v c h a i n . T h e r is k p r e m i u m f o r e q u i t y is R e - R r, ap a r a m e t e r t h a t i s u s e d i n t h e te st.

4 . The r e s u l t sT h e p a r a m e t e r s d e f i n in g p r e f er e n c e s a r e a a n d f l w h i le t h e p a r a m e t e r s

d e f i n i n g t e c h n o l o g y a r e t h e e l e m e n t s o f [ ~ ij ] a n d [ h i] . O u r a p p r o a c h i s to


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1 5 4 R. Mehra and E.C. Prescott, The equity prem ium

a s s u m e t w o s t a t e s f o r t h e M a r k o v c h a i n a n d t o r e s t r i c t t h e p r o c e s s a s f o l l o w s :~x=1+~+6, h2=1+~-6,1 # 1 1 = 1 # 2 2 = 1 # ' 1 # 1 2 = 1 # 2 1 = (1 - 1#).

T h e p a r a m e t e r s g , 1#, a n d 6 n o w d e f i n e t h e t e c h n o l o g y . W e r e q u i r e 6 > 0 a n d0 < 1# < 1 . Th i s p a r t i c u l a r pa ra m e te r i z a t ion w a s se l e c t e d be c a use i t pe rm i t t e du s t o i n d e p e n d e n t l y v a r y t h e a v er a g e g r o w t h r a t e o f o u t p u t b y c h a n g i n g g , t h ev a r i a b i l i t y o f c o n s u m p t i o n b y a l t e r i n g 6 , a n d t h e s e r i a l c o r r e l a t i o n o f g r o w t hr a t e s b y a d j u s t i n g 1#.T h e p a r a m e t e r s w e r e s e l e c t e d s o t h a t t h e a v e r a g e g r o w t h r a t e o f p e r c a p i t ac o n s u m p t i o n , t h e s t a n d a r d d e v i a t io n o f t h e g r o w t h r a t e o f p e r c a p it a c o n s u m p -t i o n a n d t h e f i r s t - o rd e r s e ri al c o r r e l a ti o n o f t h i s g r o w t h r a t e , a l l w i t h r e s p e c t t ot h e m o d e l ' s s t a t i o n a r y d i s t r i b u t i o n , m a t c h e d t h e s a m p l e v a l u e s f o r t h e U . S .e c o n o m y b e t w e e n 1 8 8 9 -1 9 7 8 . T h e s a m p l e v a lu e s f o r t h e U .S . e c o n o m y w e r e0 . 01 8 , 0 .0 3 6 a n d - 0 . 1 4 , r e s p ec t iv e l y . T h e r e s u l ti n g p a r a m e t e r ' s v a l u e s w e r e= 0 .018, $ = 0 .036 an d 1# = 0 .43. G ive n these va lues , the n a tu re o f the tes t ist o s e a r c h f o r p a r a m e t e r s a a n d f l f o r w h i c h t h e m o d e l ' s a v e r a g e d r is k - f re e r a t ea n d e q u i t y r i sk p r e m i u m m a t c h t h o s e o b s e rv e d f o r th e U .S . e c o n o m y o v e r t h isn i n e t y - y e a r p e r io d .T h e p a r a m e t e r a , w h i c h m e a s u r e s p e o p l e s ' w i l l i n g n e s s t o s u b s t i t u t e c o n -s u m p t i o n b e t w e e n s u cc es si ve y e a r l y t i m e p e ri o d s is a n i m p o r t a n t o n e i n m a n yf i e l d s o f e c o n o m i c s . A r r o w ( 1 9 7 1 ) s u m m a r i z e s a n u m b e r o f s t u d i e s a n dc o n c l u d e s t h a t r e l a ti v e r is k a v e r s i o n w i t h re s p e c t t o w e a l t h is a l m o s t c o n s t a n t .H e f u r t h e r a r g u e s o n t h e o r e ti c al g r o u n d s t h a t a s h o u l d b e a p p r o x i m a t e l y o n e .F r i e n d a n d B l u m e (1 97 5) p r e s e n t ev i d en c e b a s e d u p o n t h e p o r tf o l i o h o l d i n g so f i n d i v i d u a l s t h a t a i s l ar g e r, w i t h t h e i r e s t i m a t e s b e i n g i n t h e r a n g e o f t w o .K y d l a n d a n d P r e s c o t t ( 1 9 8 2 ) , i n t h e i r s t u d y o f a g g r e g a t e f l u c t u a t i o n s , f o u n dt h a t t h e y n e e d e d a v a lu e b e t w e e n o n e a n d t w o t o m i m i c th e o b s e r v ed r e l a ti v ev a r i a b i l i t i e s o f c o n s u m p t i o n a n d i n v e s t m e n t . A l t u g ( 1 9 8 3 ) , u s i n g a c l o s e l yr e l a t e d m o d e l a n d f o r m a l e c o n o m e t r i c te c h n iq u e s , e s ti m a t e s th e p a r a m e t e r t ob e n e a r z e r o . K e h o e (1 98 4) , s t u d y i n g t h e r e sp o n s e o f s m a l l c o u n t r i e s b a l a n c e o ft r a d e t o t e r m s o f t r a d e s h o c k s , o b t a i n e d e s t i m a t e s n e a r o n e , t h e v a l u e p o s i t e db y A r r o w . H i l d r e t h a n d K n o w l e s (1 98 2) i n t h e ir s t u d y o f t h e b e h a v io r o ff a r m e r s a l s o o b t a i n e s t im a t e s b e t w e e n o n e a n d t w o . T o b i n a n d D o l d e (1 97 1),s t u d y i n g l if e c y c le sa v i n gs b e h a v i o r w i t h b o r r o w i n g c o n s t r a i n t s , u s e a v a l u e o f1 .5 to f i t t he obs e rve d l i f e c yc l e sa v ings pa t t e rns .A n y o f t h e a b o v e c i t e d s t u d i e s c a n b e c h a l l e n g e d o n a n u m b e r o f g r o u n d sb u t t o g e t h e r t h e y c o n s t it u t e a n a p r i o r i j u s t i f i c a ti o n f o r r e s t r i c ti n g t h e v a l u e o fo t t o b e a m a x i m u m o f t e n , a s w e d o i n t h i s s t u d y . T h i s i s a n i m p o r t a n tr e s t r i c ti o n , f o r w i t h l ar g e ot v i r t u a l l y a n y p a i r o f a v e r a g e e q u i t y a n d r i sk - f r e er e t u r n s c a n b e o b t a i n e d b y m a k i n g s m a l l c h a n g e s i n t h e p ro c e ss o n c o n s u m p -


11/17

R. M ehra and E.C. Prescott , T he equi ty prem ium 155A v e r a c 3 e

IR,sk P r e m i 8(percent}Re - R ~

Aclr~,ssL b le R e ~ i o n

0 I ~, 3 N ( p e r c e n t )A v e r a ~ 3 e R ~ s k F r e e R a t e

Fig. 4. Set of admissible average equity risk prem ia and real returns.

t i o n . 5 W i t h a l e ss t h a n t e n , w e f o u n d t h e r e s u l ts w e r e e s s e n t i a ll y t h e s a m e f o rv e r y d i f f e r e n t c o n s u m p t i o n p r o c e s se s , p r o v i d e d t h a t t h e m e a n a n d v a r i an c e s o fg r o w t h r a t e s e q u a l e d t h e h i s t o ri c a ll y o b s e r v e d v a lu e s. A n a d v a n t a g e o f o u ra p p r o a c h i s t h a t w e c a n e a s i l y t e s t t h e s e n s i t i v i t y o f o u r r e s u l t s t o s u c hd i s t r i b u t i o n a l a s s u m p t i o n s .

T h e a v e r a g e r e a l r e t u r n o n r e l a t i v e l y r is k le s s, s h o r t - t e r m s e c u r i t ie s o v e r th e1 8 8 9 - 1 9 7 8 p e r i o d w a s 0 .8 0 p e rc e n t . T h e s e s e c u ri ti e s d o n o t c o r r e s p o n d p e r -f e c t l y w i t h t h e r e a l b i ll , b u t i n s o f a r a s u n a n t i c i p a t e d i n f l a t io n i s n e g l i g ib l ea n d / o r u n c o r r e la t e d w i th t h e g r o w th r a te x t + 1 c o n d i t i o n a l u p o n i n f o r m a t i o na t t i m e t , t h e e x p e c t e d re a l r e t u r n f o r th e n o m i n a l b i ll w il l e q u a l R [ . L i t t e r m a n( 1 9 8 0 ), u s i n g v e c t o r a u t o r eg r e s si v e a n a ly s is , f o u n d t h a t t h e i n n o v a t i o n i n t h ei n f l a t io n r a t e i n t h e p o s t - w a r p e r i o d ( q u a r t e r l y d a t a ) h a s s t a n d a r d d e v i a t i o n o fo n l y o n e - h a l f o f o n e p e r c e n t a n d t h a t h i s in n o v a t i o n is n e a r ly o r t h o g o n a l t o th es u b s e q u e n t p a t h o f t h e r ea l G N P g r o w t h r at e. C o n s e q u e n t l y , t h e a v e ra g er e a l i z e d r e al r e t u r n o n a n o m i n a l l y d e n o t e d s h o r t - t e r m b il l sh o u l d b e c lo s e t ot h a t w h i c h w o u l d h a v e p r e v a i le d f o r a r e a l b il l i f s u ch a s e c u r i t y w e r e t r a d e d .T h e a v e r a g e r e a l r e t u r n o n t h e S t a n d a r d a n d P o o r ' s 5 0 0 C o m p o s i t e S t o c k

Sin a private com munication, Fischer Black using the M erton (1973) continuous tim e modelwith investment opportunities constructed an example with a curvature parameter (a) of 55. W ethank him for the example.


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156 R. M ehra and E.C. Prescott , The equi ty pre miu m

Index over the ninety years considered was 6.98 percent per annum. This leadsto an average equity premium of 6.18 percent (standard error 1.76 percent).Given the estimated process on consumption, fig. 4 depicts the set of values

of the average risk-free rate and equity risk premium which are both consistentwith the model and result in average real risk-free rates between zero and fourpercent. These are values that can be obtained by varying preference parame-ters a between zero and ten and fl between zero and one. The observed realreturn of 0.80 percent and equity premium of 6 percent is clearly inconsistentwith the predictions of the model. The largest premium obtainable with themodel is 0.35 percent, which is not close to the observed value.4.1. Ro bu stne ss o f results

One set of possible problems are associated with errors in measuring theinflation rate. Such errors do not affect the computed risk premium as theybias both the real risk-free rate and the equity rate by the same amount. Apotentially more serious problem is that these errors bias our estimates of thegrowth rate of consumption and the risk-free real rate. Therefore, only if thetests are insensitive to biases in measuring the inflation rate should the tests betaken seriously. A second measurement problem arises because of tax consider-ations. The theory is implicitly considering effective after-tax returns whichvary over income classes. In the earlier part of the period, tax rates were low.In the latter period, the low real rate and sizable equity risk premium hold forafter-tax returns for all income classes [see Fisher and Lofie (1978)].We also examined whether aggregation affects the results for the case thatthe growth rates were independent between periods, which they approximatelywere, given that the estimated 4, was near one-half. Varying the underlyingtime period from one one-hundredths of a year to two years had a negligibleeffect upon the admissible region. (See the appendix for an exact specificationof these experiments.) Consequently, the test appears robust to the use ofannum data in estimating the process on consumption.

In an attempt to reconcile the large discrepancy between theory and ob-servation, we tested the sensitivity of our results to model misspecification. Wefound that the conclusions are not at all sensitive to changes in the parameter#, which is the average growth rate of consumption, with decreases to 1.4percent or increases to 2.2 percent not reducing the discrepancy. The sensitivityto 6, the standard deviation of the consumption growth rate, is larger. Theaverage equity premium was roughly proportional to 6 squared. As thepersistence parameter 0 increased (qb = 0.5 corresponds to independence overtime), the premium decreased. Reducing 0 (introducing stronger negativeserial correlation in the consumption growth rate) had only small effects. Wealso modified the process on consumption by introducing additional states thatpermitted us to increase higher moments of the stationary distribution of the


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R . M e h r a a n d E . C : P r e sc o t t , T h e e q u i ( y p r e m i u m 157growth rate without varying the first or second moments. The maximal equitypremium increased by 0.04 to 0.39 only. These exercises lead us to theconclusion that the result of the test is not sensitive to the specification of theprocess generating consumption.That the results were not sensitive to increased persistence in the growthrate, that is to increases in ~, implies low frequency movements or non-stationarities in the growth rate do not increase the equity premium. Indeed,by assuming stationarity, w~ biased the test towards acceptance.

4 .2 . E f fec ts o f f i rm leoerageThe security priced in our model does not correspond to the common stocks

traded in the U.S. economy. In our model there is only one type of capital,while in an actual economy there is virtually a continuum of capital types withwidely varying risk characteristics. The stock of a typical firm traded in thestock market entitles its owner to the residual claim on output after all otherclaims including wages have been paid. The share of output accruing tostockholders is much more variable than that accruing to holders of otherclaims against the firm. Labor contracts, for instance, may incorporate aninsurance feature, as labor claims on output are in part fixed, having beennegotiated prior to the realization of output. Hence, a disproportionate part ofthe uncertainty in output is probably borne by equity owners.The firm in our model corresponds to one producing the entire output of theeconomy. Clearly, the riskiness of the stock of this firm is not the same as thatof the Standard and Poor's 500 Composite Stock Price Index. In an attempt tomatch the two securities we price and calculate the risk premium of a securitywhose dividend next period is actual output less a fraction of expected output.Let 0 be the fraction of expected date t + 1 output committed at date t by thefirm. Eq. (7) then becomes

p e ( c , i ) = [ ~ d p i j ( ~ k j C p e j ) - t - C ~ k j - - O C a . ( 1 5 )j - 1

As before, it is conjectured and verified that pC(c, i ) has the functional formwic. Substituting w ic for pC(c, i ) in (15) yields the set of linear equations

[ ]i = ~ j ~ l * i j ~ j a ~ k j l 4 ~ -~ - ~ k j - - 0 k - 1 ~ i k X k , ( 1 6 )for i = 1 .... n. This system was solved for the equilibrium w; and eqs. (10),(11), and (14) used to determine the average equity premium.


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15 8 R. Mehra and E .C. Prescott, The equity prem ium

As the corporate profit share of output is about ten percent, we set 0 = 0.9.Thus, ninety percent of expected output is committed and all the risk is borneby equity owners who receive ten percent of output on average. This increasedthe equity risk premium by less than one-tenth percent. This is the casebecause financial arrangements have no effect upon resource allocation and,therefore, the underlying Arrow-Debreu prices. Large fixed payment commit-merits on the part of the firm do not reverse the test's outcome.4.3. Introducing production

With our structure, the process on the endowment is exogenous and there isneither capital accumulation nor production. Modifying the technology toadmit these opportunities cannot overturn our conclusion, because expandingthe set of technologies in this way does not increase the set of joint equilibriumprocesses on consumption and asset prices [see Mehra (1984)]. As opposed tostandard testing techniques, the failure of the model hinges not on theacceptance/reject ion of a statistical hypothesis but on its inability to generateaverage returns even close to those observed. If we had been successful infinding an economy which passed our not very demanding test, as we expected,we planned to add capital accumulation and production to the model using avariant of Brook's (1979, 1982), Donaldson and Mehra's (1984) or Prescottand Mehra's (1980) general equilibrium stationary structures and to performadditional tests.5 . Con c lu s ion

The equity premium puzzle may not be why was the average equity return sohigh but rather why was the average risk-free rate so low. This conclusionfollows if one accepts the Friend and Blume (1975) finding that the curvatureparameter a significantly exceeds one. For a = 2, the model's average risk-freerate is at least 3.7 percent per year, which is considerably larger than thesample average 0.80 given the standard deviation of the sample average is only0.60. On the other hand, if a is near zero and individuals nearly risk-neutral,then one would wonder why the average return of equity was so high. This isnot the only example of some asset receiving a lower return than that impliedby Arrow-Debreu general equilibrium theory. Currency, for example, isdominated by Treasury bills with positive nominal yields yet sizable amountsof currency are held.

We doubt whether heterogeneity, per se, of the agents will alter the conclu-sion. Within the Debreu (1954) competitive framework, Constantinides (1982)has shown heterogeneous agent economies also impose the set of. restrictionstested here (as well as others). We doubt whether non-time-additivity separablepreferences will resolve the puzzle, for that would require consumptions near in


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R. M ehra and E.C. Prescott, The equity prem ium 159

t i m e t o b e p o o r e r s u b s ti t u te s t h a n c o n s u m p t i o n s a t w i d e l y s e p a r a t e d d a t e s.P e r h a p s i n t r o d u c i n g s o m e f e a tu r e s th a t m a k e c e r t ai n t y p e s o f i n te r t e m p o r a lt r a d e s a m o n g a g e n t s i n f e a s i b l e w i ll re s o l v e t h e p u z z l e . I n t h e a b s e n c e o f s u c hm a r k e t s , t h e r e c a n b e v a r i a b il i ty in i n d i v i d u a l c o n s u m p t i o n s , y e t li t tl e v a r ia b i li -t y i n a g g r e g a t e c o n s u m p t i o n . T h e f a c t t h a t c e r t a i n ty p e s o f c o n t r a c t s m a y b en o n - e n f o r c e a b l e is o n e r e a so n f o r t h e n o n - e x is t en c e o f m a r k e ts t h a t w o u l do t h e r w i s e a r i s e t o s h a r e r i s k . S i m i l a r l y , e n t e r i n g i n t o c o n t r a c t s w i t h a s y e tu n b o r n g e n e r a t io n s is n o t f ea si bl e. 6 S u ch n o n - A r r o w - D e b r e u c o m p e t i t iv ee q u i l i b r i u m m o d e l s m a y r a t i o n a l i z e t h e l a r g e e q u i t y r i s k p r e m i u m t h a t h a sc h a r a c t e r i z e d t h e b e h a v i o r o f t h e U . S . e c o n o m y o v e r t h e la s t n i n e t y y e a r s. T ot e s t s u c h t h e o r i e s i t w o u l d p r o b a b l y b e n e c e ss a ry t o h a v e c o n s u m p t i o n d a t a b yi n c o m e o r a g e g r o up s .

AppendixT h e p r o c e d u r e f o r d e t e r m i n i n g t h e a d m i s s i b l e r e g i o n d e p i c t e d i n f ig . 4 is. a s

f o l l o w s . F o r a g i v e n s e t o f p a r a m e t e r s # , 8 a n d ~ , e q s. ( 1 0 ) - ( 1 4 ) d e f i n e a na l g o r i t h m f o r c o m p u t i n g t h e v a l u e s o f R e, R r a n d R e - R f f o r a n y ( a , f l ) p a i rb e l o n g i n g t o t h e s e t

x = ( ( a , f l ) : 0 < a < 1 0 , 0 < fl < 1 , a n d t h ee x i s t e n c e c o n d i t i o n o f s e c t i o n 3 is s a t i s f i e d } .

L e t t i n g R f = h l ( o t , f l ) an d R e - R f f h 2 ( c t , f l ) , h : X - * R 2, t h e r a n g e o f h i st h e r e g i o n d e p i c t e d i n f i g . 4 . T h e f u n c t i o n h w a s e v a l u a t e d f o r a l l p o i n t s o f af i n e g r i d i n X t o d e t e r m i n e t h e a d m i s s i b l e r e g i o n .

T h e e x p e r i m e n t s t o d e t e r m i n e t h e s e n s it i v it y o f t h e r e su l ts t o t h e p e r i o dl e n g t h h a v e m o d e l t im e p e r io d s n = 2 , 1 , 1 / 2 , 1 / 4 , 1 / 8 , 1 / 1 6 , 1 / 6 4 a n d 1 / 1 2 8y e a r s . T h e v a l u e s o f t h e o t h e r p a r a m e t e r s a r e # = 0 . 0 1 8 / n , 8 = 0 . 0 3 6/ x /n " a n d

= 0 .5 . W i t h t h e se n u m b e r s t h e m e a n a n d s t a n d a r d d e v i a ti o n o f a n n u a lg r o w t h r a t e s a r e 0 . 0 1 8 a n d 0 .0 3 6 r e s p e c t i v e l y a s i n t h e s a m p l e i a er io d . T h i sf o l l o w s b e c a u s e ~ = 0 .5 i m p l ie s i n d e p e n d e n c e o f g r o w t h r a t e s o v e r p e ri o d s .T h e c h a n g e i n t h e a d m i s s ib l e re g i o n w e r e h u n d r e d t h s o f p e r c e n t a s n v a ri e d.

T h e e x p e r i m e n t s t o t e s t t h e s e n s i t i v i t y o f t h e r e s u l ts t o # c o n s i d e r ~ ffi 0 . 01 4 ,0 .016 , 0 .018 , 0 .020 a nd 0 .022 , ~ = 0 .43 a nd 8 = 0 .036 . As f o r the pe r iod l e ng th ,t h e g r o w t h f a t e ' s e f fe c ts u p o n t h e a d m i s si b l e r e g i o n a r e h u n d r e d t h s o f p e r ce n t .

T h e e x p e r i m e n t s t o d e t e r m i n e t h e s e n s i ti v i t y o f re s u l t s t o 6 s e t ~ = 0 . 4 3 , ~ =0 . 0 1 8 a n d 8 - - 0 . 2 1 , 0 .2 6 , 0 .3 1 , 0 .3 6 , 0 .4 1 , 0 . 4 6 a n d 0 .5 1 . T h e e q u i t y p r e m i u mv a r i e d a p p r o x i m a t e l y w i t h t h e s q u a r e o f 8 i n t h is r an g e .

6See Wallace (1980) for an expositionon the use of the overlappinggenerations mo del and theimp ortance o f lega l constraints in explaining rate o f return anomalies.


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16 0 IL Mehra and E.C. Prescott, The equity prem iumS i m i l a r l y , t o t e s t th e s e n s i t i v i ty o f th e r e s u l t s t o v a r i a t i o n s i n t h e p a r a m e t e r

~ , w e h e l d ~ f i x e d a t 0 . 03 6 a n d / z a t 0 .0 1 8 a n d v a r i e d ~ b e t w e e n 0 . 0 0 5 a n d 0 .9 5i n s t e p s o f 0 .0 5 . A s ~ i n c r e a s e d t h e a v e r a g e e q u i t y p r e m i u m d e c l i n e d .

T h e t e s t f o r t h e s e n si t i v i t y o f r e s u lt s t o h i g h e r m o v e m e n t s u s e s a n e c o n o m yw i t h a f o u r - s t a t e M a r k o v c h a i n w i t h t r a n s i t i o n p r o b a b i l i t y m a t r i x~ / 2 ~ / 2 1 - ~ / 2 1 - ~ / 2 ]~ / 2 ~ / 2 1 - ~ / 2 1 - ~ / 2 /

1 - ~ / 2 1 - ~ / 2 ~ / 2 ~ / 2 | "1 - ~ / 2 1 - ~ / 2 ~ / 2 ~ / 2 J

T h e v a l u e s o f t h e ?~ a r e h 1 = 1 + / ~ , h 2 = 1 + / ~ + 8 , ~ 3 = 1 + # , a n d X 4 = 1 + / ~- 8 . V a l u e s o f / ~ , 8 a n d ~ a r e 0 . 0 1 8 , 0 . 0 51 a n d 0 . 3 6 , r e s p e c t i v e l y . T h i s r e s u l t si n t h e m e a n , s t a n d a r d d e v i a t i o n a n d f i r s t - o r d e r s e ri a l c o r r e l a t i o n s o f c o n s u m p -t i o n g r o w t h r a t e s f o r t h e a r t i f i c i a l e c o n o m y e q u a l i n g t h e i r h i s t o r i c a l v a l u e s .W i t h t h is M a r k o v c h a i n , th e p r o b a b i l i t y o f a b o v e a v e r a g e c h a n g e s i s s m a l l e ra n d m a g n i t u d e o f c h a n g e s l a rg e r . T h i s h a s t h e e ff e ct o f i n c r e a s in g m o m e n t sh i g h e r t h a n t h e s e c o n d w i t h o u t a l t e r i n g th e f ir st o r s e c o n d m o m e n t s . T h i si n c r e a s e s t h e m a x i m u m a v e r a g e e q u i ty p r e m i u m f r o m 0 .3 5 p e r c e n t t o 0 .3 9p e r c e n t .

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1345-1370.LeRoy, S.F. and CJ. LaCivita, 1981, Risk-aversion and the dispersion of asset prices, Journal of"Business 54, 535-548.LeRoy, S.F. and R.D. Porter, 1981, The present-value relation: Tests based upon implied variancebounds, Econometrica 49, 555-574.Litterman, R.B. 1980, Bayesian procedure for forecasting with vector autoregressious, Workingpaper (M.IT, Cambridge, MA).Lucas, R.E., Jr., 1978, Asset prices in an exchange economy, Econometrica 46, 1429-1445.Lucas, R.E., Jr., 1981, Methods and problems in business cycle theory, Journal of Money, Credit,and Banking 12, Part 2. Reprinted in: R.E. Lucas, Jr., Studies in business-cycle heory (M1TPress, Cambridge, ]viA).Mehra, R., 1984, Recursive competitiveequilibrium: A parametric example, Economics Letters 16,273-278.Mehra, R. and E.C. Prescott, 1984, Asset prices with nonstationary consumption, Working paper(Graduate School of Business,Columbia University, New York).Merton, R.C., 1973, An intertemporal asset pricing model, Econometrica 41, 867-887.Prescott, E.C. and R. Mehra, 1980, Recursive competitive equilibrium: The case of homogeneoushouseholds, Econometrica 48, 1365-1379.Shiller, R3., 1981, Do stock prices move too much to be justified by subsequent changes individends?, American Economic Review 71,421-436.Tobin, J. and W. Dolde, 1971, Wealth, liquidity and consumption, in: Consumer spending andmonetary policy: The linkage (Federal Reserve Bank of Boston, Boston, MA) 99-146.Wallace, N., 1980, The overlapping generations model of fiat money, in: J.H. Kareken and 1'4.Wallace, eds., Models of monetary economies (Fedhral Reserve Bank of Minneapolis,Minneapolis, MN) 49-82.

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