Equity PrTHE EQUITY PREMIUM A Puzzleemium a Puzzle

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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 M

A P u z z l e *

R a j n i s h M E H R A

Columbia University, Ne w York, N Y 10027, USA

E d w a r d C . P R E S C O T T

Federal 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 n

H 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 8

t 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 n

p 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 n

a 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 s

c 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 u

s 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 s

c 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. Earl ier 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 niversi ty , NBER Conference on Intertem poral Puzzles in Macroeconomics, and theAm erica n Finan ce M eetings. W e wish to thank the workshop pa rt icipants, George Coustantinides,Eugene Fa m a, M erton Miller, and part icula 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 ofMinneapo l is .

0304-3923/85/$3.30©1985, Elsevier Science Publishers B.V. (North-Holland)



146 R. Mehra and E .C Prescott, The equitypremium

f 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 r g e a v e r a g e e q u i t y

p 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 e

e q u i l i b r i u m g r o w t h r a t e p ro c e ss 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 s

a r e s t a t i o n a r y . A t t e n t i o n i s r e s t r ic t e 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 i ty o f

s 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 r

t + 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 a r e c o n s t r u c t e d to 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 n

g 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 e

o 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 t h a t f o r s u c h

e c o n o m i e s , th 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 s

o 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 - t e r m d e b t , i n s h a r p c o n t r a s t t o t h e s i x

p e r c e n t p r e m i u m o b s e r v e d . O u r r e s u l t s a r e r o b u s t t o n o n - s t a t i o n a r i t i e s i n t h e

m 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 e

q 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 ss u e s, i n p a r t i c u l a r i s s u e s

s u c h a s t h e v o l a t i l i t y 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 is n o t a n

e 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 y

e 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 d

t 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 io 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 e

r 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 l iz e d in a p e r f e c t m a r k e t

f r a m e w o r k i s a s f o l l o w s : W i t h r e a 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 y

t 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 e

y 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 i c ie n t ly s m a l l t o y ie l d t h e

s 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 s o y i e l d r e a l r a t e s o f r e t u r n f a r i n e x c es s

o 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 k

a v e r s i o n e f f e c t i 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 w

r 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 e

c o n s u m p t i o n w i ll p r o b a b l y e x c e ed 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 l

u 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 l

i 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 r a g e .

T h i s p a p e r i s o r g a n i z e d a s f o 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 ic a l

e 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 sp e c if ie s t h e s e t o f

e 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 m

d 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 e

s 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 s

t h e p a p e r .

1Th ere are other interesting eatures of time series and procedures for testing them. The variance

bound 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. Th ey d id indicate that consumptionrisk was imp ortan t [see Grossm an and Shiller(1981) and Le Ro y and LaC avita 1981)].

2See Luc as (1980) for an articulationof this methodology.



K Meh ra and E.C. Prescott, The equitypremium

Table I

147

growth rate of ~ real return on aper capita r e a l relat ivelyisldess • real return on

consumption security % risk premium S&P 500Time Standard Standard Standard Standardperiods Mean deviation Mean deviation Mean deviation Mean deviation

1.83 3.57 0.80 5.67 6.18 16.67 6.98 16.541889-1978 (Std error (Std error (Std error (Std error

0.38) ffi 0.60) = 1.76) = 1.74)

1889-1898 2.30 4.90 5.80 3.23 1.78 11.57 7.58 10.02

1899-1908 2.55 5.31 2.62 2.59 5.08 16.86 7.71 17.21

1909-1918 0.44 3.07 - 1.63 9.02 1.49 9.18 - 0.14 12.81

1919-1928 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 27.90

1939-1948 2.19 2.52 - 5.82 4.05 8.89 14.23 3.07 14.67

1949-1958 1.48 1.00 -0.81 1.89 18.30 13.20 17.49 13.08

1959-1968 2.37 1.00 1.07 0.64 4.50 10.17 5.58 10.59

1969-1978 2.41 1.40 -0.72 2.06 0.75 11.64 0.03 13.11

2. Data

The data used in this study consists of five basic series for the period

1889- 1978. 3 The first four are ide ntical t o those used by Gros sma n and Shiller

(1981) i n their study. The series are individu all y described below:

( i) S e r i e s P : An nu al average Standard and Poor's Composi te Stock Price

Inde x divided by the Cons ump tio n Deflator, a plot of which appears in

Grossman and Shiller (1981, p. 225, fig. 1).

(ii) S e r i e s D : Real a nnua l dividends for the Stan dard and Poor's series.(iii) S e r i e s C : Kuznets-Kendr ik -USNIA per cap i ta rea l consumpt ion on

non -du ra ble s and services.

(iv) S e r i e s P C : Consumption deflator series, obtained by dividing real con-

sum pti on in 1972 dollars on non-d urabl es and services by the nomi nal

cons umpt ion on non-durables and services.

(v) S e r i e s RF: Nominal yield on relatively riskless short-term securities over

the 18 89-1978 period; the securities used were nine ty-d ay gover nment

Treasury 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).



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

- 1 1 -

- . ,~E

t 8 8 Q

/ : \m

I 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 0 0 , 1 8 8 9 - 1 9 7 8 ( p e r c e n t ) .

1 9 2 0 - 1 9 3 0 p e r i o d a n d s i x t y - d ay t o n i n e t y - d a y P r i m e C o m m e r c i a l P a p e r

p 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 re 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 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 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 d

o 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 (P t+ 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 n

t 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 i od . M o d e l p a r a m e t e r s w e r e

r 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 f

r 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 ly

r 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 o

b 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 d

a 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 e

R 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 (1 9 63 ) a n d I b b o t s o n a n d S i n g u ef ie l d ( 19 7 9) .



P~ Mehra and E.C. Prescott, Th e equity prem ium 1 4 9

l S

1 8 8 9] l I 1

1 9 ~ 7 1 9 2 S 1 9 4 2 1 9 0 9

F 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 1

1 8 8 9 1 9 9 7 1 9 2 5 1 9 4 2 1 . 9 1 ~ 8 I9 7 8

F 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 ) .



150 R. M ehra and E.C . Prescott, Th e equity prem ium

3 . T h e e c o n o m y , a s s e t p r i ce s 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 growth

rate of the endowment follows a Markov process. This is in contrast to theassumption in Lucas' model that the endowment leoel follows a Markov

process. Our assumption, which requires an extension of competitive equi-

librium theory, enables us to capture the non-stationarity in the consumption

series associated with the large increase in per capita consumption that

occurred in the 1889-1978 period.

The economy we consider was judiciously selected so that the joint process

governing the growth rates in aggregate per capita consumption and asset

prices would be stationary and easily determined. The economy has a single

representative '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 increasing

concave utility function. To insure that the equilibrium return process is

stationary, the utility function is further restricted to be of the constant relative

risk aversion class,

c 1-a - 1U(c,a)= 1 -a ' O<a<oo. (2)

The parameter a measures the curvature of the utility function. When e( is

equal to one, the utility function is defined to be the logarithmic function,

which is the limit of the above function as a approaches one.

We assume that there is one productive unit producing the perishable

consumption good and there is one equity share that is competitively traded.

Since only one productive unit is considered, the return on this share of equity

is also the return on the market. The firm's output is constrained to be less

than or equal to Yr It is the firm's dividend payment in the period t as well.

The growth rate in y, is subject to a Markov chain; that is ,

Yt+l ~- X t + l Y t ' ( 3 )



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 rowth r a te , and

Pr{ x t+ 1 = h i ; x , = h i} = ~ / j . (4 )

I t i s a l so as sum ed tha t the M arko v cha in i s e rgod ic. Th e h i a re a l l pos i t ive and

Yo > 0 . T he r an do m var iab le Y t i s obse rved a t the beg inn ing o f the pe r iod , a t

wh ich t ime d iv idend paymen ts a r e made . A l l s ecu r i t i e s a r e 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 i s s tab le ; tha t i s, li ra A m as m ~ co i s ze ro . In M ehra and P resco t t

(1984) i t i s shown tha t th i s i s neces sa ry and su f f i c i en t fo r expec ted u t i l i ty to

e x i s t 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 every pe r iod . T hey a l so de f ine and

es tab l i sh th e ex i s t ence o f a D eb reu (1954) comp et i t ive equ i l ib r ium wi th a p r i ce

s 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 t io n u n d e r t h is c o n d i ti o n .

N e x t w e f o r m u l a t e e x p r es s io n s f o r t h e e q u i li b ri u m t i m e t p r i c e o f t h e e q u i t y

share and the r i sk - f r ee b i l l . We fo l low the conven t ion o f p r i c ing s ecu r i t i e s

e x - d i v i d e n d o r e x - i nt e re s t p a y m e n t s a t t i m e t , i n t e r m s o f t h e t im e t c o n s u m p -

t i o n g o o d . F o r a n y s e c u r it y w i t h p r o c e s s { d , } o n p a y m e n t s , i ts p r i c e in p e r i o d

t 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 e

s y s t e m h a s a d o t p r o d u c t r e p r e s e n t a t i o n .

T h e d i v i d e n d p a y m e n t p r o c e s s f o r t h e e q u i t y s h a re i n t hi s e c o n o m y i s { 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 t h 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 suf f ic ient re la t ive to the ent i re h is tory of shocks up

to , and inc lud ing , t ime t fo r p red ic t ing f l ae sub sequ en t evo lu t ion o f the

e c o n o m y . T h e y t h u s c o n s t i t u te l e g i ti m a t e s t a te v a r i a b le s f o r t h e m o d e l . S i n ce

Y s = Y t " x t + t . . . . . x s , t h e p r i c e o f t h e e q u i t y s ec u r i ty is h o m o g e n e o u s o f d e g r e e

o 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 e

e q u i l i b r i u m v a l u e s o f t h e e c o n o m i e s b e in g s t u d i e d a r e t i m e i n v a r i a n t f t m e t io n s

o f t h e s t a t e ( x t , Y t ) , t h e s u b s c r i p t t c a n b e d r o p p e d . T h i s i s a c c o m p l i s h e d b y

r e d e f i n i n g t h e s t a t e t o b e t h e p a i r ( c , i ) , i f y t = c a n d x t = h ~ . W i t h t h i s



1 5 2 K M ehra and E.C. Prescot t, The equi ty pre m 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 e e o n e i n c , w e

r 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 t

g 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 e

p 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 e

e 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 s

s 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 e r ta i n ty .



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 s

R [ = 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 f

s 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 s t 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 n

a 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 - 1

T 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 e

e 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, a

p 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 s

T 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



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 re 6 > 0 a n d

0 < 1# < 1 . T h i s p a r t i cu l a r p a r am e te r i za t io n was se l ec ted because i t pe r m i t t ed

u 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 ge g ro 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 e

v 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 h

r 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 a

c 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 th e g r o w t h r a te o f p e r c a p i ta c o n s u m p -t ion an d the f i r s t - o r de r se r ia l co r r e l a t i on o f t h i s g r ow th r a t e , a l l w i th r e spec t t o

t 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 al u es f o r t h e U . S. e c o n o m y w e re

0 . 01 8 , 0 .0 3 6 a n d - 0 . 1 4 , r e sp e c ti v e ly . T h e r e s u l t i 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 a nd 1# = 0 .43. G ive n the se va lues , the na tu re o f the tes t i s

t 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 ra g e d r is k - fr e e r a t e

a n d e q u i t y r is k 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 th i s

n 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 y

f 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 d

c o n c l u d e s t h a t r e la t iv e ri s k a v e r s io n w i t h r e s p e ct 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 re t ic a l 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 e v i d e n c e b a s e d u p o n t h e p o r t f o l i o h o l d i n g s

o 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 d

t 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 rv e d r e la t iv e

v 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 y

r e l a te 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 , es t im a t e s t h e p a r a m e t e r t o

b 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 th e r e s p o n s e o f s m a l l c o u n t r ie s b a l a n c e o f

t 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 d

b 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 i r s t u d y o f th e b e h a v i o r o f

f a r m e r s a l s o o b t a i n e s t i m 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 9 7 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 f

1 .5 t o f i t t he obs e r ved l i fe cyc l e sav ings pa t t e r ns .

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 s

b u t t o g e t h e r t h e y c o n s ti t u te a n a p r i o r i j u s t i f ic a t i o n f o r r e s t r ic t i n g t h e v a l u e o f

o 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 t

r e s t r ic t i o n , f o r w i t h l a rg 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 re e

r 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 -



R. M ehra and E.C. Prescott , T he equi ty pre m ium 155

A v e r a c 3 e

IR,sk P r e m i 8(percent}

Re - R ~

Acl r~ ,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 r

v 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 f

g 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 r

a 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 h

d 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 i es o v e r t h e

1 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 e

a 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 , th e e x p e c t e d r ea l r e t u r n f o r t h e n o m i n a l b i ll w i ll 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 e

i 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 f

o 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 e

s 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 e

r 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 i ll s h o u l d b e c l o s e t o

t 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.



156 R. M ehra an d E.C. Prescott , The equi ty pre m ium

Index over the ninety years considered was 6.98 percent per annum. This leads

to 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 four

percent. 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 real

return of 0.80 percent and equity premium of 6 percent is clearly inconsistent

with the predictions of the model. The largest premium obtainable with the

model 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 the

inflation rate. Such errors do not affect the computed risk premium as they

bias both the real risk-free rate and the equity rate by the same amount. A

potentially more serious problem is that these errors bias our estimates of the

growth rate of consumption and the risk-free real rate. Therefore, only if the

tests are insensitive to biases in measuring the inflation rate should the tests be

taken seriously. A second measurement problem arises because of tax consider-

ations. The theory is implicitly considering effective after-tax returns which

vary 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 for

after-tax returns for all income classes [see Fisher and Lofie (1978)].

We also examined whether aggregation affects the results for the case that

the growth rates were independent between periods, which they approximately

were, given that the estimated 4, was near one-half. Varying the underlying

time period from one one-hundredths of a year to two years had a negligible

effect upon the admissible region. (See the appendix for an exact specification

of these experiments.) Consequently, the test appears robust to the use of

annum 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. We

found that the conclusions are not at all sensitive to changes in the parameter

#, which is the average growth rate of consumption, with decreases to 1.4

percent or increases to 2.2 percent not reducing the discrepancy. The sensitivity

to 6, the standard deviation of the consumption growth rate, is larger. The

average equity premium was roughly proportional to 6 squared. As the

persistence parameter 0 increased (qb = 0.5 corresponds to independence over

time), the premium decreased. Reducing 0 (introducing stronger negative

serial correlation in the consumption growth rate) had only small effects. We

also modified the process on consumption by introducing additional states that

permitted us to increase higher moments of the stationary distribution of the



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 157

growth rate without varying the first or second moments. The maximal equity

premium increased by 0.04 to 0.39 only. These exercises lead us to the

conclusion tha t the result of the test is not sensitive to the specification of the

process generating consumption.That the results were not sensitive to increased persistence in the growth

rate, 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 leoerage

The 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 with

widely varying risk characteristics. The stock of a typical firm traded in the

stock market entitles its owner to the residual claim on output after all other

claims including wages have been paid. The share of output accruing to

stockholders is much more variable than that accruing to holders of other

claims against the firm. Labor contracts, for instance, may incorporate an

insurance feature, as labor claims on output are in part fixed, having been

negotiated prior to the realization of output. Hence, a disproportionate part of

the uncertainty in output is probably borne by equity owners.

The firm in our model corresponds to one producing the entire output of the

economy. Clearly, the riskiness of the stock of this firm is not the same as that

of the Standard and Poor's 500 Composite Stock Price Index. In an attempt to

match the two securities we price and calculate the risk premium of a security

whose 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 the

firm. 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 form

wic. 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.



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 borne

by equity owners who receive ten percent of output on average. This increased

the 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 is

neither capital accumulation nor production. Modifying the technology to

admit these opportunities cannot overturn our conclusion, because expanding

the set of technologies in this way does not increase the set of joint equilibrium

processes on consumption and asset prices [see Mehra (1984)]. As opposed to

standard testing techniques, the failure of the model hinges not on the

acceptance/rejection of a statistical hypothesis but on its inability to generate

average returns even close to those observed. If we had been successful in

finding an economy which passed our not very demanding test, as we expected,

we planned to add capital accumulation and production to the model using a

variant of Brook's (1979, 1982), Donaldson and Mehra's (1984) or Prescott

and Mehra's (1980) general equilibrium stationary structures and to perform

additional tests.

5 . Conclusion

The equity premium puzzle may not be why was the average equity return so

high but rather why was the average risk-free rate so low. This conclusion

follows if one accepts the Friend and Blume (1975) finding that the curvature

parameter a significantly exceeds one. For a = 2, the model's average risk-free

rate is at least 3.7 percent per year, which is considerably larger than the

sample average 0.80 given the standard deviation of the sample average is only

0.60. 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 is

not the only example of some asset receiving a lower return than that implied

by Arrow-Debreu general equilibrium theory. Currency, for example, is

dominated by Treasury bills with positive nominal yields yet sizable amounts

of 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. restrictions

tested here (as well as others). We doubt whether non-time-additivity separable

preferences will resolve the puzzle, for that would require consumptions near in



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 t u re s t h 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 l

t 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 h

m 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 e

n 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 d

o 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 t

u 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 e

e 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 s

c 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 o

t 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 y

i n c o m e o r a g e g r o up s .

Appendix

T 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 re 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 n

a 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 r

b 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 e

e 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 s

t 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 a

f i n e g r i d in X t o d e t e r m i n e th 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 si t iv i ty o f t h e r e s u lt s t o t h e p e r i o d

l 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 8

y 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 l

g 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 s

f 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 r io 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 . 0 1 6 , 0 .0 1 8 , 0 . 0 2 0 a n d 0 . 0 2 2 , ~ = 0 . 4 3 a n d 8 = 0 .0 3 6 . A s f o r t h e p e r i o d l e n g t h ,

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 . 43 , ~ =

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 m

v 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 m odel and theimp ortance of legal constraints in explaining rate o f return anomalies.



16 0 IL Mehra and E.C. Prescott, The equity prem ium

S 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 5

i n s t e p s o f 0 .0 5 . A s ~ i n c r e a s ed 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 en s i ti v i t y o f r e s u lt s to h ig 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 i s M a r k o v c h a in , t h 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 r

a 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 c t o f i n c r e a s in g m o m e n t s

h 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 te r in g t h e f ir st o r s e c o n d m o m e n t s . T h i s

i 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 9

p e r c e n t .

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