Full text of Working Papers (Federal Reserve Bank of Chicago) : The Equity Premium Puzzle and the Risk-Free Rate Puzzle at Long Horizons, Working Paper 1996-4

The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.

Working Paper Series



T h e E q u ity P r e m iu m P u z z le a n d t h e
R i s k - F r e e R a te P u z z le a t L o n g H o r iz o n s
Kent Daniel and David Marshall

l i b r a r y

F E B 219960
FE E A R SE V
DR L E R E
bank O C IC G
F H AO

Working Papers Series
Issues in Financial Regulation
Research Department
Federal Reserve Bank of Chicago
January 1996 (WP-96-4)

FEDERAL R ESER V E BANK
O F CHICAGO

T h e E q u ity P r e m iu m P u z z le a n d t h e R is k -F re e R a te P u z z le a t
L o n g H o r iz o n s t
Kent Daniel* and David Marshall**

A bstract
The failure of consumption based asset pricing models to match the stochastic properties
of the equity premium and the risk-free rate has been attributed by some authors to frictions,
transaction costs or durability. However, such frictions would primarily affect the higher fre­
quency data components: consumption-based pricing models that concentrate on long-horizon
returns should be more successful.
We consider three consumption-based models of the asset-pricing kernel: time-separable
utility, and the models of Abel (1990) and Constantinides (1990). We estimate a vector ARCH
model that includes the pricing kernel and the equity return, and use the fitted model to
assess the model’s implications for the equity premium and for the risk-free rate. We find
that time-separable preferences fail at all horizons, and none of the models perform well at
the quarterly horizon. When consumption is measured as nondurables plus services, the Abel
and the Constantinides models show modest improvement at the one- and two-year horizon.
However, when consumption is measured either as expenditures on nondurables or as total
consumption purchases, versions the Abel and the Constantinides match the mean and the
variance of the observed equity premium at the two-year horizon, capture a good deal of
the time-variation of the equity premium in post-war data, and have more success matching
the first and second moments of the observed risk-free rate. A major unresolved issue is to
understand why the measured consumption services series performs so poorly in these models.

t T h e views expressed in th is p ap er are strictly those of the au th o r. T hey do not necessarily represent
th e position of th e Federal Reserve B ank of Chicago, or th e Federal Reserve System . We th a n k Steve
C ecchetti, L arry C h ristian o , Jo h n C ochrane, George C onstantinides, L arry E pstein and Lars H ansen
for helpful discussions, and we acknowledge G lenn McAfee for su p erlative research assistance. Daniel
gratefully acknow ledges research su p p o rt from the the C enter for R esearch in Security Prices (C R SP)
a t th e U niversity of Chicago. Som e results in this paper were previously circulated in a working paper
en titled , ” C onsu m p tio n -B ased M odeling of Long-Horizon R etu rn s.”
" ra d u a te School of Business, U niversity of Chicago
‘G
"‘"'Federal Reserve B ank of C hicago




1

In tro d u c tio n

Most research on consumption-based asset pricing focuses on short-horizon returns. The start­
ing point is the familiar intertemporal Euler equation
1— Et

(LI)

[ m t + Tr t + T]

where r[+r denotes the gross real cumulative equity return from date t to date

t + r

and 7n[+r

denotes the intertemporal marginal rate of substitution (IMRS) between wealth at date t and
wealth at date

t + r.

A model of preferences is then posited that delivers

mt
r

as a function of

aggregate consumption. Implications of equation (1.1) are tested for return horizons r equal
to one month or one quarter. Typically, the tests reject the model, often decisively.
Of particular interest are the implications of (1.1) for the equity premium and the risk-free
rate. Let

rff

denote the gross real risk-free return between dates

t

and

t + r.

Equation (1.1)

implies the following characterizations of the equity premium and the risk-free rate:
E t r T+ T
t

r ft

=

-co vt

r ft

r ft

The

e q u ity p r e m i u m p u z z l e ,

m t + T.
T

(1.2)
(1.3)

t+ T

as defined by Mehra and Prescott (1985), is the claim that the

mean of — Tj* f 7 rJi is much bigger than —
E
models of

E tm

(m[+T,r t+T)

[covt

(m[+T, r[+T)], for plausible consumption-based

In the context of equation (1.2) we can also define the

p r e d i c t a b i l i t y p u z z le

as the claim that, for plausible consumption-based models of m[+r, there is insufficient timevariation in cout (m[+r, r[+T) to explain the observed variability of
r is k -fr e e r a te p u z z l e

—i-.1 Finally, the

is the claim that, for consumption-based models of ra[ that come within

striking distance of resolving the equity premium puzzle, both the mean2 and the variance3
!T h e p red ictab ility of excess re tu rn s to equities and to o th er financial assets is discussed in F am a
and French (1988), B ekaert and H odrick (1992), C utler, P o terba, and Sum m ers (1991), and Froot
(1990), am ong o thers. T hese observations are anom alous according to the trad itio n a l random -w alk
ch aracterizatio n of m ark et efficiency. T hey could be explained if, in equilibrium , risk prem ium s required
by investors vary th ro u g h tim e. E q u atio n (1.2) is a precise ch aracterizatio n of the needed variation in
risk prem ium s, in te rm s of th e tim e-series properties of m [ .
2See Weil (1989) and C ochrane and H ansen (1992).
3See th e discussion in C ochrane and Hansen (1992, p.137).




1

C o r r ( R ( t ) , N D -t-S

C o n s u m p tio n

G r o w th (t-la g )), Q u a r te r ly

H o r iz o n

This plot presents the correlation between quarterly real VW index returns at t and the quarterly
growth rate of real non-durable and services consumption at (t —la g ) for lag = -8 through + 8 quarters.
The vertical lines in the center of the graph indicate the contemporaneous correlation.
Figure 1 . 1 : Correlation between Real Q uarterly VW Index R eturns at
Q uarterly Non-Durable and Services Consumption Grow th

t

and Real

of -gr—— are too high to match the corresponding moments of r / tT. Cecchetti, Lam and
7
Mark (1993) estimate a representative-agent, time-separable model where consumption and
dividends are governed by a bivariate model Markov switching model, and find that while they
can match the first moments of the equity and risk-free return data, they cannot match both
the first and second moments.
The equity premium puzzle and the predictability puzzle are both reflections of a more
fundamental problem: the low correlation between consumption-growth and equity returns at
short horizons. Consider Figure 1 .1 , which displays correlations between real quarterly stock
returns quarterly and quarterly consumption growth at various leads and lags. (In this figure,
consumption is measured as purchases of nondurables plus services.) The contemporaneous
correlation between quarterly returns and quarterly consumption growth is small (less than
0.15), and the largest correlation at any lead/lag (when returns lead consumption growth by
two quarters) is less than 0 .2 . Cochrane and Hansen (1992) call this low correlation between
the return on market proxies and consumption growth the “correlation puzzle.” A num­
ber of factors have been proposed to account for the low correlations between stock returns
and aggregate consumption growth at short-horizons, including uninsurable cross-sectional




2

C o r r ( R ( t) , N D + S C o n s u m p t i o n G r o w th ( t - la g ) ) , O n e - Y e a r H o r iz o n
0 .4 - 0
0 .3 2
0 .2 4 -

0.1 6
O.OS
0.00
- 0 .0 8

-0.1 6
- 0 .2 4
- 0 .3 2
la g = r -8 t o 8 Q u a r t e r s

T h is p lo t presents the correlation betw een real cum ulative one-year V W index retu rn s a t t an d the
one-year grow th ra te o f real n o n -d u rab le and services consum ption a t (t — lag) for lag = -8 th ro u g h + 8
q u arters. T h e vertical lines in th e center of th e grap h indicate the contem poraneous correlation

Figure 1.2: Correlation between Real One-Year VW Index R eturns at
One-Year Non-Durable and Services Consum ption Growth

t

and Real

heterogeneity,4 fixed costs of adjusting consumption,5 costs of portfolio adjustment,6 and even
small deviations from perfect rationality.7
While these factors could substantially affect the co-movements of asset returns and aggre­
gate consumption at high frequencies, they should be less disruptive to the theory at longer
horizons. Figure

1 .2

suggests that there may be merit in this argument. Figure

1 .2

is analogous

to Figure 1 . 1 , except that the correlations in Figure 1.2 are between cumulative stock returns
over a o n e

year

horizon and

val is one quarter). Figure

one year
1.2

consumption growth (i.e., ct+4 /ct, where the timing inter­

suggests that Cochrane and Hansen’s (1992) correlation puzzle

is less pronounced for one-year cumulative returns. While the contemporaneous correlation
between consumption growth and returns at the one-year horizon is only slightly higher than
that shown in Figure

1.1

for the one-quarter horizon, the maximal correlation over all leads

and lags is much higher. In particular, the correlation between one-year consumption growth
and one-year returns lagged by two quarters is 0.35, almost twice as high as the maximal
4 See, e.g., C o n stan tin id es an d Duffie (1992).

5G rossm an and L aroque (1990), M arshall (1994), M arshall and P arekh (1994).
6L u ttm e r (1995), He an d M odest (1995).
7C ochrane (1989).




3

correlation found between quarterly returns and quarterly consumption growth.8
These results suggest that consumption-based models of the equity premium may have
more success if they focus on longer horizon returns. Specifically, if the higher
co rre la tio n s

displayed in Figure

1 .2

imply higher

c o n d itio n a l c o v a ria n c e s

u n c o n d itio n a l

between returns and

consumption-growth, then the equation (1 .2 ) should provide a better fit to observed data as
the horizon increases.9 However, it is unclear how the longer horizon horizon will affect the
risk-free rate. In fact, Cochrane and Hansen (1992) find that lengthening the horizon actually
exacerbates the risk-free rate puzzle. 10
In this paper, we explore these questions directly. For several models of ra[, we estimate
a model of the vector process (rt+T,m[+T) that allows for time-varying conditional second
r
moments. We use this model to generate estimates of the conditional moments £*tr[+r, E t m t +T,
T
and c o v t (ra[+r, r[+T) and we then use these estimates to evaluate ( 1 .2 ) and ( 1 .3 ) for investment
,
horizons r ranging from three months through three years.
Most empirical studies of consumption-based pricing measure consumption as purchases
of nondurables plus services, as measured by the Bureau of Economic Analysis. We see two
potential problems with this measure of consumption. First, the BEA’s data series on con­
sumer services includes the imputed rental value of owner-occupied housing. The quantity of
these housing services consumed is roughly proportional to the stock of housing. Given the
substantial transaction costs in the residential housing market, it is unlikely that changes in
investment opportunities induce agents to vary their consumption of housing services to the
extent predicted by the consumption-based paradigm. 11 Second, it is not clear that the omis­
sion of consumer durables is appropriate when studying longer-horizon returns. If the durable
good depreciates quickly enough, the distinction between durables purchases and the durables
stock becomes less important as the horizon lengthens. For example, if the half-life of the
durable good were about the same as the decision interval, fluctuations in durables purchases
8T h e frequency-dom ain analysis in D aniel and M arshall (1995) delivers a sim ilar result.
9Som e form of tim e-n o n sep arab ility m u st be incorporated to account for th e tw o -q u arter lag in th e
m ax im al correlation: if u tility is tim e-n o n sep arable, lagged co nsum ption and co n d itional m o m en ts of
fu tu re co n su m p tio n will en ter in to th e contem poraneous m f+T process.
10 However, C ecchetti, L am and M ark (1994) use annual equity and bond retu rn s from 1890 to 1987,
and find th a t, ta k in g account of sam p lin g variability, volatility bounds are satisfied.
11 T h is insight is m odeled form ally by G rossm an and L aroque (1990).




4

would capture a good deal of the variability in the durables stock. To omit durables in a
such a model may potentially represent as big a specification error as the inclusion of durables
purchases.
These considerations suggest that our results may be sensitive to the way consumption
is measured. We start by using the conventional measure of consumption as purchases of
nondurables plus services. We then perform our analysis using only nondurables purchases as
our measure of consumption. Finally, as a somewhat crude check on the importance of durables,
we also replicate our results with consumption measured by total consumption expenditures.
According to our empirical results, none of the consumption-based models fit the equity
premium or the risk-free rate at the quarterly horizon, regardless of the way consumption
is measured. When consumption is measured by nondurables plus services, there is a some
improvement in the ability of the Abel (1990) and Constantinides (1990) models to match
moments of the equity premium when the horizon is lengthened to one or two years, but the
models still fail to capture the time-series properties of the observed equity premium series. In
contrast, if we discard the data on consumer services, and measure consumption as purchases of
nondurables, both the Abel (1990) and Constantinides (1990) models perform remarkably well
at the two-year horizon. In particular, versions of these models replicate both the mean and
the standard deviation of the observed equity premium, and the theoretical equity premium
series generated by these models shows some ability to track the observed Equity premium
through time. We obtain similar results when consumption is measured by total consumption
purchases (including purchases of durables). In our tests of the risk-free rate equation (1.3), we
find that the Abel (1990) and Constantinides (1990) models provide a better fit to the mean
of the risk-free rate as the horizon is lengthened, but only if we measure consumption either
as nondurables or as total consumption expenditures. Interestingly, the excessive variability of
the risk-free rate implied by these models at the quarterly horizon completely disappears when
the horizon is set between two and three years. We conclude that consumption-based models
can match important features of the observed equity premium and risk-free rate processes at
longer horizons, provided that data on consumer services is excluded. The critical question to
be resolved is why the BEA’s measure of consumer services performs so poorly as an input




5

into these models.

The remainder of the paper is organized as follows: Section

2

describes the models of

preferences we use for the remainder of the paper. Section 3 describes the time-series model
we use to investigate ( 1 .2 ) and (1.3). Section 4 presents our empirical results, and section

5

summarizes.
2

M o d e ls o f P r e f e r e n c e s

2.1

T im e S ep arab le Pow er U tility

The most widely-studied (and widely-rejected) preference specification in the consumptionbased pricing literature is time-separable power utility. In this specification, agents solve the
following maximization problem:
1 — 'Y

.

max

0

u

7
j=0
subject to the usual budget constraint. The r-period IMRS is:

(2.1)

(2.2)
Cochrane and Hansen (1992) find that the performance of time-separable utility actually
terio ra tes

de­

as the horizon lengthens. The problem is that aggregate consumption is a (stochas­

tically) growing series. In the time-separable model, agents seek to transfer some of the high
future consumption to the present by borrowing. A counterfactually high risk-free rate is
needed to discourage this borrowing. (Recall that net borrowing must equal zero in equilib­
rium.) In principle, this effect could be countered by a strong precautionary motive for saving:
agents may wish to insure against the possibility of consumption downturns. However, the
probability of a consumption downturn gets smaller as the horizon lengthens: Cochrane and
Hansen (1992) note that there is no five-year period in post-war US data over which aggregate
consumption declines. As a result, the time-separable model predicts a lower precautionary
demand for savings, and a higher equilibrium risk-free rate, as the horizon lengthens.
What is needed, then, is a reason why the precautionary motive for saving remains strong
at longer horizons. One possible reason is that agent’s within-period utility-of-consumption




6

changes through time. In particular, suppose agents seek protection, not against an absolute
decline in consumption, but against a decline in consumption

r e la tiv e to s o m e r e f e r e n c e p o i n t ,

where the reference point itself grows at the same rate that consumption grows. In such a
model, the precautionary motive for saving would not become attenuated as the horizon grows.
Preference specifications with this property include Constantinides’ (1990) habit-formation
preferences and Abel’s (1990) “catching-up-with-the-Joneses” preferences. In the following
section, we formalize the Abel (1990) and Constantinides (1990) preference specifications.
2.2
Let

A b e l’s (1990) ” C a tc h in g -U p -W ith -th e -Jo n e se s” P references
ct

denote the per-capita consumption at date

t.

The agent solves

i ( ct+ j ~ h t + j )1'
U = E t J" (3j

max
o

(2.3)

1- 7

U

subject to the usual budget constraint, where
ht =

»> 0 0< 5< 1
?
,
.

(2.4)

i= 1

The interpretation is that agents compare their consumption to the consumption of their
neighbors (the ’’Joneses”) in the recent past. In the formal model, the neighbors’ consumption
is represented by ct, and agents have a subsistence point equal to
of the per-capita consumption levels over the past

m

7]

times a weighted average

periods. Notice that agents treat

ht

as

exogenous: the marginal utility of a fixed level of consumption inherits the upward trend in
but agents cannot alter the

ht

process by their own actions. (Of course, in equilibrium

ct

=

ct .)

With Abel preferences,
m t+r

Let the value function

V (W ,h )

_

^t+r)

be defined as the maximum value of the objective function

that can be attained given initial wealth

W

coefficient of relative risk aversion (denoted




(2.5)

(ct - ht y

and an initial subsistence point
R R A t)

RRAt = -W

We define the

by

V w w (W ,h )
v w

7

h.

(W, h )

(2 .6 )

With the time-separable preferences described in section 2 . 1 , R R A t =

, for all

7

t.

With Abel

preferences,
RRAt = 7

—
ct —ht

(2.7)

so the coefficient of relative risk aversion is time-varying, and everywhere exceeds 7 . The model
parameters we use are
2.3

7/ =

0.8,

5 =

0.7,

n = 8 , fi = 1 .

We vary

7

between

2

and 14.

C o n sta n tin id es (1990) H ab it-F o rm a tio n P referen ces

Constantinides (1990) models agents as maximizing an objective function of the same form as
(2 .3 ) with the following alternative specification for
h t=

s ~~ j S

ht:

1 £ $ict- *’ v > °’ 0 < s < l
■
*=1

(2-8)

The difference between (2.4) and (2.8) is that in (2.8) the stochastic subsistence point
a function of the agent’s own consumption

ht

is

rather than the per-capita consumption. The

c t)

marginal rate of substitution is now
m T+
t

,TM U t+T
T

where the marginal utility of consumption
m v

,

s

(2.9)

MUt
MUt

(c, - h , r -

is defined by
±

m

11
=

e

,

[c+, - < ,+j) r
>

(2 .10)

Terms involving conditional expectations appear in equation (2.10) because agents consider
the effect of their current consumption on future values of h t . These conditional expectations
must be computed when we construct m[. We do this as follows. First, define the variable

Dt

by:
A =
The variable

Dt

(c,.„

-

h , . mr

'

1

1

behaves as a stationary stochastic process. Equations (2.9) and (2.11) imply

that, in the Constantinides model,




m

T (^-t+r
t+

^C+r) ^t+T^it+T+m
(ct —h t) E t D t+m
8

(2.12)

Since

Dt

is stationary, we can fit an autoregressive time-series model for this variable: we use

the fitted values as our estimate of
testing

lags against

n

n —

E t D t+ m .

For most models, the likelihood ratio statistics

1 lags in the autoregression for

four lags. We estimate a fourth-order autoregression in
m

Dt

D t,

(for

n

between

1

and 5) favor

and project the fitted regression

periods into the future. We estimated {m T} only for values of 7 that do not imply negative
t

marginal utilities (as defined by equation (2.10)) for any observations. 123 As with the Abel
1
model, we set 7/ = 0.8,

S

= 0.7,

n

= 8 , (3 = 1.

We consider Constantinides preferences separately from Abel preferences for two reasons.
First, many empirical applications of time-nonseparable preferences use the Abel model, rather
than implementing true habit-formation, because the Abel model is much easier to solve: there
is no need to compute the conditional expectation terms in (2.10).13 It is of interest, therefore,
to see whether the Abel model does function as a good empirical proxy for the less-tractable
Constantinides model. A second reason is that habit formation preferences do not accentuate
risk aversion in the way that Abel’s preferences do. 14 In response to a wealth shock at date
£, the agent with habit-formation preferences adjusts her state-contingent plans for future
consumption so as to optimally adjust

, i = 1,..., m. This attenuates the impact of a given

wealth shock on the objective function, as compared to the Abel specification. In particular,
higher

77

does not increase

RRAt

as much as in (2.7). While

RRAt

cannot be computed

analytically for our model of habit-formation, Constantinides (1990) and Constantinides and
Ferson (1991) obtain closed-form solutions for

RRAt

in the context of a simpler model. They

show that, for preference parameters similar to ours, the mean coefficient of relative risk
aversion is not too far above
3

7 . 15

A V e c to r A R C H M o d e l o f C o n d itio n a l C o v a ria n c e s

12W hen co n su m p tio n is m easured as nondurables-plus-services, th e m ax im u m usable value of 7 is 12.
W ith n o n d u rab le co n su m p tio n and to ta l consum ption expenditures, th e m axim um value of 7 is 9 and
1 1 , respectively.
13For exam ple, C am p b ell an d C ochrane (1995) is entitled, ” By Force of H a b it” . However, these
au th o rs m odel the h a b it stock as a function of the p er-cap ita consum ption process, so the m odel is
actu ally a v arian t of the A bel m odel.
14T h is p o in t is extensively discussed by C o n stantinides (1990), and Ferson and C o n stantinides (1991).
We th a n k L arry C h ristia n o for p o in tin g this o u t to us.
15 B oldrin, C h ristian o , and Fisher (1995) rep o rt a sim ilar result.




9

3.1

T h e B asic S et-U p

In this section, we describe the time-series model we use to evaluate (1.2) and (1.3). Since there
is no observable asset with a risk-free real payoff over a multi-year horizon, we examine the
implications of ( 1 . 1 ) for nominal returns. Let

Pt

denotes the price level at date t , and let

denote the nominal cumulative equity return from date t to date

R t+ T
T

(so RJ+T = rt+T [^ ^j)*
r

t + r

Equation ( 1 . 1 ) then implies:
i=

Et

[m

;+ t r

;+t]

(3.1)

where Mtr, r is the marginal-rate-of-substitution in nominal wealth between t and
M

— m t+ r

t+ r

r

t + r:

Pt

(3.2)

L 1 t+ T

Let

RFf

denote the risk-free nominal return from t to t + r. The observable analogue to

RFf

is the return on a r-period zero-coupon dollar bond. Equation (3.1) implies the analogue s to
(1 .2 ) and (1.3):
E^

= _ CWt (MtV,

±r Z M

r f

: =

(3.3)

RI+t)

(3.4)

E t M t\ T

To test (3.3), we need a model of the conditional first moment of
second moments of the joint {Mtr+r,

Rt
T

and of the conditional

process. We use the following vector ARCH model.

Let X t denote an (N —2) x 1 vector of variables that is useful in predicting
let

Ytf

= (/?[, Mtr , X { ) . We assume that the (T x l)-dimensional process
V

and
Yt

follows a vector

autoregression:
^*+1 — A 0 + A i Y t

where

A0

is an

N

x

1

+

A 2Y t _ i

+ ... +

vector of constants, A, , i = 1 , ...,

A p Y t_p+i

p

+

are T x
V

u t+ i

N

(3.5)

matrices, and

u(+1 = L t+lvt+i , vt l ~ i.i.d.Af(0,I),
+

and

L t+ i




is a lower triangular matrix such that
—Ht+i —Et (ut+1ut+i)
10

(3.6)

We now must specify the law-of-motion for
N XN

symmetric matrix E let

v e c ( E)

H t+ 1 .

We use the following notation: For any

stack the distinct elements of E into a —E t l) x
(.

Following the ARCH approach of Engle (1982), we assume that

v e c ( H t+ i)

\

vector.

can be approximated

by a linear function of squared residuals dated t and earlier (i.e., elements of the matrices
u t _ qu't _ q).

That is,

v e c ( H t + 1)

=

B0

+

B i v e c ( u t u [)

+
where
N(N

B0

is an

N(N

u t u [,

+ l)/2 x

1

+

B 2v e c ( u t „ l u t _ l )

+ ...

(3.7)

B q v e c ( u t _ q + lu [ _ q + l )

vector of constants, and St, i = 1 , . . . , <, are
7

N(N

+ l)/2 x

+ l)/2 matrices. The parameters in (3.7) can be estimated by fitting the regression
uec(nt+i'u,+1) =
t

S0 +

+ S9uec(tzt_g+1zz;_g+1) +
where wt+1 is an

+ l)/2

i.i.d, N ( N

...

B i v e c ( u t u't ) + B 2v e c ( u t -.iu't _ 1) +

(3.8)

w t+ i

1 vector process. The linear model (3.5) - (3.7) allows

X

for easy computation of the r-step-ahead conditional first and second moments of Y t: Let
us write (3 .5 ) in first-order “companion” form by defining

A

by:

A'Q =

{Y ^ Y ^ ,

(A'0, 0 (NxN)}..., 0 {N xN )) . We define the

• o

0

x

Np

=

coefficient

0
0

•
•
O

o

i

.• o

•
•

•

I

Np

U[

---- 1

1---

a
.

A p - i

0

...,

i
--

A i
I

O

matrix

OtvxW , and
(
))

-- ,

( u t , 0 {NxN), ...,

=

y[

Equation (3.5) can now be written
Y t +1

—A o +

AY t

+ I4t+1

so
E t y t+r = ( I - A r ) ( i

- A ) ~ l Ao + A Ty t

(3.9)

and




r— 1
v a r ty t+T

= ^ A iE t

i=0
11

W

(3.10)

In (3.10),
HLi

Et

B kL k,

ls

computed from (3.8): If we use the notation that

B (L )

=

(3.8) implies
j - i

E tvec

[ut+iu't+J.]

= B0^2
k =

[£(!)]* +

[ B ( L ) f v e c [ u t u '] .
t

(3.11)

0

Equations (3.9) and (3.10) are used to evaluate the conditional moments in (3.3) and
(3.4):
and

E t R t+T
T

covt

principle,

and

are the first and second elements of the vector

E tM f+ T

(Mtr+r, /?J+T is the (2,l)t/l element of the matrix
)
covt

(Mtr+r,

depends on

R t + T)
T

all

v a r t y t+ T

E ty t+ r

in (3.9),

in (3.10). Notice that, in

elements of the matrices

Et

[ut+r_tVt+r_t] , i =

l , . . . , r - 1.
We use the linear model (3.5) - (3.8) because it provides the straightforward analytic
expressions (3.9) and (3.10) for

E t R ^ T) E t M l + T,

and

covt

(Mtr+T, /t!J+r). However, the linear

model is not without drawbacks. First, it tends to generate a large number of free parameters.
For example, if no ex-ante restrictions were placed on the matrices
equation (3.8) were treated as an unrestricted VAR in the —
would be a total of
grows at rate

TV4.

- - + n2

+

pN 2

+

(that is, if

elements of Ut+i^t+i), there

j free parameters to be estimated This number

q

It is easy to see that modest values of

TV, p,

and

q

can give severe degrees-

of-freedom problems.
Given this problem of parameter proliferation, we experimented with rather ruthless zerorestrictions on the coefficient matrices

We arrived at the following specification:

First, we exclude all cross-terms (of the form

u itt^ kUj)t- k ) k =

0,1 ,...,<7 —1 , i

^ j)

from the

right-hand side of (3.8). Second, in those equations of (3.8) where the dependent variable is a
squared residual of the form w?t+1, only own lagged dependent variables (i.e., u 2t ,
i
u i , t - q+ 1)

are use(l

3s

form ui|t+1«iit+i,i #

u 2t_ l ,

...,

regressors. Third, where the dependent variable is a cross-term of the
j,

only lagged squared residuals

are used as regressors. (That is,

u \ t_ n, k

/

u? t_ , +1

u i t , u ) t u i t_
2
2

is never used as an explanatory variable for

^M+i^j.t+i-) These restrictions were loosely patterned after the constant-correlation model,
which also excludes cross-terms as explanatory variables and only uses lagged dependent vari­
ables as explanatory variables in the squared-residual equations. These restrictions reduce the
number of free parameters to —
empirical work, we set




TV =

4,

p

+ (p +

= 1, and

q) TV2,

q — 8 , so

12

a number which grows at rate

TV2.

In our

the total number of free parameters is 158.

A second drawback of our linear model is that it does not guarantee positive-definiteness
of the

v a r t Yt+ T

matrix. Positive-definiteness is a nonlinear restriction, so multivariate models

with time-varying second moments that impose positive-definiteness necessarily must introduce
nonlinearities either into the model structure or into the estimation procedure. These sorts of
nonlinearities substantially increase the computational burden in estimating and solving the
model. For example, a widely-used multivariate model that guarantees positive-definite condi­
tional covariance matrices is the constant-correlation model of Bollerslev (1990). This model is
not suitable for our purposes, since, for i > 1 , the elements of the matrix U t +
functions of the innovations

w t + 1 , tut+2,

As a result, the matrices

Et

{ U

are

n o n lin ea r

[Z t+r-t^+r-t] in
V

equation (3.10) cannot be computed as a linear projection, as in equation (3.11). For

r

> 1,

computing these conditional covariance matrices would require integrating out the innovations
•••)

& computationally burdensome task. An alternative way of imposing

positive-definiteness is the diagonal GARCH model of Bollerslev, et. al (1988). This model
delivers a linear model of the general form (3.8), but guarantees positive-definiteness by im­
posing a nonlinear restriction on the coefficient matrices

B 2> B

q.

As such, this model

must be estimated using nonlinear techniques, such as maximum likelihood. Due to the large
number of parameters in the models we use, nonlinear estimation would be extremely burden­
some. Furthermore, it is not clear how much our inference is distorted by our failure to impose
positive-definiteness as a restriction. For these reasons, we estimate (3.5) and (3 .8 ) by OLS.
We report the number of violations of positive-definiteness for each model studied, and we use
the number of such violations as a check for model mis-specification.
3.2

Im p le m e n ta tio n of th e M odel

We include two predictor variables in
inflation rate

X

t: the term spread and the default spread. 16 The

is constructed from the deflator associated with the consumption series being

used. For each model of raj-, vector process {T*} is constructed, the first vector autoregression
16We in itially included th e dividend yield on the C R S P value-w eighted portfolio as a th ird predictor
variable in X t . We found, however, th a t for m ost preference specifications the dividend yield was
insignificant in the eq u atio n s for M tr+1 and R (+1, according to sta n d a rd F -tests. T his result is consistent
w ith F am a and French (1989). In th e interest of parsim ony, we therefore exclude the dividend yield
from X t. (Recall th a t th e n u m b er of free p aram eters grows a t rate TV2.)




13

(3.5) is estimated by OLS, vector process

{ v e c (u t u £)} is

constructed from the residuals of (3.5),

and the second regression (3.8) is estimated by OLS.
We used the multivariate Schwartz and Akaike Information Criteria to determine the ap­
propriate order

p

of the first VAR, equation (3.5). In most models, these criteria favored a

single lag, so we set

p

=

1

for all models. It is unclear how relevant these information-based

criteria are for the second regression (3.8), since the elements of v e c

( u t u ' )are
t

generated from a

smaller number of distinct information sources. Instead, we use a more informal procedure to
choose the order q of regression (3.8). We seek to maximize the variability of c o v t (Mtr+r, R l + T)
while keeping the number of non-positive-definite estimates for
els, we found that

q

=

8

v a r t Y t+ T

low. For most mod­

worked well according to this standard. For each model, Table 3.1

reports the number of times that our proxy for

v a r t Y t+T

failed to be a positive-definite matrix.

Failures of positive-definiteness are distressingly frequent at the quarterly horizon (r = 1 ), but
are infrequent or nonexistent at longer horizons. These results could be interpreted as evidence
of misspecification at when

r —

1: our linear time-series model (3.5) - (3.8) may simply be

inappropriate for modeling conditional second moments of the {r[, m[} process for very small
r ’s. Alternatively, the problem may be that the true eigenvalues of v a r tY t + i are very close to
zero at the quarterly horizon. When taking a linear approximation to

v a r t Y t + 1,

it would not

be surprising that the smallest eigenvalue of the approximate covariance matrices frequently
falls below zero. 17
Using the estimated values for param eters^!,
premium series

B j, j

= 1 ,..., 8 }, we construct the equity-

EPf
e p

;

*

.

(3.12)

t
R F tT

the ’’theoretical equity-premium” series implied by the particular model, which we denote
EP]

=

-co vt

(MtT
+r,

R [+ t )

and the ’’theoretical risk-free rate” series, which we denote
1

Et
M;+r

=

RFt

EP

t:

,
R F t:

(3.13)

17In su p p o rt of th is in te rp re ta tio n , we note th a t when positive-definiteness fails, th ere is usually only
one negative eigenvalue, an d its abso lu te value is usually several orders of m ag n itu d e sm aller th a n th e
o th er three eigenvalues of th e conditio n al covariance m a trix estim ate.




14

4

R esu lts

4.1 Implications for the Equity P r e m i u m

In this section we examine the implications of the models for equation (3.3). In order to make
our results comparable across different time horizons, we compute annualized continuouslycompounded equity premiums, denoted

E P fA

and

-- T^

EPt

o

as follows:18

E P tTA = - l o g [ E P ; + 1 ]

(4.1)

T

E P tA = ~

log [ E P t + l]

(4 .2 )

If a model of m tT and the time-series model (3.5) - (3.8) together described the data perfectly,
we would find

E P [A

=

EPt

for every date

t.

No one would expect such an outcome even

for a successful model. Rather, we wish to see whether, for any of the pricing models,

------- - T A

EPt

approximates some of the key properties of E P f A. In particular, we ask whether the following
hold:
mean [EP^A] « mean

(4.3)

var [EPfA] ~ var

(4.4)

corr
4.1.1

[ E P tTA, E P t ) »
T

0

(4.5)

Consum ption M easured by Nondurables Plus Services

Table 4.1 summarizes our results for (4.3) and (4.4) when we measure consumption by
purchases of nondurables plus services. The first line of the table gives our estimate of the
mean and the variance of the equity premium

E P fA

at horizons equal to one quarter, one

year, two years, and three years. The remainder of the table gives the corresponding moments
Rr

18The annualization in (4.1) is appropriate, since E P t -f 1 = E t
, where both
gross rates of return. Also, recall that the horizon r is in units of quarter-years.




15

R[

and

RF^T

are

of the theoretical equity premium

implied by the three preference models. The num__ a

E P t

bers in parentheses are asymptotic p-values testing whether

( m e a n [ E P f A]

—m e a n [ E P t ]) and

___ a

(v a r [ E P [ A] —v a r [ E P t ]) are significantly different from zero. In the case of the means, we use
the test statistic

t

t

EL

(

e p

;

-

a

A

e p x

(4.6)

Zmp.an —
std

(EP?* - E P ^

In the case of the variances, we use the test statistic

12
>
--- A
t
EPt - p

[EPlA - A 2 ~

± EL,

Z Va —
r
- -

std

where

ji

b

EL

[E P ?a

-

EPt

(4.7)

2
>

TA

- n

--‘
~
A
and p are constants set equal to the sample means of E P [ A and E P tTA , respectively.

Under the hypothesis that

E [ E P ^ A]

=

E

as a standard normal variate; similarly, if

—

tA

EPt

, statistic

v a r [ E P f A]

=

Z mean

var

is asymptotically distributed

'A
t
EPt

, statistic

Z var

is asymp-

totically standard normal. We compute the standard deviations in the denominators of (4.6)
and (4.7) using 12 Newey-West lags for the quarterly, yearly, and two-year horizons, and 16
lags for the three-year horizon. 19 Note that we treat

E P fA

and

___ a
EPt

as known data series,

not generated series, so the uncertainty in estimating the VAR parameters in 3.5 and 3.8 is
not taken into consideration. As a result the standard deviations used in constructing
and

Z var

Z mean

are understated.

The time-separable model exhibits both the equity-premium puzzle and the predictability
puzzle at all horizons. The observed annualized equity premiums have means between 4.5%
19We w ant the n u m b er of N ew ey-W est lags to equal the m ax im u m of th e a p p ro p ria te lag-lengths for
_ A
_

___A

an d for E P t . A ccording to eq u atio n (3.8), E P t is a function of eight lagged regressors, each of
which is serially co rrelated , so the a p p ro p ria te lag-length for th is variable is a t least 9. T h e r-h o rizo n
eq u ity p rem iu m involves r overlapping observations, so th e a p p ro p ria te equ ity -p rem iu m lag-length is a t
least t + 1 . We then ex p erim en ted lag-lengths above m a x ( 9 , r + 1) u n til th ere were no large changes
in th e s ta n d a rd d ev iatio n s.
EPfA




16

CONSTANTINIDES.

N O N D U R .

OO*1
.00 .3

SERV.: M E A N S

...... ...
.

1
00 .1 j
oo . oi
*
1

___------ ----- “

CBAMMA
B
CONSTANTINIDES.

1m

m

"t I quTt»r.
nu n
i
a
j
r
PMMIUM
I M O I M D U R . -- S E R V . :
1

S T D D E V

T his figure displays m eans (to p panel) and sta n d a rd deviations (b o tto m panel) of th e annualized equity

--tA

p rem iu m E P £ A and th e annualized theoretical equity p rem ium E P t im plied by th e C o n stan tin id es
m odel w ith 7 = 7 (lowest line), 9, 11, and 12 (highest line), and w ith co nsum ption m easured by
expen d itu res on n o n d u rab les plus services. T he sta n d a rd deviations are all scaled by ( ^ ) 2 as in (4.4),
to facilitate com parison across horizons. T h e horizons are r = 1, 4, 8 , an d 12 q u arters.

Figure 4.1: Equity Prem ium vs. Theoretical Equity Prem ium : Constantinides
Model, N ondurables Plus Services
and 6.5%. The time-separable model has difficulty generating a mean equity premium in excess
of one percent for any horizon. Furthermore, the equity-premium variance generated by the
model is an order of magnitude too small at all horizons.
Let us now turn to the Abel and Constantinides models. Figure 4.1 plots the results
reported in Table 4.1 for the Constantinides model.

(The pattern for the Abel model is

similar.) The upper panel plots the mean of E P f A (heavy lines) at the four horizons,along
— tA—
with the mean of E P t for four different values of 7 . The lower panel in each figure displays
___a

the analogous plots for the standard deviations of

E P tA

and

EPt

, scaled by

1
( ^ ) 2 . 20

The

20In eq u atio n (4.4), we m u ltip ly by ( - ) 2 to cou n teract the effect of an n u alizatio n on th e sta n d a rd
d ev iatio n o f E P [ A . If log [ E P f -f l]were the sum of r i.i.d. ran d o m processes, th en s t d ( E P ^ A ) would
decline a t ra te ri as r increases, b u t ( J ) 2 s t d { E P ^ A ) would be co n stan t. W hile definition (3.12) of
E P J does n o t im ply th a t log [ E P { -f 1] is determ ined in precisely this way, we find th a t, in practice,
( J ) 2 s t d { E P ^ A ) is a p p ro x im ately c o n stan t in r.




17

P R E M I U M

vs. C O N D I T I O N A L C O V A R I A N C E ,

C o n s i s t n t i n i d e s ,

N o n d u r . - t- S & r \ s _

C

o n s u m

p t i o n ,

g a m

m

a ^

T A U = 4
1 2

T h is figure displays tim e-series p lo ts o f th e annualized equity p rem iu m E P £ A (dashed lines) and th e

— tA

an n u alized th eo retical eq u ity p rem iu m E P t (solid lines) im plied by th e C o n stan tin id es m odel w ith 7
= 1 2 . C o n su m p tio n is m easu red by e x p en d itu res on non d u rab les plus services, and the horizon r = 4
q u a rte rs.

Figure 4.2: Equity Prem ium vs. Theoretical Equity Prem ium (Constantinides
Model) Using N ondurables Plus Services: Time-Series Plots
Constantinides model does not fare much better than the time-separable model at the shortest
__ a

and longest horizons: the means of E P t are less than

2%

when

r = 1

and r = 12, even with

extremely high risk-aversion. According to Table 4.1, these point estimates are significantly
below the mean equity premium. However, the model performs somewhat better at the oneyear horizon: when the Constantinides model is implemented using one-year returns with

7

=

_____ _ y\

12, the mean of

EPt

is about 3%, which is somewhat closer to the mean value of E P tA than

was obtained using quarterly returns. According to Table 4.1, equality of the means of
and

——

A

EPt

is not rejected at the 5% significance level. The point estimate for the standard

EPA

deviation is substantially below that of the observed equity premium. However, equality of the
variances cannot be rejected at the

1%

level for the one-year horizon, and cannot be rejected

at any conventional significance level at the two year horizon. Similar results obtain for the
Abel model with

7

= 14.

While the Constantinides model shows some improved ability to fit conditions (4.3) and
(4.4) at the one-year horizon, it fails to satisfy equation (4.5). Figure 4.2 plots the timeseries for

E F fA




(dotted lines) and

___ 4a
EPt

(solid lines) generated by the Constantinides model
18

with

7

= 12. If (3.3) held exactly, the two series would be identical. As can be seen from

the figure, the two series appear to be totally unrelated. As a more formal test, we regress
EP?A

on the

EPt

series implied by the Constantinides model, along with the linear and

quadratic trends.21 In order to account for the high serial persistence in these series, we
compute standard errors using 12 Newey-West lags. For both models, the slope coefficient on
- ^^
E P t is insignificantly different from zero (p-value of .43), and the point estimate is actually
negative.22 This regression evidence confirms the visual impression of Figure 4.2.
To summarize, when consumption is measured as nondurables plus services: ( 1 ) there is
a modest improvement in the models’ ability to match the mean and variance of the equity
premium if the horizon is lengthened from one quarter to one year, but (2 ) time-series variation
in the observed equity premium is explained by the model to any significant degree. This would
appear to be a rather disappointing result for consumption-based pricing models. In the next
section, we consider whether this result is due to the use of nondurables plus services as our
measure of consumption.
4.1,2

A lternative Specifications of Consum ption

In the previous section measure consumption as consumer expenditures on nondurable
goods plus consumer services, as measured by the Bureau of Economic Analysis. This is the
consumption series used in most consumption-based asset pricing research since the original
work of Hansen and Singleton

(1 9 8 2 ).

As discussed in the introduction, we believe that our

results may not be robust to alternative specifications of consumption. To look at this possi­
bility, we first replicate our analysis with consumption measured by nondurables only. Table
4 .2

is analogous to Table 4 .1 , except that consumption is measured by purchases of nondurable

consumption goods. (That is, the services component is omitted from the consumption data.)
21 We include the tren d term s to accom m odate th e slight ” U” shape in E P * A . T h is p a tte rn is due
alm o st entirely to th e secular rise in the nom inal tw o-year risk free ra te over this period. (N om inal
equity re tu rn s do n o t display any pronounced tren d in post-w ar d a ta .) B o th the linear an d q u a d ra tic
tre n d term s enter significantly. As in T able 4.1, we do n o t take into consideration the fact th a t we are
using generated regressors, so th e sta n d a rd errors are u n d erstated .
--- 4A

22T h e Abel m odel yields sim ilar results. W hen we regress E P ? A on th e E P t

series im plied by the
- 4A
-

’’b e st” A bel m odel (7 = 14), along w ith the linear and q u ad ra tic tren d s, th e coefficient on E P t
negative, and insignificantly different from zero.




19

is

OONSXANXIIMIDES,

N O N D U R A B L E

CONSUMPTION:

M E A N S

MEAN

1
OONSTANTINIDES.

■ m u

In

quartvra

1
N O N D U R A B L E

( A M M A
^ --- <---H»
«- *
**
_____________ __________________________ ______ • — a — •
O ___________________________ ■A-.V.V..V.-.V. p r e m i u

m

-r
j
____________ ______________________________

CONSUMPTION:

3 T D D E V

T h is figure displays m eans (top panel) an d sta n d a rd dev iatio n s (b o tto m panel) of the annualized equity
a
.
.
■ 'Tj4
—
prem iu m E P ^
and th e annualized theo retical equity prem iu m E P t im plied by th e C o n stan tin id es
m odel w ith 7 = 1 (lowest line), 5, 7, and 9 (highest line), an d w ith con su m p tio n m easured by purchases
of consum er n ondurables. T h e s ta n d a rd d ev iatio n s are all scaled by ( ^ ) 2 as in (4.4), to fa cilitate
com parison across horizons. T h e horizons are t — 1, 4, 8 , and 12 qu arters.

Figure 4.3: Equity Prem ium vs.
Model, N ondurables

Theoretical Equity Prem ium : Constantinides

The results for the time-separable model improve somewhat at the highest levels of risk aver­
sion for the one- and two year horizons. With the Abel and Constantinides models, however,
the difference in model performance is striking. Figure 4.3 plots the point estimates reported in
Table 4.2 for the Constantinides model. Note that the theoretical equity premium matches the
mean equity premium observed in the data at both the one- and two-year horizons with
(Comparable results obtain in the Abel model with

7

7

=

7.

= 10. ) 23 According to the p-values in

Table 4.2, the means of the theoretical equity premiums for these models are insignificantly
different from the means of the observed equity premiums at any conventional significance level
The variances are insignificantly different at the two-year horizon. Figure 4.4 plots the

E P fA

23In th e A bel m odel, these results require extrem ely high risk aversion: 7 = 10 im plies a m ean
coefficient of relativ e risk aversion of ap p ro x im ately 49. In c o n trast, th e level of risk-aversion im plied by
7 = 7 in th e C o n stan tin id es m odel is less extrem e. As noted in section 2.3, the ste a d y -sta te coefficient
of relative risk aversion in th e C o n stan tin id es m odel is n o t too m uch higher th a n 7 .




20

P R E M I U M

vs. C O N D I T I O N A L . C O V A R I A N C E ,

G o n s ta n tin ic i&

s ,

M o r tc J u r .

C D o n s u m p ttio r n ,

g a m

m

a

TAU==S

— 7^

T h is figure displays tim e-series plots of th e annualized equity prem ium E P f A (dashed lines) and the
—

y

annualized th eo retical equity p rem iu m E P t (solid lines) im plied by th e C on stan tin id es m odel w ith
7 = 7. C o n su m p tio n is m easured by ex p en d itures on consum er n ondurables, and th e horizon r = 8
qu arters.

Figure 4.4: Equity Prem ium vs. Theoretical Equity Prem ium : Time-Series Plots,
N ondurable Consum ption
against the

-— 8A

EPt

series implied by the Constantinides model with

7

= 7. Unlike Figure 4.2,

these plots appear to display clear (albeit imperfect) co-movement between the theoretical and
observed equity premium series. The main discrepancy is that the theoretical equity premium
- - TA
-

EPt

does not capture the secular decline in the observed equity premium from 1954 through

1980: the theoretical series is too low in the 1950’s and too high in the early 1980’s. However,
our construct for the theoretical premium does appear to capture some of the cyclical fluctua­
tion in the equity premium: note equity-premium peaks in 1956, 1965, and 1976-77, as well as
the sharp fall-offs in 1977-78 and 1988. On the whole, Figure 4.4 provides some evidence that
consumption-based models can generate time-varying risk premiums appropriate to the observed data. When

E F fA

is regressed on this

time-trend, the coefficient on

EPt

- —8A
EPt

process, along with a linear and quadratic

is 0.364, with a standard error of 0.161 (significant at the

5% marginal significance level) . 24
We conclude from this evidence that the rather tepid performance of our consumption-based
24W hen th e co m p arab le regression is perform ed for the Abel m odel w ith 7 = 10, the coefficient is
0.313, w ith a sta n d a rd error of 0.129. A gain, the coefficient is significantly positive.




21

Corr(R(t), S e r v i c e s C o n s u m p t i o n Growth(t-lag)), T w o - Y e a r H o r i z o n
0.1 4
0 .0 7

0.00
- 0 .0 7

-0 .1 4

-0.21
-0 .2 8
-0 .3 5
la g = -8

to

8 Q u a rte rs

This plot presents the correlation between two-year real VW index returns at t and the two-year growth
rate of real and services consumption at (t —
lag) for lag = -8 through + 8 quarters. The vertical lines
in the center of the graph indicate the contemporaneous correlation
Figure 4.5: C orrelation between Real Two-Year VW Index R eturns at
Two-Year Services Consum ption G row th

t

and Real

models with the standard measure of consumption (at least at one-and two-year horizons) is
due to the consumption-services component of the consumption measure. This should not
be all that surprising, since the consumption services data are less well-correlated with equity
returns than are the data on purchases of nondurables. Consider Figures 4.5 and 4.6, which plot
the correlations between two-year equity returns and the two-year growth rates of consumption
services and consumption of nondurables, respectively. According to Figure 4.5, the maximum
contemporaneous correlation between the growth of consumer services and real equity returns
at any of the horizons tested is 0.09 (at the one-year horizon). The maximum correlation at any
lead or lag is 0 . 2 2 (also for the one-year horizon, when growth of consumer services leads returns
by two quarters). In contrast, the contemporaneous correlation between nondurables growth
the equity return series is 0.26 for the one-year horizon (0.30 for the two-year horizon), and the
maximum correlation (again, when consumption-growth has a two-quarter lead) is above .40.
Evidently, the data on consumer services provided by the BEA has little explanatory power
for equity returns.
We conclude this section by replicating the analysis with consumption measured by total
consumption purchases. For completeness, we include results for horizons ranging from one
quarter to three years. As discussed above, this measure of consumption is inappropriate




22

Z^orr(R(t), N o n d u r a b l e s C o n s u m p t i o n Growth(t-lag)), T w o - Y e a r Horizor
0.^5
0 .3 6
0 .2 7
0 .1 8
0 .0 9

-0.00
-0 .0 9
-0 .1 8
-0 .2 7
la g = -8 to 8 Q u a r t e r s

T h is p lo t presents th e correlatio n betw een tw o-year real V W index retu rn s a t t and th e tw o-year grow th
ra te of real an d services co n su m p tio n a t (t —
lag) for lag = -8 th ro u g h 4-8 q u arters. T h e vertical lines
in th e center of th e g rap h in d icate th e contem poraneous correlation

Figure 4.6: Correlation between Real Two-Year VW Index R eturns at
Two-Year Services Consum ption Growth

t

and Real

for short horizons, but may give some indication of the role of durables in longer horizon
returns. These results are displayed in Table 4.3, which is analogous to Table 4.2, and in
Figure 4.7, which is analogous to Figure 4.3. The results look remarkably like the results
when only nondurables purchases are used. The Constantinides model comes quite close to
replicating both the mean and the variance of the equity premium at the two-year horizon when
7

=

11;

The Abel model behaves similarly when

7

= 14. Table 4.3 shows that these results

are statistically significant. For both of these models, at both the one- and two-year horizons,
neither the mean nor the variance of

___ a
EPt

is significantly different from the corresponding

moments of E P A .
As is the case when only nondurables are used, the time-series properties of E P { A (dotted
-

lines) are replicated to some extent by the behavior of

T

A

EPt

in these models. Figure 4.8

is analogous to Figure 4.4. Again, the theoretical equity premium series generated by the
Constantinides model cannot replicate the long-run secular movement of the observed equity
premium, but does appear to mimic the cyclical movements. Unlike the models that use only
nondurable consumption, the patterns in Figure 4.8 do not emerge as significant when tested
___ QA

using formal statistical methods. When we regress

E P fA

with the linear and quadratic trends, the coefficients on




23

on

EPt

__ j
A
EPt

from either model, along

are positive, but insignificant

CONSTANTINIDES.

TOTAL

C O N S U M P T I O N :

-ri
P P I H M I U M

C O N

M E A N S

*»

UMPTION:

S T D D E V

2^2 1

This figure displays means (top panel) and standard deviations (bottom panel) of the annualized equity
premium EP^ and the annualized theoretical equity premium E P t implied by the Constantinides
model with 7 = 1 (lowest line), 5, 9, and 11 (highest line), and with consumption measured by total
consumption expenditures. The standard deviations are all scaled by ( ^ ) 2 as in (4.4), to facilitate
comparison across horizons. The horizons are r = 1, 4, 8 , and 12 quarters.
Figure 4.7: Equity Prem ium vs.
Model, Total Consum ption
at the

4.2

10%

Theoretical Equity Prem ium : Constantinides

marginal significance level.25

Implications for the Risk-Free Rate

We now test equation (3.4) for the models studied in the previous section. We ask whether
the means and standard deviations of R F t match those of R F f , and whether these two series
have substantial positive correlation. As with the equity premium, we annualize by setting
r f

;

a

= -

log { r f ; )

T

_ 8A
-

—

25For the Constantinides model, the estimated coefficient on E P t is 0.137 with a standard error of
0.105. The same regression for the Abel model yields a coefficient estimate of 0.137 with a standard
error of 0.104. As before, we compute standard errors using 12 Newey-West lags.




24

P R E M I U M

vs. C O N D I T I O N A L

(D o n s ta tn tin id & s ,

T o ta l

C O V A R I A N C E ,

(D o n s u m p > tio n ,

g a m

m

T A U = 8

a — 7 1

This figure displays time-series plots of the annualized equity premium E P £ A (dashed lines) and the
- TA
annualized theoretical equity premium E P t (solid lines) implied by the Constantinides model with 7
= ll. Consumption is measured by total consumption expenditures, and the horizon r — 8 quarters.
Figure 4.8: Equity Prem ium vs. Theoretical Equity Prem ium : Time-Series Plots
RFt

4.2.1

= - log ( R F t )

Consum ption M easured by Nondurables Plus Services

Table 4.4 displays our results for the risk-free rate when consumption is measured by
expenditures on nondurables plus services. The first line of the table gives our estimate of
the mean and the variance of the observed nominal risk- free rate at the four horizons. The
remainder of the table gives the corresponding moments of the theoretical risk-free rate

--- tA

RFt

implied by our three preference models. The numbers in parentheses are asymptotic p-values
___

testing whether (m e a n [ R F [ A]

— m ea n [R F t

-- y\

]) and (v a r [ R F f A]

— var[R F t

]) are significantly

different from zero.26
Our results for the time-separable model clearly illustrate the risk-free rate puzzles: For all
7

’s up to 50, all horizons, and both measures of consumption, the means and variances of the

risk-free rates implied by the models vastly exceed the values observed in the data. The Abel
26As in Table 4.1, we construct the p-values using 12 Newey-West lags for the quarterly, yearly, and
--- A
two-year horizons, and 16 lags for the three-year horizon. The fact that R F t is a generated series is
not taken into consideration in computing the p-values.




25

T h e to p panel p lo ts th e m ean of th e annualized risk-free ra te R F ? A a t th e four horizons r = 1, .4, 8 ,

-— -"4
- 7y

an d 12 q u a rte rs (heavy line), along w ith th e m eans of th e annualized th eo retical risk-free ra te R F t

for

th e C o n stan tin id es m odel w ith 7 = 7, 9, 11, and 12. T h e b o tto m panel plo t ( ^ ) 2 tim es the sta n d a rd
_____ T

>
4

d ev iatio n of R F £ A a t th e four horizons (heavy line) ag ain st th e corresponding s ta tistic for R F t
th e four values of 7 . C o n su m p tio n is m easured by ex p en d itu res on n o n durables plus services.

for

Figure 4.9: Risk-Free R ate vs. Theoretical Risk-Free Rate: Constantinides Model,
Nondurables Plus Services Consum ption
and Constantinides models do not perform much better at matching the mean risk-free rate.
In Figure 4.9, we plot the point estimates from Table 4.4 for the Constantinides model. (The
patterns for the Abel model are similar.) The top panels in Figure 4.9 plots the mean of R F f A
——

t

A

(heavy lines) at the four horizons, along with the means of R F t for the Constantinides model
using four values of 7 . The bottom panels plot (^)* times the standard deviation of
against the corresponding statistic for

- TA
-

RFt

R F {A

. As demonstrated by Figure 4.9, the model fails

to match the mean of the risk-free rate (although the failure is less dramatic than with the
time-separable model). As with the time-separable model, lengthening the horizon increases
the mean theoretical risk-free rate counterfactually.
While these models have difficulty matching the mean of the risk-free rate at any horizon,
the excessive volatility of the theoretical risk-free rate does become less of a problem at longer
horizons with the Abel and Constantinides models. This is of considerable interest, since these




26

models have been criticized for implying counterfactually high interest-rate variability at short
--- t A

horizons. Figure 4.9 shows that, for all values of
approximates the variance of the observed

R F fA

7

studied, the variance of

RFt

actually

for a horizon r somewhere between two and

three years.
It would appear, then, that the problem of excessive variability in the risk-free rate implied
by time-nonseparable models is primarily a short-horizon problem. The intuition behind this
result is that the variability of

RFt

is determined by the variability of -■( E t M

. (See

^ T)

equation (3.4).) Let us use the decomposition
M[+T = f[M;+l.
ii
=

For purposes of exposition, let us ignore Jensen’s inequality, and consider
±Et

[Iog(MtT
+T)-1j =

[log(M/+i) ] .

(4.8)

According to (4.8), the variability of £ E t [log(Mtr+r)_1] is determined by the
of M/+i, the one-period marginal rate of substitution prevailing

i

p red icta b ility

periods in the future. If

Mt1 is highly predictable (for example, if l o g ( M ^ i ) were a random walk), then the variability
+t
of

Et

[log(Mt1 would not decline substantially as r gets big. If, on the other hand,
fi)]

displays rapid mean-reversion, then we would find that
moderately large values of

i.

E t [ \ o g ( M l + i )] =

r

E [ l o g ( M ^ i )]

for

In that case, most of the terms in the right-hand side of (4.8)

would be approximately non stochastic, and the variance of
rapidly as

M ?+i

^Et

[log(Mtr+r)] would drop off

increases. A similar intuition holds for the observed risk-free rate

R F f.

Let us

assume, for purposes of exposition, that long interest rates satisfy the following version of the
expectations hypothesis:
log R F & = ±

[log(J2Ft‘ |.)].
+

(4.9)

'1 1
=

According to (4.9), the variability in
one-period interest rates
be explained if, as

i

RF^+ i.

R F {A
T

is determined by the predictability of the future

The patterns displayed in the bottom panel of Figure 4.9 can

increases, the predictability of

RF^+i

attenuates more slowly than the

predictability of M/+i.
In Figure 4.10, we compare the i-step ahead predictability of lo g

(R F ^ )

with the z-step ahead

predictability of the lo g (M/) series implied by the Abel model. (The log transformation makes




27

Autoc o r r e l a t i o n s of L o g Quarterly R a t e s
A^oc<&/ f J m r 9 *
\oi-£/
(so/ic/) stncJ /^/s/c-Fr®©
C
c/as/iec/)

This figure plots the first thirty autocorrelations of the log nominal quarterly marginal rate of substi­
tution log (M/) in the Abel model, and of the observed log quarterly risk-free rate log (R F ^ ) (dashed
line). In computing the marginal rates of substitution, we measure consumption as expenditures on
nondurables plus services.
Figure 4.10: A utocorrelations of the Risk-Free R ate and of the M arginal R ate of
Substitution (Abel Model)
the autocorrelations invariant to 7 .) For each series, we display the first thirty autocorrelations.
Notice that the autocorrelations for
While

lo g (R F t)

lo g (R F * )

die out much more slowly than for

behaves as a near-random walk,

These results explain why

v a r (jR F t

log (M ? )

lo g (M ^ ).

displays no long-run predictability.

^ falls so much more rapidly than

v a r (jR F t

J as r is

lengthened from one quarter to twelve quarters.

4,2.2

A lternative Specifications of Consum ption

In tables 4.5 - 4.6, and in Figures 4.11 - 4.12, we display implications of the models for
the risk-free rate when consumption is measured either as expenditures on nondurables or
as total consumption expenditures. The qualitative behavior of

v a r (^ R F t

^ is similar to

the case of nondurables plus services: For all models, this variance is decreasing in r, but
this effect is far more pronounced in the time-nonseparable models than with time-separable
preferences. However, the behavior of m e a n
Constantinides preferences,

m ean y R F t

j

[RFt

) is rather different. With either Abel or

is much lower with either nondurable consumption

or total consumption than when the standard nondurables-plus-services measure is used. At




28

These figures

d splay m eans and s ta n d a rd d ev iatio n s of th e annualized risk-free ra te
i
--tA

RFfA

and the

annualized th eo retical risk-free ra te R F t im plied by the C o n stan tin id es m odel w ith 7 = 1, 5, 7, and 9,
and w ith co n su m p tio n m easured by consu m p tion ex p enditures on nondurables. T h e sta n d a rd deviations
are all scaled by ( ^ ) 2 as in (4.4), to fa c ilita te com parison across horizons. T h e horizons are r — 1, 4,
8 , and 12 q u a rte rs.

Figure 4.11: Risk-Free R ate vs. Theoretical Risk-Free Rate: Constantinides Model,
Nondurable Consum ption
the quarterly horizon, this mean is actually

b e lo w

the mean of the observed R F [ A series for most

specifications, and is extremely negative for the higher 7 ’s. As with nondurable-plus-services
consumption,

m ean

^R F t ^ tends to increase with the horizon r, approaching

m e a n (^RFTA ^
t j

at a horizon between two and three years.
The explanation for these results is that

m ean

^ R F t ^ is increasing in the mean growth

rate of consumption (higher consumption growth increases the incentive of agents to borrow
from the future) and decreasing in the variability of the consumption growth rate (higher vari­
ability accentuates the precautionary motive for saving). The mean quarterly growth rate of
expenditures on nondurables from 1947 to 1994 is 0.26%, and the standard deviation of this
growth rate is 0.80%. In contrast, the mean growth rate for nondurables plus services over
this period is 0.45%, with a standard deviation of only 0.56%, so both effects tend to reduce
m ean

^ R F t ^ when expenditures on services are eliminated from the consumption measure.




29

T hese figures display m ean s and sta n d a rd d eviations of th e annualized risk-free ra te R F f A an d th e
-

TA

annualized th eo retical risk-free ra te R F t im plied by th e C o n stan tin id es m odel w ith 7 = 1, 5, 9, and
1 1 , an d w ith co n su m p tio n m easured by expenditures on n o n d u rab les plus services, four versions of the
C o n stan tin id es m odel w ith co n su m p tion m easured by to ta l co n su m p tio n exp en d itu res. T h e sta n d a rd
d ev iatio n s are all scaled by ( ^ ) 2 as in (4.4), to facilitate com parison across horizons. T h e horizons are
r = 1 , 4, 8 , an d 12 q u arters.

Figure 4.12: Risk-Free R ate vs. Theoretical Risk-Free Rate: C onstantinides Model,
Total Consum ption
With total consumption expenditures, the mean growth rate is 0.49%, with a standard devia­
tion of 0.77%, so the reduction in

m ean

^ R F t ^ when this series is used is due solely to the

enhanced precautionary demand.
While the alternative measures of consumption allow the models to match the mean and
standard deviation of the risk-free rate for a horizons between two and three years, the models
do not capture any of the time-series variation in the observed risk-free rate. As in section 4.1,
we regress the observed risk- free rate

RF^

on the theoretical risk-free rate

-- tA
RFt

along with

a constant and linear and quadratic time-trends. Our point estimates for the coefficient on
-

TA

RFt

are in all cases negative; the estimates are insignificant for the two- year horizon, but

are significantly negative for the three year horizon. We conclude that these models do not
succeed in replicating the time-series movement of the nominal risk-free rate.




30

5

C onclusions

In this paper, we ask whether consumption-based models are better able to match observed
equity premiums and risk-free rates as the horizon lengthens. We find that time separable
utility fails at all horizons with all measures of consumption. When we follow the prevailing
literature and measure consumption as expenditures on consumer nondurables plus services,
the Abel and Constantinides models display a marginal improvement in fit when the horizon
is lengthened to one year. However, significant features of the data are not captured by the
models. In contrast, when consumption is measured either as expenditures on nondurables
or as total consumption purchases, versions the Abel and the Constantinides can can match
the mean and the variance of the observed equity premium at the two-year horizon, and can
capture a good deal of the time-variation of the equity premium in post-war data. In addition,
these measures of consumption allow the models more success matching the first and second
moments of the observed risk-free rate. The time-variation in the risk-free rate is not captured
by any of the models at any horizon.
These results are intruiging. They suggest that the equity premium and risk-free rate
puzzles are less puzzling for one-year cumulative equity returns than for quarterly returns, and
can be substantially resolved for two-year returns. However, this conclusion emerges only if we
throw out the BEA series for expenditures on consumer services, or if we include expenditures
on durables in the measure of consumption. If one is convinced that measured expenditures
on nondurables plus services is the correct measure of consumption, then this result would
simply constitute a rejection of the consumption-based pricing paradigm. Alternatively, our
results could be evidence that consumer services are separable from nondurables in agents’
preference orderings. As described in the introduction, however, we are inclined to view our
results as indicating that the BEA data on consumer services is flawed, and that these data
do not provide a good empirical analogue of the consumption concept appropriate to these
models. We regard this issue as an important topic for future research.
The improvement in performance when total consumption expenditures is used suggests




31

that the durables component of consumption is important for longer-horizon returns. A more
careful treatment would be desirable, in which the service flow from consumer durables is
explicitly modeled.27 Finally, a major puzzle is why all models dramatically fail to match the
equity-premium at the three-year horizon. Market frictions could disrupt the linkage between
asset returns and the consumption-based pricing kernel at short horizons. It is not clear what
economic model would similarly disrupt this linkage at the very long horizons.

27E ichenbaum and H ansen (1990) m odel preferences over du rab les in this way, b u t they only consider
th e o n e-m o n th in v estm en t horizon.




32

R eferences
Abel, Andrew B, 1990, Asset prices under Habit Formation and Catching Up with the Joneses,
A m e r i c a n E c o n o m i c R e v i e w 80, 38-42.
Bekaert, G. and R. Hodrick, 1992, Characterizing Predictable Components in Excess Returns
on Equity and Foreign Exchange Markets, J o u r n a l o f F i n a n c e 47, 467-509.
Boldrin, M., L. Christiano, and J. Fisher, 1995, Asset Pricing Lessons for Modeling Business
Cycles, manuscript.
Bollerslev, Tim, 1990, Modeling the Coherence in Short-Run Nominal Exchange Rates: A
Multivariate Generalized ARCH Approach,” R e v i e w o f E c o n o m i c s a n d S t a t i s t i c s 72,
498-505.
Bollerslev, Tim, Robert F. Engle and Jeffrey M. Wooldridge (1988), “A Capital Asset Pricing
Model with Time Varying Covariances,” J o u r n a l o f P o l i t i c a l E c o n o m y 96, 116-131.
Breeden, D.T., 1979, An Intertemporal Asset Pricing Model with Stochastic Consumption and
Investment Opportunities, J o u r n a l o f F i n a n c i a l E c o n o m i c s 7, 265-96.
Cecchetti, Stephen G., Pok-Sang Lam and Nelson C. Mark, 1993, The Equity Premium and the
Risk-Free Rate: Matching the Moments, J o u r n a l o f M o n e t a r y E c o n o m i c s 31, 21-46.
Cecchetti, Stephen G., Pok-Sang Lam and Nelson C. Mark, 1994, Testing Volatility Restrictions
on Intertemporal Marginal Rates of Substitution Implied by Euler Equations and Asset
Returns, J o u r n a l o f F i n a n c e 49, 123-152.
Cochrane, J.H., 1989, The Sensitivity of Tests of the Intertemporal Allocation of Consumption
to Near-Rational Alternatives, A m e r i c a n E c o n o m i c R e v i e w 7 9 , 319-337.
Cochrane, John and Lars Hansen, 1992, Asset Pricing Explorations for Macroeconomics, 1 9 9 2
N B E R M a c r o e c o n o m i c s A n n u a l , Olivier Blanchard and Stanley Fischer, editors. Cam­
bridge: MIT Press 1992.
Constantinides, George M., 1990, Habit Formation: A Resolution of the Equity Premium Puz­
zle, J o u r n a l o f P o l i t i c a l E c o n o m y 98, 519-543.
Daniel, Kent, and David Marshall, 1995, Consumption-Based Modeling of Long-Horizon Re­
turns, University of Chicago, manuscript.
Eichenbaum, Martin, and Lars P. Hansen, 1990, Estimating Models with Intertemporal Substi­
tution Using Aggregate Time Series Data, Journal of Business and Economic Statis­
tics 8 , 53-69.
Ferson, Wayne E., and George M. Constantinides, 1991, Habit Persistence and Durability in
Aggregate Consumption; Empirical Tests, J o u r n a l o f F i n a n c i a l E c o n o m i c s 29, 199240.




Constantinides, George M. and Darrell Duffie, 1992, Asset Pricing with Heterogeneous Con­
sumers, University of Chicago manuscript.
Cutler, David M., James M. Poterba and Lawrence H. Summers, 1991, Speculative Dynamics,
R e v i e w o f E c o n o m i c S t u d i e s 58, 529-546.
Engle, R., 1982, Autoregressive Conditional Heteroskedasticity with Estimates of the Variance
of U.K. Inflation, E c o n o m e t r i c a 50, 987-1008.
Fama, Eugene F. and Kenneth French, 1988, Permanent and Temporary Components of Stock
Prices, J o u r n a l o f P o l i t i c a l E c o n o m y 96, 246-273.
Fama, Eugene F. and Kenneth French, 1989, Business Conditions and Expected Returns on
Stocks and Bonds, J o u r n a l o f F i n a n c i a l E c o n o m i c s 25, 23-50.
Froot, K., 1990, Short Rates and Expected Asset Returns, manuscript.
Grossman, Sanford and Guy Laroque, 1990, Asset Pricing and Optimal Portfolio Choice in the
Presence of Illiquid Durable Consumption Goods, E c o n o m e t r i c a 58, 25-51.
Grossman, S.J., and R.J. Shiller, 1982, Consumption Correlated ness and Risk Measurement in
Economies with Non-Traded Assets and Heterogeneous Information, J o u r n a l o f F i n a n ­
c i a l E c o n o m i c s 1 0 , 195-210.
He, Hua, and David M. Modest, 1995, Market Frictions and Consumption-based Asset Pricing,
J o u r n a l o f P o l i t i c a l E c o n o m y 103, 94-117.
Kandel, Shmuel and Robert Stambaugh, 1989, Modeling Expected Stock Returns for Long and
Short Horizons, CRSP Working Paper, University of Chicago.
Kandel, Shmuel and Robert Stambaugh, 1990, Expectations and Volatility of Consumption
and Asset Returns, R e v i e w o f F i n a n c i a l S t u d i e s 3, 207-232.
Kothari, S.P., and J. Shanken, 1995, Book-to-market, Dividend Yield, and Expected Market
Returns: A Time-series Analysis, University of Rochester manuscript.
Lucas, R. E., Jr., 1978, Asset Prices in an Exchange Economy,

E c o n o m e tr ic a

46, 1426-14.

Luttmer, Erzo. G. J., 1995, Asset pricing in economies with frictions, University of Chicago
manuscript.
Marshall, David, 1994, Asset Return Volatility with Extremely Small Costs of Consumption
Adjustment, Federal Reserve Bank of Chicago Working Paper No. WP-94-23.
Marshall, David and Nayan Parekh, 1994, The Effect of Costly Consumption Adjustment on
Asset Price Volatility, Federal Reserve Bank of Chicago Working Paper No. WP-94-21.
Mehra, Rajnish and Edward Prescott, 1985, The Equity Premium: A Puzzle,
M o n e t a r y E c o n o m i c s 15, 145-161.
Telmer, Chris I., 1993, Asset Pricing Puzzles and Incomplete Markets,
48, 1803-1832.

J o u r n a l o f F in a n c e

Weil, P., 1989, The Equity Premium Puzzle and the Risk-Free Rate Puzzle,
M o n e t a r y E c o n o m i c s 24, 402-421.




34

Journal o f

Journal o f

A pp en dix

A

C on stru ction o f th e D a ta

The total quarterly real non-durable, durable, services, and total consumption series, the defla­
tors for each of the three components series, and the population series (GPOP) were extracted
from CITIBASE for the 1947:1-1994:1 period.
Monthly VW index returns were obtained from CRSP, and were cumulated to obtain quarterly
returns. One month T-Bill returns were taken from the CRSP RISKFREE file. One, two, and
three year nominally risk-free rates were computed as the returns to one-, two-, and three-year
zero-coupon bonds, computed from the Fama-Bliss data in CRSP.
The default spread, term spread, and dividend yield are calculated following Fama and French
(1989), using data supplied by Roger Ibbotsen.




35

Table 3.1: N um ber of non-positive-definite approxim ate covariance m atrices gener­
ated by the linear model (3.5) and (3.8)
For each m odel, th is gives th e n u m b er of observations for w hich we o b tain ed non-positive-definite
e stim a te s for th e m a trix
defined in 3.6. T h e colum ns labeled ’’N ondur 4- Serv.” are for th e m odels
w here co n su m p tio n is m easu red as ex p en d itu res on consum er no n d u rab les plus services; th e colum ns
labeled ” N o n d u rab les” are for th e m odels w here con su m p tio n is m easured as n o n d u rab le con su m p tio n
ex p en d itu res; th e colum ns labeled ” T o ta l” are for th e m odels w here con su m p tio n is m easured as to ta l
co n su m p tio n ex p en d itu res. In th e C o n stan tin id es m odel, th e m ax im u m values of 7 for which all m arg in al
u tilities were p ositive equaled 12 for n o n d u rab les plus services, 9 for n o n d u rab le co n su m p tio n , and 11
for to ta l co n su m p tio n .




7

20
30
40
50

7
2
6
10
14

7
1
5
7
9
11
12

Time-Separable Preferences
Nondurables
Nondur. + Serv.
Total
Horizon (Yrs.)
Horizon (Yrs)
Horizon (Yrs)
0.25 1 2
3 0.25 1 2 3 0.25 1 2 3
43 1 0
0
46 1 1 0
71 3 0 0
44 1 0
0
47 3 2 0
64 1 0 0
48 2 0
0
47 5 1 0
63 2 0 0
52 2 0
44 5 1 0
0
63 1 0 0
Abel Preferences
Nondur. + Serv.
Nondurables
Total
Horizon (Yrs.)
Horizon (Yrs)
Horizon (Yrs)
0.25 1 2
3 0.25 1 2 3 0.25 1 2 3
49 1 0
0
46 1 0 0
59 1 0 0
67 0 0 0
48 1 0
0
46 2 2 0
47 1 0
67 0 0 0
0
49 5 2 0
47 1 0
0
51 5 0 0
65 1 0 0
Constantinides D
re1'erences
Nondur. + Serv.
Nondurables
Total
Horizon (Yrs)
Horizon (Yrs.)
Horizon (Yrs)
0.25 1 2
3 0.25 1 2 3 0.25 1 2 3
57 5 0 0
49 2 0
0
53 1 0 0
0
57 1 0 0
63 1 0 0
50 1 0
0
55 1 0 0
63 3 0 0
53 0 0
54 0 0
64 2 0 0
0
52 2 0 0
0
56 0 0
66 5 0 0
0
56 1 2

36

Table 4.1: T ests of E q u ity -P re m iu m M odel: N ondurables + Services
T h is tab le displays m ean s and variances of E P £ A, along w ith the corresponding m om ents of the the_—

y

oretical equity p rem iu m E P t im plied by the tim e-separable m odel, th e Abel (1990) m odel, and the
C o n stan tin id es (1990) m odel. Each m odel is ev aluated a t four values for the cu rvature p a ra m e te r 7 ,
w ith co n su m p tio n m easured as ex p en d itu res on no n d u rab les plus services. T h e num bers in p a re n th e ­
ses are asy m p to tic p-values testin g w hether the m eans and variance of E P £ A equal the corresponding
-

y

m om en ts of E P t . Specifically, th e num bers in parentheses the tw o-sided p-values for the sta n d a rd
no rm al d istrib u tio n evalu ated a t Z mean (for th e colum ns labeled ” m e a n ” ) and Z var (for th e colum ns
labeled ” var” ), as defined in eq u atio n s (4.6) and (4.7)

HORIZON
7

0.0621
20
30
40
50

2
6
10
14
7
9
11
12

2-year
3-year
var
mean
mean
var
var
mean
Estimated Equity Premium
0.0556
0.0036
0.0075
0.0551
0.0015
0.0478
0 .0 0 1 1
Time-Separable Model, Nondurables + Services
0.00002
0.0063 0.00001
0.0043 0.00000
0.0010 0 . 0 0 0 0 0
(0.0000) (0.0000) (0.0243) (0.0000) (0.2569) (0.0000) (0.3616)
0.0084 0.00002
0.00005
0.0049 0.00000
0.0009 0.00000
(0.0000) (0.0000) (0.0247) (0.0000) (0.2571) (0.0000) (0.3616)
0.00002
0.00010
0.0100
0.0050 0.00000
0.0007 0.00000
(0.0241) (0.0000) (0.2573) (0.0000) (0.3616)
(0.0000) (0.0001)
0.00010
0.0111
0.00003
0.0049 0.00000
0.0006 0.00000
(0.0000) (0.0001) (0.0256) (0.0000) (0.2573) (0.0000) (0.3616)
Abel Model, None urables + Services
0.0032 0.00000
0.00000
0.0043 0.00000
0.0010 0.00000
(0.0002) (0.0000) (0.0342) (0.0000) (0.2300) (0.0000) (0.3076)
0.00008
0.0130 0.00003
0.0101
0.00001
0.0032
0.00000
(0.0002) (0.0352) (0.0000) (0.2361) (0.0000) (0.3081)
(0.0002)
0.00020
0.0215
0.00008
0.0170 0.00005
0.0044
0.00000
(0.0002) (0.0026) (0.0371) (0.0003) (0.2512) (0.0000) (0.3087)
0.00040
0.0310 0.00018
0.0255 0.00012
0.0049
0.00000
(0.0002) (0.0250) (0.0415) (0.0054) (0.2832) (0.0000) (0.3093)
Constantinic es Model, Vondurables + Services
0.00008
0.0155 0.00003
0.0139 0.00003
0.0039
0.00000
(0.0001) (0.0005) (0.0209) (0.0000) (0.2457) (0.0000) (0.2813)
0.00014
0.0203 0.00005
0.0187 0.00005
0.0046
0.00000
(0.0001) (0.0025) (0.0223) (0.0002) (0.2612) (0.0000) (0.2816)
0.0257 0.00008
0.00020
0.0246 0.00009
0.0052 0.00000
(0.0002) (0.0123) (0.0248) (0.0028) (0.2867) (0.0000) (0.2821)
0.00024
0.0282 0.00012
0.0288 0.00011
0.0055 0.00000
(0.0002) (0.0267) (0.0269) (0.0103) (0.3053) (0.0000) (0.2825)

quarterly
mean
var

0.0037
(0.0000)
0.0051
(0.0001)
0.0063
(0.0001)
0.0072
(0.0001)
0.0023
(0.0003)
0.0069
(0.0008)
0.0109
(0.0019)
0.0144
(0.0036)
0.0087
(0.0018)
0.0109
(0.0028)
0.0130
(0.0043)
0.0140
(0.0052)




1-year

37

Table 4.2: Tests of E q u ity -P re m iu m M odel: N ondurable C o n su m p tio n

This table displays means and variances of E P ^ A , along with the corresponding moments of the theoretical equity premium E P t implied by the time-separable model, the Abel (1990) model, and the
Constantinides (1990) model. Each model is evaluated at four values for the curvature parameter 7 ,
with consumption measured as expenditures on nondurables services.
The numbers in parentheses are asymptotic p-values testing whether the means and variance of E P { A
--- A
equal the corresponding moments of E P t : Specifically, the numbers in parentheses the two-sided pvalues for the standard normal distribution evaluated at Z m ean (for the columns labeled ’’mean”) and
Z var (for the columns labeled ”var”), as defined in equations (4.6) and (4.7)
_____ T

t

HORIZON
7

quarterly
mean
var
0.0621

20
30
40
50

2
6
10
14

1
5
7
9

0.0045
(0.0001)
0.0062
(0.0001)
0.0076
(0.0001)
0.0084
(0.0002)
0.0033
(0.0004)
0.0096
(0.0014)
0.0151
(0.0041)
0.0198
(0.0095)
0 .0 0 2 1

(0.0006)
0.0116
(0.0035)
0.0173
(0.0094)
0.0277
(0.0472)




1-year
2-year
3-year
mean
var
mean
var
mean
var
Estimated Equity Premium
0.0075
0.0556
0.0036
0.0551
0.0015
0.0478
0 .0 0 1 1
Time-Separable Model, Nondurable Consumption
0.00007
0.0107 0.00002
0.00002
0.0105
0.0030
0 .0 0 0 0 0
(0.0000) (0.0000) (0.0150) (0.0000) (0.2737) (0.0000) (0.3456)
0.00004
0.00015
0.0173 0.00004
0.0161
0.0040
0.00000
(0.0000) (0.0004) (0.0157) (0.0000) (0.2848) (0.0000) (0.3458)
0.00026
0.0250 0.00009
0.00008
0.0049
0.0236
0.00000
(0.0000) (0.0044) (0.0172) (0.0009) (0.3056) (0.0000) (0.3459)
0.00017
0.00039
0.0348 0.00017
0.0351
0.0058
0.00000
(0.0000) (0.0490) (0.0200) (0.0353) (0.3525) (0.0000) (0.3460)
Abel Model, Nondurable Consumption
0.00002
0.0066
0.00000
0.0054
0.00000
0.0011
0.00000
(0.0000) (0.0323) (0.0000) (0.2399) (0.0000) (0.3105)
(0.0001)
0.00014
0.0225
0.00011
0.0221
0.00007
0.0058
0.00000
(0.0001) (0.0038) (0.0373) (0.0017) (0.2712) (0.0000) (0.3112)
0.00039
0.0435 0.00043
0.0523
0.00000
0.00033
0.0109
(0.0001) (0.2509) (0.0779) (0.7616) (0.4036) (0.0004) (0.3129)
0.00086
0.0766
0.00145
0.1199
0.00294
0.0178
0.00000
(0.0002) (0.2418) (0.3493) (0.0000) (0.8962) (0.0044) (0.3166)
Constantinides Model, IN
ondurable Consumption
0.0033 0.00000
0.0027
0.00001
0.00000
0.0002
0.00000
(0.0002) (0.0000) (0.0110) (0.0000) (0.1971) (0.0000) (0.2693)
0.0241
0.0239 0.00008
0.00031
0.00008
0.00000
0.0063
(0.0002) (0.0165) (0.0118) (0.0092) (0.2342) (0.0002) (0.2698)
0.00067
0.0414 0.00016
0.0462
0.00022
0.0102
0.00000
(0.0004) (0.3481) (0.0149) (0.6804) (0.3215) (0.0009) (0.2706)
0.00143
0.0803 0.00082
0.1016 0.00081
0.0177
0.00000
(0.0015) (0.0322) (0.0603) (0.0000) (0.7086) (0.0120) (0.2723)

38

Table 4.3: Tests of E quity-Prem ium Model: Total Consum ption Expenditures
T h is tab le displays m eans and variances of EP£A , along w ith the corresponding m om ents of the the—

y

oretical equity p rem iu m E P t im plied by the tim e-separable m odel, the Abel (1990) m odel, and the
C o n stan tin id es (1990) m odel. Each m odel is evaluated a t four values for the cu rv atu re p aram eter 7 , w ith
co n su m p tio n m easured as to ta l co n su m p tio n expenditures. T h e num bers in parentheses are asy m p to tic
p-values testin g w hether th e m eans and variance of E P ^ equal the corresponding m om ents of E P t :
Specifically, th e n u m b ers in parentheses th e two-sided p-values for the sta n d a rd norm al d istrib u tio n
ev aluated a t Z mean (for th e colum ns labeled ’’m ean ” ) and Z var (for the colum ns labeled ” var” ), as
defined in eq u atio n s (4.6) an d (4.7)

HORIZON
7

quarterly
mean
var
0.0621

20
30
40
50

0.0075

0 .0001

0.00006
(0.0000)
0.00013
(0.0000)
0.00024
(0.0000)
0.00040
(0.0001)

(0.0000)
0.0010
(0.0000)
0.0029
(0.0000)
0.0057
(0.0002)

2

0.0005
(0.0002)

6

0.0001

10
14

1

5
9
11

(0.0002)
0.0034
(0.0001)
0.0110
(0.0000)
0.0003
(0.0002)
0.0000

(0.0002)
0.0046
(0.0001)
0.0094
(0.0000)




0.00003
(0.0004)
0.00020
(0.0004)
0.00059
(0.0006)
0.00126
(0.0014)
0.00000

(0.0002)
0.00016
(0.0002)
0.00055
(0.0003)
0.00088
(0.0004)

2-year
1-year
mean
var
mean
var
Estimated Equity Premium
0.0556
0.0036
0.0551
0.0015
Time-Separable Model, Total Consumption
0.0074 0.00002
0.0048
0.00000
(0.0000) (0.0340) (0.0000) (0.2649)
0.0095
0.00003
0.0058
0.00001
(0.0000) (0.0352) (0.0000) (0.2667)
0.0109 0.00005
0.00002
0.0065
(0.0000) (0.0366) (0.0000) (0.2695)
0.0121
0.00006
0.0071
0.00003
(0.0001) (0.0382) (0.0000) (0.2723)
Abel Model, Total Consumption
0.0056
0.00000
0.0039 0.00000
(0.0000) (0.0347) (0.0000) (0.2236)
0.0171
0.00006
0.0135
0.00004
(0.0008) (0.0399) (0.0000) (0.2405)
0.0298
0.00017
0.0283
0.00021
(0.0264) (0.0520) (0.0090) (0.3178)
0.0465
0.00048
0.0561
0.00082
(0.3917) (0.1084) (0.9786) (0.6614)
Constantinides Mode , Total Consumption
0.0026
0 .0 0 2 2
0.00000
0.00000
(0.0000) (0.0202) (0.0000) (0.2122)
0.0146
0.00007
0.0142
0.00005
(0.0006) (0.0240) (0.0000) (0.2354)
0.0287 0.00026
0.0339
0.00027
(0.0392) (0.0422) (0.0486) (0.3628)
0.0381
0.00047
0.0520
0.00061
(0.2159) (0.0750) (0.9417) (0.5963)

39

3-year
mean
var
0.0478

0 .0 0 1 1

0.0013
(0.0000)
0.0013
(0.0000)
0.0013
(0.0000)
0.0012
(0.0000)

0 .0 0 0 0 0

0.0012
(0.0000)
0.0038
(0.0000)
0.0061
(0.0000)
0.0087
(0.0002)
0.0003
(0.0000)
0.0034
(0.0000)
0.0055
(0.0000)
0.0065
(0.0001)

(0.3377)
0.00000

(0.3379)
0.00000

(0.3381)
0.00000

(0.3381)
0.00000

(0.3075)
0.00000

(0.3086)
0.00000

(0.3107)
0.00000

(0.3147)
0.00000

(0.2699)
0.00000

(0.2709)
0.00000

(0.2734)
0.00000

(0.2752)

Table 4.4: Tests of R isk fre e R a te M odel: N ondurables + S ervices
T h is ta b le d isp lay s m ean s and variances of th e nom inal risk-free ra te R F f A , along w ith th e corre——

t

A

sp o n d in g m o m e n ts o f th e th eo retical risk-free ra te R F t im plied by th e tim e-sep arab le m odel, th e Abel
(1990) m odel, an d th e C o n stan tin id es (1990) m odel. Each m odel is ev aluated a t four values for th e
cu rv a tu re p a ra m e te r 7 , w ith co n su m p tion m easured as ex p en ditures on n o n d u rab les plus services. T h e
n u m b ers in p aren th eses are asy m p to tic p-values testing w hether th e m eans and variance of R F £ A equal

—

th e co rresp o n d in g m o m e n ts o f R F t : Specifically, the num bers in parentheses th e tw o-sided p-values
for th e s ta n d a rd n o rm a l d istrib u tio n ev aluated analogously to Z mean (for th e colum ns labeled ’’m ean ” )
an d Z var (for th e colu m n s labeled ” v ar” ), as defined in eq uations (4.6) an d (4.7)

HORIZON
7

quarterly
mean
var
0.0541

20
30
40
50

2
6
10
14

7
9
11
12

0.4089
(0.0001)
0.5783
(0.0000)
0.7388
(0.0000)
0.8905
(0.0000)
0.0793
(0.0120)
0.1252
(0.0041)
0.1352
(0.0573)
0.1094
(0.3815)
0.1285
(0.0163)
0.1312
(0.0549)
0.1226
(0.1703)
0.1135
(0.2827)




1-year
2-year
3-year
mean
var
mean
var
mean
var
Observed Risk-Free Rate
0.00084
0.0608 0.00096
0.0630 0.00097
0.0664
0.00088
Time-Separable Model, Nondurables + Services Consumption
0.3970
0.0184
0.0253
0.3949
0.0061
0.4055
0.0026
(0.2155) (0.0000) (0.2160) (0.0000) (0.6061) (0.0000) (0.8592)
0.0617
0.5546
0.0450
0.5425
0.0139
0.5585
0.0053
(0.1680) (0.0000) (0.1619) (0.0000) (0.5164) (0.0000) (0.8073)
0.1184
0.7001
0.0868
0.6685
0.0239
0.6899
0.0076
(0.1406) (0.0000) (0.1319) (0.0000) (0.4661) (0.0000) (0.7941)
0.8348
0.0354
0.1999
0.1503
0.7713
0.7977
0.0087
(0.1209) (0.0000) (0.1111) (0.0000) (0.4297) (0.0000) (0.8000)
Abe Model, b ondurables + Services Consumption
0.0782
0.0009
0.0043
0.0766
0 .0 0 0 1
0.0790
0 .0 0 0 0
(0.0031) (0.0394) (0.9453) (0.0999) (0.4508) (0.1354) (0.5159)
0.1242
0.0445
0.1220
0.0099
0.1368
0.0006
0 .0 0 0 0
(0.0000) (0.0001) (0.0320) (0.0000) (0.7987) (0.0000) (0.5513)
0.1354
0.1307
0.0293
0.1437
0.1757
0.0018
0 .0 0 0 1
(0.0000) (0.0063) (0.0024) (0.0000) (0.7089) (0.0000) (0.6315)
0.0585
0.2936
0.1039
0.1343
0.1958
0.0036
0.0002
(0.0000) (0.2362) (0.0001) (0.0000) (0.3752) (0.0000) (0.7249)
Constantinides Mot el, Nondurables + Services Consumption
0.1264
0.1157
0.0633
0.0178
0.0009
0.1446
0 .0 0 0 0
(0.0000) (0.0077) (0.0043) (0.0000) (0.9827) (0.0000) (0.6540)
0.1290
0.1090
0.1148
0.0299
0.0015
0.1612
0 .0 0 0 1
(0.0000) (0.0430) (0.0010) (0.0000) (0.7203) (0.0000) (0.6955)
0.1030
0.0449
0.1205
0.0024
0.1722
0.1703
0.0002
(0.0000) (0.2069) (0.0002) (0.0000) (0.4813) (0.0000) (0.7298)
0.0534
0.1116
0.2078
0.0926
0.0029
0.1754
0.0002
(0.0000) (0.3963) (0.0000) (0.0000) (0.3604) (0.0000) (0.7423)

40

Table 4.5: Tests of R isk -F re e R a te M od el: N ondurable Consum ption
T h is tab le displays m eans and variances of the n o m inal risk-free ra te R F ^ A , along w ith the corre---r>4

spo n d in g m o m en ts of th e th eo retical risk-free ra te R F t im plied by the tim e-separable m odel, the Abel
(1990) m odel, and th e C o n stan tin id es (1990) m odel. Each m odel is evaluated a t four values for the
cu rv atu re p a ra m e te r 7 , w ith con su m p tio n m easured as ex p enditures on nondurables. T he num bers in
parentheses are a sy m p to tic p-values testin g w hether th e m eans and variance of R F £ A equal the cor—

y yj

resp o n d in g m o m en ts o f R F t : Specifically, the num bers in parentheses the two-sided p-values for the
sta n d a rd n o rm al d istrib u tio n ev aluated analogously to Z mean (for the colum ns labeled ’’m ean ” ) and
Z var (for th e colum ns labeled ” v ar” ), as defined in eq u atio n s (4.6) and (4.7)

HORIZON
7

quarterly
mean
var
0.0541

20
30
40
50

2
6
10
14

1
5
7
9

0.2250
(0.0001)
0.2873
(0.0000)
0.3292
(0.0000)
0.3508
(0.0002)
0.0554
(0.9097)
0.0259
(0.3806)
-0.0893
(0.0153)
-0.2905
(0.0003)
0.0489
(0.3889)
0.0211
(0.1881)
-0.0618
(0.0021)
-0.3320
(0.0000)




1-year

2-year
3-year
var
mean
var
mean
var
Observed Risk-Free Rate
0.00084
0.0608 0.00096
0.0664 0.00088
0.0630 0.00097
Time-Separable Model, Nondurable Consumption
0.2062
0.0054
0.0424
0.0250
0.2100
0.2159
0.0016
(0.0003) (0.0000) (0.0051) (0.0000) (0.2570) (0.0000) (0.8393)
0.2324
0.2500
0.0631
0.0128
0.1039
0.2646
0.0033
(0.0001) (0.0000) (0.0024) (0.0000) (0.1133) (0.0000) (0.6770)
0.0234
0.2675
0.1283
0.2280
0.2001
0.2870
0.0045
(0.0000) (0.0000) (0.0030) (0.0000) (0.0572) (0.0000) (0.5999)
0.2407
0.1876
0.1907
0.0400
0.3395
0.2849
0.0046
(0.0004) (0.0002) (0.0000) (0.0300) (0.0000) (0.5692)
(0.0000)
Abel Model, Nondurable Consumption
0.0524
0.0016
0.0524
0.0127
0 .0 0 0 1
0.0563
0 .0 0 0 0
(0.0000) (0.3302) (0.5888) (0.1937) (0.4896) (0.3312) (0.5201)
0.0197
0.0344
0.0017
0.1308
0.0201
0.0678
0 .0 0 0 0
(0.0000) (0.0316) (0.0000) (0.0034) (0.5994) (0.7912) (0.5169)
0.0652
-0.0876
-0.0551
0.0074
0.4030
0.0488
0 .0 0 0 0 0
(0.0000) (0.0000) (0.0000) (0.0000) (0.0017) (0.7188) (0.5270)
-0.2656
0.1537
-0.2273
0.0237
0.9323
0.0027
0 .0 0 0 0 0
(0.0009) (0.0000) (0.0000) (0.0000) (0.0098) (0.0000) (0.5498)
Constantinides Model, b ondurable Consumption
0.0462
0.0025
0.0007
0.0480
0 .0 0 0 0
0.0500
0 .0 0 0 0
(0.0633) (0.0210) (0.8545) (0.0200) (0.5030) (0.0461) (0.5738)
0.0014
0.0921
0.0040
0.0217
0.0226
0.0618
0 .0 0 0 0
(0.0000) (0.0016) (0.0000) (0.0000) (0.6653) (0.4573) (0.5600)
-0.0452
-0.0791
0.0477
0.0041
0.2071
0.0470
0 .0 0 0 0
(0.0000) (0.0000) (0.0000) (0.0000) (0.0563) (0.0141) (0.5662)
-0.2104
0.0114
-0.2929
0.0998
0.4651
-0.0031
0.0002
(0.0000) (0.0000) (0.0000) (0.0000) (0.1001) (0.0000) (0.6362)
mean

41

Table 4.6: Tests of R isk -F re e R a te M odel: Total C o n su m p tio n E x p e n d itu re s
T his tab le d isplays m eans and variances of the nom inal risk-free ra te R F { A, along w ith th e corre-

TA

sponding m o m en ts of th e th eo retical risk-free ra te R F t im plied by th e tim e-sep arab le m odel, th e Abel
(1990) m odel, an d th e C o n stan tin id es (1990) m odel. Each m odel is ev aluated a t four values for th e
c u rv atu re p a ra m e te r 7 , w ith co n su m p tio n m easured as to ta l co nsum ption expenditures. T h e nu m b ers
in parentheses are a sy m p to tic p-values testing w hether th e m eans and variance of R F £ A equal th e
-- rA
-

corresponding m o m en ts of R F t : Specifically, the num bers in parentheses th e tw o-sided p-values for
th e sta n d a rd n o rm al d istrib u tio n ev alu ated analogously to Zmean (for th e colum ns labeled ’’m e a n ” ) and
Zx
,ar (for th e colum ns labeled ” var” ), as defined in eq uations (4.6) an d (4.7)

HORIZON
7

quarterly
mean
var
0.0541

20
30
40
50

2
6
10
14

1
5
9
11

0.00084

0.4176
(0.0000)
0.5776
(0.0000)
0.7189
(0.0000)
0.8421
(0.0000)

0.0601
(0.0360)
0.1503
(0.0211)
0.2991
(0.0132)
0.5302
(0.0085)

0.0745
(0.1552)
0.0833
(0.4863)
0.0147
(0.6104)
-0.2080
(0.0642)

0.0157
(0.0000)
0.1658
(0.0002)
0.6241
(0.0101)
0.9726
(0.0000)

0.0589
(0.5283)
0.0807
(0.2581)
0.0311
(0.9376)
-0.0381
(0.6183)

0.0035
(0.0050)
0.1095
(0.0000)
0.4438
(0.0000)
0.7769
(0.0000)




1-year

2-year
mean
var
mean
var
Observed Risk-Free Rate
0.0608
0.00096
0.0630 0.00097
Time-Separable Model, Total Consumption
0.3934
0.3906
0.0329
0.0091
(0.0000) (0.1160) (0.0000) (0.5175)
0.5287
0.0744
0.5125
0.0187
(0.0000) (0.0916) (0.0000) (0.4504)
0.5967
0.6336
0.1291
0.0281
(0.0000) (0.0784) (0.0000) (0.4160)
0.7070
0.1961
0.6459
0.0353
(0.0000) (0.0720) (0.0000) (0.3934)
Abel Model, Total Consumption
0.0729
0.0023
0.0728
0.0001
(0.2225) (0.3384) (0.2395) (0.4924)
0.0012
0.0202
0.0801
0.0929
(0.3724) (0.0002) (0.0020) (0.8911)
0.0029
0.0430
0.0465
0.0029
(0.0546) (0.0000) (0.1528) (0.2025)
-0.1695
0.0595
-0.0751
0.0060
(0.0000) (0.0000) (0.0000) (0.0255)
Constantinides Mode , Total Consumption
0.0010
0.0602
0.0583
0.0001
(0.6193) (0.9352) (0.4034) (0.5000)
0.0775
0.0252
0.0880
0.0018
(0.3748) (0.0003) (0.0280) (0.5846)
0.0221
0.0796
0.0382
0.0057
(0.5190) (0.0000) (0.0481) (0.0237)
-0.0446
0.1198
0.0094
-0.0256
(0.0908) (0.0000) (0.0000) (0.0001)

42

3-year
mean
var
0.0664

0.00088

0.3942
(0.0000)
0.5191
(0.0000)
0.6095
(0.0000)
0.6679
(0.0000)

0.0034
(0.8049)
0.0069
(0.7461)
0.0097
(0.7316)
0.0106
(0.7436)

0.0777
(0.1858)
0.1212
(0.0000)
0.1262
(0.0000)
0.0949
(0.0027)
0.0616
(0.4374)
0.1102
(0.0000)
0.1179
(0.0000)
0.1046
(0.0000)

0.0000

(0.5196)
0.0000

(0.5433)
0.0000

(0.5780)
0.0001

(0.6167)
0.0000

(0.5647)
0.0001

(0.5995)
0.0001

(0.6038)
0.0001

(0.5908)