HD28
.M414
no.
3jOO
'
^1
On
the Volatility of Stock Market Prices
Rajnish Mehra'
March 19S9
Revised December 1990
University of California. Santa Barbara
*I
am
especiaUy indebted to Edward C. Prescott for his helpful suggestions.
I
would
like to
thank
John Donaldson, Daniel Holland. Rodolfo Manuelli, Robert Taggart and Danny Quah for helpful
I also thank the Wharton School of the University of Pennsylvania for a productive
sabbatical during which part of this research was completed. The usual caveat applies.
discussions.
1
Introduction
The majority
of studies to date dealing with stock price volatility have been micro-studies.*
This line of research has
origins in the important early
its
and Porter (1981), which found evidence of excessive
reported that,
in his
and LeRoy
volatility of stock prices relative to
model, the volatility of actual stock prices exceeded the
upper bound by a factor of
theoretical
Shiller (1981)
Using data for a hundred years, Shiller (1981)
the underlying dividend/earnings process.
in particular,
work of
5.59.
These studies use a constant
interest rate,
an
assumption subsequently relaxed by Grossman and Shiller (19S1) who addressed the issue
of varying interest rates.
Shiller's
They concluded
that although this reduced the excess volatility,
conclusion could not be overturned for reasonable values of the coefficient of relative
risk aversion.
The conclusions
above cited studies have been challenged
of the
in recent years,
most
notably by Flavin (1983), Kleidon (1986) and Marsh and Merton (1986). These challenges
appear to have merit. The essence of their criticism
intervals
wide and sensitive to trend.
movements
bound
in
dividends.
that the tests are biased, the confidence
They emphasize the importance of low frequency
and LeRoy (1990)
Gilles
in their critical
literature point out that Shiller's volatility tests are likely to
process generating dividends
later variance
bound
is
tests of
of sampling variability.
The
is
if
the stochastic
inappropriate.
The
West (1986) and Mankiw, Romer and Shapiro (1985) are
interested reader
(19S9) for a detailed overview. Gilles and
robust
review of the variance
be biased
such that the detrending procedure
unbiased but essentially inconclusive because,
is
is
like Shiller's tests,
is
referred to Gilles
LeRoy conclude
".
.
.
they leave open the question
and LeRoy (1990) or
Shiller
This finding of excess volatility
....''
This paper shifts the focus of analysis from the firm to the aggregate level and comple-
ments the work by Grossman and
we choose
to
examine
Shiller (1981).
Rather than studying individual
issues of volatility utilizing aggregate stock
gate after-tax net cash flows as a ratio to National Income.
'a notable exception
is
Grossman and
Shiller (1981).
securities,
market values and aggre-
Our approach
is
in the tradition of
the infinitely-lived classical growth model of Solow, where the behavior of capital, consumption
and investment are studied
as shares of output, bearing in
model
regularities of their ratios (Solow (1970)). This
sively by.
among
a central construct in contemporary
is
and business cycle theory^ and
finance, public finances
mind the well-documented
its
variants have been used exten-
and Kotlikoff (1987), Barro and Becker
others, .A,bel et al (19S9), .Auerbach
and Ross (1985), Lucas (1988), Kydland and Prescott
(19SS), Brock (1979), Cox. IngersoU
(1982) and Merton (1971).
This paradigm has several advantages:
The
partial equilibrium
micro studies cited earlier
ignore the interaction of consumption growth and interest rates, implicitly assuming their in-
dependence. In contrast, the neoclassical growth model explicitly captures their interaction.
Secondly, examining aggregate values relative to National Income
ical setting since
The
detrending
is
we observe
Income has moved by a factor
1974.
Furthermore, there
is
of
economy
National Income
In this paper,
in
a fair
in 196S.
amount
years 1946-
in
1948
dropping down to 0.53 of National Income in
of persistence in the
1 ).
plo.t
of the ratio of
market
During the same period, the share
been relatively stable (approximately 2.5%), ranging from 2.67% of
1948 to 2.91% in 1968 and 2.02%.
we analyze the behavior of equity
the question as to whether this behavior
is
in
1974 (see figure
consists of
2).
as a ratio of National Income.
a) consistent with the
construct; b) confirms the challenges to Shiller (1981) and
The study
War
about three, from a low of 0.48 of National Income
value of equity to National Income vs. time (see figure
of claims to equity has
for the post
that the value of equity in the U.S. as a ratio of National
Income
to a high of 1.33 of National
natural in this theoret-
not a problem as these series appear to be co-integrated.
principal results of our study cover the U.S.
1987. During this period
is
We address
standard neoclassical growth
LeRoy and Porter
two main parts. To build some intuition, we
first
(1981).
address these
issues in a deterministic steady state context.
Next we extend our analysis to stochastic
models with low frequency movements to gauge
if
of Equity (e) to National
Income
(y).
In addition,
-Barro and Becker (1988) provide a justification
tion of a dynastic utility function.
this implies large
we
movements
will also discuss
for the infinitely lived
in the ratio
the implied relation
family construct in their formula-
between growth rates and e/y. The paper
is
organized as follows: Section 2 summarizes
the U.S. historical experience for the period 1946-87.
Section 3 deals with deterministic
steady state models. Section 4 examines the extension to stochastic economies and section
5 concludes the paper.
Data
2
The data used
in this
paper consists of a
set of series for the
period 1946-1987. These are
individually described below.
(i)
(ii)
Series y: .\ational
Series ry: Real per capita National Income. This
and the
(iii)
Income data; obtained from The Economic Report of
Series
GNP
e:
deflator from the
is
series y divided
the President.
by the population
Economic Report of the President.
Market Value of Equity: obtained from the Board of Governors publication,
Flow of Funds Accounts Financial Assets and
of Governors data for 1982-1987
is
Liabilities
Year-End Values. The Board
obtained by multiplying the value of equity traded
on major exchanges by 1.25 to adjust for privately held corporations. For the period
1946-19S1. the
SEC
privately held stocks.
supplied the estimates for the total value of equity including
.After
1981 (when the
SEC
discontinued supplying the data),
the Board of Governors used the ratio of total value of equity to the market value of
traded equity (which was 1.25 in 1981) to adjust subsequent
of the present study, which uses data only until 1987, the
held stocks,
it
in
For the purpose
above may be an adequate
approximation. However, with the proliferation of Leveraged
1988 and the subsequent change
data."'
Buy Out
activity since
the ratio of publicly traded equity and privately
probably understates the true value. The ratio of e/y
is
plotted in figure
1.
(iv)
Series xe:
Extended Market value of Equity. For the period 1945-1987 the values were
taken from the Board of Governors publication:
Flow of Funds Accounts Financial
•'We thank Judy Ziobro, an economist with the Board of Governors, for this information.
&
Assets
Year-End Values and are
Liabilities
identical to series e.
For the period
1929-1944 the values were taken from Holland and Myers (19S4) after an adjustment
discussed below.
Holland and Myers report equity values from 1929-1981
Since there
mean
The
overlapping data from the period 1945-1981 (36 years), we calculated the
value of the ratio of the Holland-Myers data to the Flow of Funds year end data.
value
is
0.644,
the Holland-Myers data
i.e..
with mean 0.644 and variance .00377.
We
data from 1929-1944. The ratio xe/y
is
(v) Series d:
(vi)
is
for nonfinancial corporations.
Series ne:
is
systematically biased
downward
used this value to adjust the Holland-Myers
plotted in figure
4.
Dividend data; obtained from The Economic Report of the President.
Net
New
Equity Issues; obtained from the Board of Governors publication,
Flow of Funds.
(vii)
Series x: .After-Tax
is
(viii)
Cash Flow
to Equity;
computed
Series
c:
Consumption
of
(f
—
ne.
The
ratio of
x/y
Xondurables and Services; obtained from The Economic
to the President.
The study commences with information from 1946
3
=
plotted in figure 2.
Report
'e' is
as x
since reliable data for the series 'ne'
and
unavailable prior to that year.
Deterministic Steady State Analysis
In this
paper we consider the neoclassical growth model with labor-augmenting technological
progress.''
unit orders
The economy we
its
consider has a single representative 'stand-in' household. This
preferences over consumption paths by
f^/3'u{c^),
^In the
Growth Literature
this
is
0</?<l
often referred to as a Harrod neutral technical change.
(3.1)
where
q
R+
R
—*
is
the per capita consumption,
is
the increasing, continuously differentiable concave utility function.
restrict the utility function to
^
the subjective tinie discount factor and u
is
be of the constant relative
u{c,a)
c^-° —
=
1
where the parameter q measures the curvature of the
defined to be logarithmic which
function
is
period
there
We
t,
is
a single good,
y,,
that
is
(1
+
7)
>
1.
(3.2)
When q =
utility function.
,
the utility
A-,,
and
labor,
/<.
effective supply of labor units each period
production function /(t,
In addition the
I
the limit of the above representation. In each
assume that technological change increases the
by the factor
oo
produced using two inputs, capital,
is
further
risk aversion class
< Q <
,
—a
1
We
:
has capital and effective labor units as arguments,
I)
:
R+ x R+
—*
R+ which
strictly increasing, strictly concave,
is
continuously differentiable and exhibits constant returns to scale. Then, letting
i
denote the
gross investment, the optimal growth problem can be written as
TS'^A^IZ}!
max
(P)
{(c.....^.+.)}~o fr'o
s.t.
/,+i
<
=
^"i+i
=
+
Ct
it
^"0, /o
If
we renormalize
all
+
variables by (1
riY
(1
f{ktJt)
{l
+
n)lt
h
-a)
alU
alU
3.11 t
>
given
and assume that
/i(l
+
'?)'""
<
1,
the above
becomes a "standard" optimal growth problem. (See Stokey, Lucas and Prescott
The
solution to (P)
is
thus equivalent to finding a value function
v{k^J,)
=
max
{u(/(A'„/,)
-
i,)
+
3r(7„(I
i-(/:,,/{)
+
rj)l,)}
(1989).)
where
.
(3.3)
0<i,</(fc,,/,)
The
solution to (3.3)
is
a pair of optimal policy functions c(-) and
optimal consum.ption and investment
in
application of these policy functions, the
rate; that
T]
is,
in the
every state.
economy
will
It
is
well
i{-)
known
which determine
that by repeated
converge to a steady state growth
steady state, consumption and investment will each be growing at a rate
every period.
cU^
= {l+r]K
(3-4)
and
=
'7+1
Given
problem being analyzed
(3.4) the
Prescott (1985) and
Mehra
(1
+
'/)';•
isomorphic to the one studied in Mehra and
is
and analysis
(19S8). Using the results
pute the steady state equilibrium risk free
(3.5)
To do
rate.
this, first
paying one unit of consumption each period. The expression
of this security
is
we
in
these papers,
we com-
price the risk free security
for the
equilibrium time
price
given bv
or
E
=
po
^'
Vc?(l+r?)-V
fr{
or
po
-
Such a security can also be expressed
rrj(TT^
(3-^)
•
relative to its implied rate of interest.
Since for a
perpetuity
= -
Po
where
r is
the interest rate in the economy,
r
(3.7)
and
Expression (3.9)
time. This result
is
is
1
by
e"''
-
(3.9)
1
'/)-
the discrete time analogue of a well
known
relationship in continuous
easily derived since
ln(l
/?
+
(3.S).
i
Replacing
we can obtain
=
^(1
by equating
(3.8)
r
+
+
r
r)
= r'i^ + nr
=
r,
=
-ln/3
+
Q7?c
.
(continuous compounding) we obtain
r,
=
p
+
aTic.
(3.10)
The
subscript c denotes continuous compounding. This result can be found in
Kurz (1970), Solow (1970)
Arrow and
we retain the continuous time
or Dixit (1976). For convenience
interest rate expression.
What observable quantities
oped here?
.^t
best correspond to the theoretical valuation expressions devel-
the firm level, the value of a stock
frequently represented as the discounted
is
present value of future dividends. This representation has been used in the work of Shiller
(19S1) and others cited earlier. However, the value of the equity of a firm
present value of
where
cq
is
future dividends,
all
The
not equal to the
i.e.
the current value of the equity of the firm and
dividend paid out at time
is
dt
is
the value of the aggregate
t.
correct expression
is*
—
dt
net
E'^
where ne,
when
is
the net
new equity financing between time
t
—
and
I
t.
Only
in the special case
a firm finances using only retained earnings and neither issues nor repurchases shares,
does (3.11) hold with equality.®
Since data on stock issues and repurchases
cash flow to equity holders
in
the
economy
X
available since 1946,
is
we
calculate the net
as
=
(f
—
ne
where we now interpret d as the aggregate dividend and ne the value of the net new equity
issues.
Hence the aggregate value
of equity in this
economy
satisfies
-fnr^.
In the neoclassical
growth model
be growing at a constant rate
ij.
in
(3.13)
steady state along a balanced growth path, capita]
will
Let capital k be divided into two components, debt capital
^For a comprehensive dbcussion see Miller and Modigliani (1961).
*At the aggregate level, this implies no net stock issue or repurchase
for the firms in the
economy.
b
and equity capital
e,
so that k
and hence the claims to equity
=
will
b
+
e.
If
the debt equity ratio
is
be growing at a constant rate
tj,
specified, then equity
i.e.
xt+i
=
+
j((l
rj).
Substituting in equation (3.13)
xdl + ri)
r-T]
=
Cf
(3.14)
or
=
e/y
r
Can the
-
(3.15)
.
jj
large changes in e/y be accounted for within this standard neoclassical growth
model? In this model, by varying parameters that are exogenous to the miodel, we can have
different values for e/k.
kjy and
r.
Let us e.xamine the effects of each of these on e/y as a
possible explanation.
a)
In
steady state along a balanced growth path,
with capital {k) fixed, then e/y
6
+
=
1
+
6
e
is
high
k
= -,
e
e
.\s
the debt/equity ratio {hie)
low. Since
is
-
a high 6/e implies a low e/t.
if
k
is
and kly
fixed
low. (Figure 5 illustrates the effect of a
e
change
in
is
a constant, this implies e/y
is
dje ratios on e/y for an economy in
steady state.)
Historically for the U.S., the debt/equity ratio (6/e) has steadily increased since 1950.
Taggart (19S5) reports that while
19S0
it
was
% 40%
(see figure 3).'
in
1945 the debt/equity ratio (6/^)
Taggart (19S5) reports,
".
.
.
^^'^s
%
10%,
in
the use of debt financing
has increased considerably in the post war period .... This trend emerges regardless
of the
method
of
measurement employed
."
.
.
.
Does
historical evidence support the
steady state result that e/y and 6/e move inversely? Debt/Equity ratios in the 1980s
were comparable to those
to National
'In a private
Debi+Eqmty
'
in
the late 20s, whereas the ratios of market- value of equity
Income (e/y) were
significantly different (see Figure 4).
During the period
communication, Robert Taggart suggested that
^^\^ was a reasonable approximation for
ratio can be easily calculated. Note that, in this study, we use the
^^^^ which the debt/equitv
quaixiative fact that this ratio has
been increasing
in
the post war period.
8
1950-1970, when 6/e was monotonically increasing, e/y was persistently high
—
in direct
contradiction to our theoretical expectation.
Some
(i)
(ii)
caveats are in order.
We
are implicitly assuming a Miller (1977) model of capital structure.
Inflation tends to lower the value of equity since assets are depreciated
on the
basis of historical cost; on the other hand, the real value of a firm's long-term debt
obligations declines, thereby raising the value of equity.
We implicitly assume that
these effects offset each other.
b) In steady state,
fixed). If bje
is
if
the capital/output ratio (t/y)
6
+
,
1
6
=
+
c)
A
is
fc
= e
account for the movement
is
trendless
and constant and cannot, there-
in e/y.
third implication of the deterministic neoclassical
growth model concerns the
action between real interest rates and consumption growth rates.
growth path,
is
high (given
r
=
p
q >
-\-
0).
t]q
Tj
To summarize,
implying that r
is
high
Hence a high growth rate
6/e are the same). This
a high
large (holding 6/e
which implies e/y and kjy are positively correlated.
Historically, the capital-output ratio [kj-y)
fore,
e
e
e
fixed,
large, then e/y
fixed, then
-
is
is
is
when the growth
(r/)
inter-
Along a balanced
rate of consumption
implies a low e/y (given i/y
not substantiated by our data. During the 1960's
and
we observe
as well as record high values of e/y.
historical
movements
in
e/y cannot be systematically accounted
for in
the
deterministic neoclassical growth model.
4
.\
Stochastic Models
deterministic, steady state analysis does not provide an adequate explanation of the ob-
served large
movements
in the
market value of equity as a share of National Income (e/y). In
an
effort to achieve greater
congruence between theory and observed data, we examine the
behavior of (e/y) in stochastic economies including those with persistent changes, generating
random-walk type behavior of cashflows
We
retain the recursive structure developed in section 3,
invariant
where the economy
is
time
and economic agents solve a similar problem each period. All relevant information
for individual decision
of state variables.
and output
flow
("dividends"').
If
for
making
{c.x.y]
in
such an economy can be characterized by a limited set
the equilibrium stochastic process of consumption, cash
is
such a homogenous consumer economy with specified preferences and
technology then we can determine an equilibrium process {c/y,x/y, y'/y, e/y}.
Consider
an infinitely lived representative individual in such an economy. This individual orders his
preferences over feasible consumption plans by
0<J<1
Eolf^3'u{ct)\
where
£'o{"}
time.
Ct
is
is
the expectation operator conditional upon information available at the present
the per capita consumption and 3 the subjective time discount factor.
period utility function u(-)
deterministic case.
agent, the price of
If
is
assumed
to be identical to the
one considered
The
earlier in the
we assume maximizing behavior on the part of the representative
any asset
(e,)
with a stochastic process {xt] as
its
claims, satisfies the
Euler equation
e.
=
+
J£,M^)
'Ct
—a
l
h^,+x,^,]^.
,,,).
Consequently the equilibrium process {c/y,x/y,y' /y and e/y}
(4.1)
will satisfy
(4.2)
Vt+i
in
!/(+iJ
addition to satisfying other restrictions imposed by technology.
If
we cannot account
for the variation in
e/y without imposing technological restrictions,
then the addition of further restrictions will not change our lesults.
formulation we do not explicitly model the technology.
is
the joint equilibrium process generated by an
economy with
technology that incorporates capital accumulation.
10
Instead,
Hence,
in
our
initial
we assume that {c^x^y}
specified preferences
and a
The
and
economy {i,j,k} follows an independent Markov
state of this
c,jk
be the values of y'/y. x/y and c/y, respectively,
correlations between these variables
where
z
=
Aj, ^,
{I'.j. A:}
and
-jt
follow a
process. Let ^.^k, Xijk
in state {i,j,k}.
To capture the
let
9ijk
=
Crjk
= a^Xj+di
Xijk
=
A,
(4.3)
+
a2Aj
03$,
(4.4)
+
Markov process and are
be the current state and
z'
(4.5)
Ik
For notational simplicity
iid.
be the next period's state.
Using
let
this notation,
equation (4.2) can be rewritten ds
e,
Calibration of the
Step
1:
=
.ij:|;r,,,(^)"'(5,,)^-'*[e.'
+
x.-]|
(4.6)
.
Economy
Calculate the mean, variance and cross covariances of the Markov process {g^^
Cj,
ir},
with respect to their stationary distribution. Match these results to the corresponding
sample moments of the U.S. Economy
calculate the values for ai, 02 and 03.
for the period
1946-1987. This
In addition, since A, 9
is
sufficient to
and 7 are represented
by symmetric 2-state transition matrices, with these calculations we can also compute
the levels of the two states.
Step
2:
To
calculate the persistence parameters (switching probabilities), for each of the
transition matrices,
them
to the
compute the
sample values
Sample Moments
for the U.S.
Variable
serial correlations for
for the U.S.
Economy
each of the variables and match
for the
Economy: 1946-87
x^
e.
y,
c,
Mean
1.016
67.00%
2.52%
83.80%
Standard Deviation
3.33%
2.34%
8.40%
24.37%
11
time period 1946-87.
1.48
X lO""
Cov(^,,^,,)
=
4.56 x 10"*
Cov(^,,x,)
= LOS
X 10"^
Cov
(c,,c,-)
=
3.94
Cov(c,.x,)
=
Cov(i„x,.)
=
5.31 x 10"*
Cov
(g,,c,)
=
6.26 X 10-^
Calibration of the model
=
=
=
0.13
d=
0.54
a,
a2
aj
rr{0)
^1
&2
q
where
p. q
=
=
=
=
economy
x
10"''
yields the following results:
0.01
0.12
A=
2.30 X lO'^
<7(A)
0.56
A,
0.51
A2
0.87
p
and
r are
1.016
=
=
=
=
7
3.33 x lO'^
(7(7)
1.0.50
7,
0.983
72
0.52
r
=
=
=
=
=
-0.05
0.79 x 10"^
-0.042
-0.058
0.8S
switching probabilities for the transition matrices for A, 6 and 7. That
is,
A,
Ai
A2
P
l-l
i
A2 V
1
.Ml the variables in equation (4.2) are in
interested in the ratios of e/y, c/y
-
p
real
p-
)
per capita terms.
and x/y we can use nominal aggregate values
the numerator and denominator without affecting the results.
where we must use the values of
mean and standard
real
The only exception
per capita National Income (Series ry).
deviations of g, are for per capita real National Income.
economic theory typically uses low values of a.
literature, but.
However, since we are
based upon
it,
In this
upper bound q by
10.
in
is
both
y'/y
Hence the
Established
study we do not challenge this vast
For an alternative view, see Kandel
and Stambaugh (1990).
Once the economy has been
calibrated the state contingent values of e/y
(etjjt)
can be
calculated from the following set of equations.
222
e.;Jt
=
-^
5Z
(=1
where
,jk<Plmn
=
9i/
'
P;m
•
H
51
-^ \~"* [e/m„ + X/„„]
/c
^2k4>lmng]~^
m=l n=l
\
rkn-
12
^:k J
(4.7)
Note that equation
and
7.
5
Results
(4.7)
is
linear in
e.^fc,
since
we have assumed
There are eight such equations. These can be solved
2 state matrices for 0, A
for the eight values of
Simulations on our calibrated economy yielded the following results:
Table
i
Q
=
=
1
.96
e.^fc.
our results: In
i.e..
{c/y)
is
the simulations,
all
low, (e/y)
correspondence
is
where consumption
in states
is
low relative to output,
correspondingly low (other variables held constant).
A
similar
observed between (x/y) and (e/y). For every scenario, in states where the
is
cashflow to equity was low relative to output, (e/y) was also low.
To
test the robustness of
We
parameters.
where
p. q
and
r
our results we did a sensitivity analysis by varying various
report results of two polar cases of interest.
=
implying that
.99.
g., c,
First
we
consider the case
and i~ almost follow a random walk.
Table 3
^ =
p
Q =
Mean
(e/y)
=
^
=
.96
r
=
.99
1
2
0.oS4
0.557
0.139
0.274
"0.40-0.77
0.21-1.04
10
4
Equilibrium does
a{ely)
not exist
Range
Xe.xt
of (e/y)
we consider the case where
p, q, r
=
.50,
implying that successive changes
in g^, c,
and X; are independent.
Table 4
/3
p
Q =
Mean
(e/y)
<7(e/y)
Range
Tables 3 and
of (e/y)
4 clearly
=
q
=
=
.96
r
=
0.50
1
2
4
10
0.5S3
0.421
0.2S2
0.181
0.022
0.032
0.043
0.066
0.56-0.61
0.39-0.46
0.23-0.33
0.11-0.26
show that introducing low frequency movements
of output greatly increases the range of values of e/y
frequency movements in any study of
volatility.
14
in the
growth rate
and emphasizes the importance of low
An
(g,)
interesting
and
and intriguing relationship
is
observed between the growth rate of output
For low levels of persistence, irrespective of the
e/y.
rate resulted in a high value for e/y.
economy which displayed
(for the
This
sample period) low
where the growth rates were low, the
a >
of the deterministic neoclassical growth
levels of persistence of g^.
1,
However,
the relationship reversed. In states
was high
ratio (e/y)
a high growth
consistent with observations for the U.S.
is
high levels of persistence and for
for sufficiently
level of a,
— consistent with the implications
model but inconsistent with observations. For the
U.S. data, the range of e/y was large and e/y was positively correlated with the growth rate.
Intuitively,
with high levels of persistence the economy behaves
switching between two growth rates
and
(high)
t/i
rjj
like
a deterministic one,
(low). In a deterministic
economy along
a balanced growth path,
e/y
where we have used equation
We
see that
p
+ {a-l)r}
and the
fact that r
<OifQ>l+/?%l.
J^'
high just as we observe
Our
(3.15)
=
in
=
p
+
arj.
Therefore, the e/y ratio will be low
when
rj
is
our simulations.
simulations indicate that
it is
the low frequency movements in the growth rate that
are important in determining the volatility of stock prices. This
is
in
sharp contrast to the
determinants of the equity premium (see Mehra and Prescott (19S5)) where high frequency
movements
in
the growth rale were crucial.
To
further capture the impHcations of low
frequency movements, consider the following thought experiment.
We
retain
all
the parameters of the calibrated
where output grows
will
at
be the implications
In this case, Aj
=
two
rates,
for our
1.03
3% and
1%,
economy except
for
that
we consider the case
an expected period of 10 years.*
What
model?
and Aj
=
1.01
and
p,
the transition probability satisfies the
relation
"Of
course,
when the process on
g^
changes, the process on consumption
change; see equations (4.3)-(4.5).
15
Cj
and cash flows x,
will also
k=l
or
p
^
=
10-1
=
.9
10
Table 5
=
.3
A,
a =
=
1.03, A2
.96
=
1.01,
p=
.90
market
ais
a share of National Income.
significantly
It
may
from the
levels implied
This suggests that the stock market
why
will
prove more useful in
there are such large movements in the stock market relative to National
Income and only small, relatively transitory movements
Income. This
diverge
by the neoclassical growth model.
be the case that an overlapping generation structure
understanding
may
may be a
fruitful
area for future research.
17
in
earnings as a share of National
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Princeton University (January).
19
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Date Due
Lib-26-67
MIT LIBRARIES
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