Finance 567. Presentation on 20.5.93 by Sanjeev
Sabhlok
DO STOCK PRICES MOVE TOO MUCH
TO BE JUSTIFIED BY SUBSEQUENT CHANGES IN DIVIDENDS?
by Robert J. Shiller, American Economic Review, 1981
INTRODUCTION:
THE EFFICIENCY PUZZLE:
The term "efficient market" was introduced into the
literature by FFJR (1969). There are three types of
efficiency which are usually tested (Fama, 1970):
b)
a)
In the weak form, share prices are thought to fully
reflect all past information. This form of efficiency
has now been re-christened by Fama (1991) as Return
Predictability. The current paper by Shiller has been
classified by Fama as a test of this form of
efficiency.
In
the semi-strong form, share prices are assumed to
reflect
all
past,
plus
all
publicly
available
information. This form of efficiency is now called
Event Studies by Fama (1991).
c)
The strong-form assumes that share prices reflect all
known
information
(including
privately
held,
or
insider information). Fama (1991) now calls this form
"Tests for private information".
The Ball (1990) review shows that the concept of
efficiency, the research designs for testing this concept,
and the evidence thereupon, are anomalous, and puzzling.
Various research designs have been
constructed, with
different criticisms of these methods. We look into a
design called VARIANCE BOUND TESTS, devised by Shiller
(1981), and Stephen LeRoy and Richard Porter (1981). In
this procedure, the bounds of different variances are
determined by theory, and then data is tested for these
bounds.
Shiller's study involves comparing the volatility of
stock prices implied by the volatility of dividends to
observed stock price volatility (Martin:279). "Consider the
following simplified stock valuation model. Define stock
price, P(t), as the present value of future expected
dividends, E(Divk).
P(t) =
_
ð
k=t+1
E(Divk)
------(1 + R)k
Eqn. i
where R s the market's required rate of return
(discount rate) for the stock.
Changes in P(t) over time can result from
a)
variations in expected future dividends
and/or
b)
changes in the discount rate used in
valuing them.
Shiller focused his tests on the former, treating R as
a constant." (Martin)
SHILLER'S PAPER:
DATA SETS USED: For his empirical
different Stock Price indices:
Data set 1:
Data set 2:
study,
he
uses
two
Standard and Poor's Annual series from 18711979.
Modified Dow Jones annual Industrial average
from 1928-1979.
(details in given in the Appendix of Shiller's paper)
Shiller is interested in finding out what explains the
movements in real stock prices. Can these movements be
explained
by
new
information
about
subsequent
real
dividends? He starts off by the assumption that the
expected real returns for the aggregate stock market are
constant through time (or approximately so).
PART I: KEEPING REAL DISCOUNT RATES CONSTANT:
VERSION I OF Efficient Markets model:
Generalising equation i,
_
Pt = ð Et Dt+k
k=0 (1+r)k+1
Eqn. ii.
where Pt is the real price of a share at the beginning
of time period t, r is the constant real interest
rate, Et is the expectation conditional on information
available at time t, and Dt+k is the real dividend
paid at time t+k. E(D) is therefore the expectation of
the real dividend. Here, the return from holding the
stock for one period is r, the real interest rate.
Writing Ó = 1/(1+r), we get
_
Pt = ð Ók+1 Et Dt+k where 0<Ó<1
k=0
where Ók+1
(Eqn. 2)
is the constant real discount factor.
VERSION II: Detrended version of Eqn. 2:
Dividends are expected to grow (at a rate g) in the
future. However, if they are expected to grow perpetually
at a rate higher than the real interest rate then the
present value of the expected future dividends would sum to
infinity (Statement A).
Let
equal 1/(1+g). This is the long-run growth
t-T,
factor of the stock. Now divide both sides of (2) by
and multiply the numerator and denominator on the RHS by
k+1.Putting Ó=
Ó,
_
pt = ð Ók+1 Et dt+k
k=0
(Eqn. 3)
where pt and dt are respectively the proportion of the
price and dividend discounted by the long-run growth
factor. By this process we have removed the growth
trend from the series in Eqn.2, or have de-trended the
series.
Now, Ó = (1+g)/(1+r). If g>r, then Ó>1 and the series in
Eqn.3 sums to infinity. Therefore Ó <1 (this proves the
above Statement A). The discount rate appropriate for the
pt, dt series (Eqn.3) is given by Ó = 1/(1+r).
VERSION III of EM model:
The above two versions can also be written as follows,
since the real stock price is equal to the present value of
rationally expected or optimally forecasted future real
dividends discounted by a constant real discount rate. This
implies,
pt = Et(pt*)
(Eqn. 4)
where pt is the real stock price (or index), Et is the
conditional expectation operator, and pt* is the ex
post rational counterpart of pt, i.e., the present
value of the actual subsequent real dividends at time
t. Here,
_
pt* = ð Ók+1 dt+k
k=0
In (4), pt is the optimal forecast of pt*. The difference
between pt* and pt measures the forecast error, ut, i.e.,
ut = pt* - pt, or pt* = ut + pt.
*
From the theory of conditional expectations it is
known that ut is not correlated with pt.
*
From statistics, it is known that the variance of
the sum of two uncorrelated variables is the sum
of their variances.
=> var (p*) = var (u) + var(p)
Now, variances cannot be negative.
=> var (p) µvar (p*), or
Í(p) µ Í(p*)
(Eqn.1)
(VARIANCE BOUND 1)
In other words, if above model of EMH holds good, then the
above equation should be true, i.e., the uncertainty of the
actually declared dividends should be greater than the
expected uncertainty of these dividends (Those who doubt
the validity of this equation are referred to footnote 3 of
Shiller).
Shiller tests these bounds with data: (Table 2)
Data set 1:
Í(p) = 50.1
Í(p*)= 8.9 (5.6 times LHS)
Data set 2:
Í(p) = 355.9
Í(p*)= 26.8 (13.3 times LHS)
Thus the
margin.
data
violates
the
inequality
No.1
by
a
long
Pictorial representation:
See Figures 1 and 2 for a pictorial representation of this
position.
Note: Problem with finite data:
One of the problems with this
pointed out by Shiller. Since
approach
has
been
_
*
pt = ð Ók+1 dt+k
k=0
and the summation goes upto infinity, one can never
observe pt* without error, except in the case when a
long dividend series is known (as used in Figures 1
and 2). Different assumptions about the present value
of dividends in 1979 lead to slightly different
measures of pt*. This uncertainty is shown in Figure
3. Even allowing for this uncertainty, however, as we
can see from Figure 1, the violation of variance bound
1 would take place.
VERSION IV of EM model:
In this version, the maths and statistics tends to
become tends to become quite advanced. We simply look at
the results. Shiller begins with:
_
_tpt = ð Ók+1 _tdt+k
k=0
where Ó<1
which is similar to Eqn.3. Here, _t is known as the
innovation operator and represents Et - Et-1. The main
purpose of considering innovation is that it is observable,
i.e., _tpt is observable. However, Shiller points out that
_tdt+k is not, since it is not known when the public gets
information of the dividends. Shiller then derives the
limit on the standard deviation of _p given the standard
deviation of d. Without going into the mathematics, it is
seen that the upper bound of Í(_p) is:
Í(_p) µ
Í(d)/ ´r2
(Eqn.11)
(VARIANCE BOUND 2)
where r is the discount rate
detrended series, r2 = (1+r)2 interest rate, which is roughly
interest rate, and _p =
p + d-1
appropriate for the
1, is the two period
twice the one-period
- rp-1
By testing the inequality No.11 using data, it was found:
Data set 1:
Í( p + d-1 - rp-1) = 25.5
Í(d)/ ´r2
= 4.7 (5.4 times LHS)
Data set 2:
Í( p + d-1 - rp-1) = 242.1
Í(d)/ ´r2
= 32.2
(7.51 times LHS)
Thus the data violates the inequality No.11 by a wide
margin.
Further, Shiller shows that the upper bound of Í(
Í(
where
p) µ Í(d)/ ´2r
p) is:
(Eqn. 13)
(VARIANCE BOUND 3)
p is the change in price, and is related to _tpt.
Testing this on data, it was found:
Data set 1:
Í( p)
=
Í(d)/ ´2r =
25.2
4.7 (5.3 times LHS)
Data set 2:
Í( p)
=
Í(d)/ ´2r =
239.5
32.5 (7.4 times LHS)
Thus the data violates the inequality No.13 by a wide
margin, too.
PART II: ALLOWING REAL DISCOUNT RATES TO VARY:
Shiller has recognised the possibility of the real discount
rate varying over time. He shows that this leads to a nonlinear model (Eqn.15, a counterpart of Eqn. 4), which is
untestable if the real discount rates move without
restriction over time.
However, if the movements in the real discount rate
are not too large, then, by using the process of
linearisation (truncating the Tailor expansion in Eqn. 15
after the linear term), around the mean dividend and
discount rate, he is able to derive a lower bound on Í(r).
Shiller spends some time showing that this process of
linearisation does not detract from the accuracy of the
estimates.
Provided that the real discount rate has a first-order
autoregressive structure and perfect negative correlation
with dividends, the lower bound on Í(r), given the
variability of
p, is shown to be :
Ö
Ì
Í(r) · °[´2E(r)]Í( p) - Í(d)° / E(d)
Û
ì
(Eqn. 17)
(VARIANCE BOUND 4)
Shiller then uses the data sets to find out how volatile r
would have had to be in order to explain the volatility of
stock prices:
Data Set 1:
r = 0.480
Í(r) · 0.0436
=>
at ± 2Í, r can range between -3.91% to 13.52%
Data Set 2:
r = 0.0456 (error in Table 2?)
Í(r) · 0.0736
=>
at ± 2Í, r can range between -8.16% to 17.27%
This variability is far higher than even the movements in
nominal interest rates during the period. Hence Shiller
tends to rule out the variability of the real interest
rates as playing a significant role in determining stock
volatility.
CONCLUSION:
Thus, all four variance bounds are violated or lead to
absurd results. Shiller concludes that there is a dramatic
failure of the efficient markets model.
CRITICISM:
a)
NPV of Expected future Cash Flows and NOT Expected
future Dividends should be considered. This depends on
many term structure variables.
A company should be valued by the present value of its
future cash flows, and dividends do not fully reflect this.
Marsh and Merton (1986) noted that managers tend to smooth
dividends. This lowers the variance in the observed series
of dividends and could possibly be inadequate for the
market's estimation of future dividends. The market
therefore uses other additional signals/ information in
forming its estimate of the value of a firm. If these other
signals are taken into account, then perhaps the volatility
of share prices will be seen not to be "excessive"
(Martin:280). Fama (1991) believes that "the tests (of
Shiller) are not informative about market efficiency (due
to the) central assumption ... that ... the variation in
stock prices is driven entirely by shocks to expected
dividends. By the end of 1970s, however, evidence that
expected stock and bond returns (tend to) vary with
expected inflation rates, interest rates, and other termstructure variables was becoming commonplace".
b)
Non-stationarity in variables, e.g., changes
market's discount rate.
in the
Shiller (1986), while responding to the Marsh and
Merton critique, argued that perhaps stock price volatility
is also a function of changes in the market's discount
rate, r
(Martin:280). Ball (1990) too, feels that such
models
say
"little
about
where
to
expect
nonstationarities, and use data and estimation techniques that
could easily disguise them" (Ball: 20). According to Fama
(1991), "the tests (of Shiller) are not informative about
market efficiency (due to the) central assumption ... that
expected
returns
are
constant...
returns vary through time."
(whereas),
expected
Short list of references:_
Shiller (1981)
Articles (given in class) by Ball (1990) and Fama
Text book by Martin and Cox (1988).
(1991).