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increases, this effects only the Sr+1 terms on the right hand side of equation
not the other discount factors 61+2, St+3, etc. However, now consider the case
is persistence in the risk premium. This means that should investors believe
premium will increase tomorrow, then they would also expect it to increase in
periods. Hence a change in risk premium today has a large effect on stock pr
rpr increases then 6t+l, 6r+2, 6r+3, etc. are all expected to increase.
The key element in Poterba and Summers™ view of why stock prices are hig
is that there can be ˜shocks™ to the economy in the current period whic
perceived risk premium for many future periods. Thus the rational valuati
for stock prices with a time varying and persistent risk premium may expla
observed movements in stock prices. If this model can explain the actual mo
stock prices, then as it is based on rational behaviour, stock prices cannot be
volatile.
Poterba and Summers use a linearised approximation to the RVF and th
to show exactly how the stock price responds to a surprise increase in th
riskiness of stocks. The response depends crucially on the degree of persistenc
by agents™ perceptions of changes in the conditional volatility of the forec
stock returns. If, on average, the degree of persistence in volatility is 0.5, then
in volatility of 10 percent is followed (on average) in subsequent periods b
increases of 5, 2.5, 1.25, 0.06, 0.03, etc. percent. Under the latter circumstan
and Summers calculate that a 1 percent increase in volatility leads to only a
fall in stock prices. However, if the degree of persistence is 0.99 then the
would fall by over 38 percent for every 1 percent increase in volatility: a size
Given that stock prices often do undergo sharp changes over a very short peri
then for the Poterba-Summers model to ˜fit the facts™ we need to find a hig
persistence in agents™ perceptions of volatility.

Poterba -Summers: Empirical Results
Poterba and Summers treat all variables in the RVF in real terms and con
the real stock price of the market portfolio which uses the S&P index over
1928-1984. As Poterba and Summers wish to focus on the impact of the ris
on P, they assume

(i) real dividends grow at a constant rate g, so that the ˜unobservable™ Et(D
+
to (1 g)™Dt and
(ii) the real risk-free rate is constant (rt = r).
The CAPM (Merton, 1973 and 1980) suggests that the risk premium on
portfolio is proportional to the conditional variance of forecast errors on eq
Eta:+l
consumers™ relative risk aversion parameters which are assumed to be consta
and Summers assume (and later verify empirically) that volatility can be rep
ARMA model(s) and here we assume an AR(1) process:
+ slat2 + V t + l
2
O>ai>l
a,+, =
where v,+l is a white noise error. The latter provides the mechanism by whi
(randomly) switch from a period of high volatility to one of low volatility a
from positive to negative. If a1 is small, the degree of persistence in volatili
For example, if a = 0, 0: is a constant (= ao) plus a zero mean random e
1
hence exhibits no persistence. Conversely if a1 is close to 1, for example a1
if vt increases by 10, this results in a sequence of future values of a2 of 9, 8
in future periods and hence a is highly persistent. If a follows an AR(1) p
: :
so will rp,:
+
rpt = Aao Aa1rpf-i
To make the problem tractable Poterba and Summers linearise (17.1) aroun
value of the risk premium p . This linearisation allows one to calculate the
response of Pt to the percentage change in a:
;




Thus the response of Pt to a change in volatility a increases with the degree of
:
al. Poterba and Summers compute an unconditional volatility measure for t
of monthly stock returns based on the average daily change in the S&P comp
over a particular month. Hence for month t:
m
1sE./m
i=
:
i= 1

where Sri = daily change in the stock index in month t and m = number of t
in the month. They then investigate the persistence in ; using a number of
?
t

ARMA models (under the assumption that 5; may be either stationary or non-
For example, for the AR(1) model they obtain a value of a1 in the range 0.6-
also use estimates of a2 implied by option prices which give estimates of
forward-looking volatilities. Here they also find that there is little persistence in
The values of the remaining variables are

(i) 7 = average real return on Treasury bills = 0.4 percent per annum = 0.
per month,
(ii) g = average growth rate of real dividends = 0.01 percent per annum
percent per month,
(iii) ˜p = mean risk premium = average value of (ERm - r ) = 0.006 per mo
implies a value of A = rp/a-2 of 3 5 .
.)
movements in stock prices.

17.2.2 Volatility and ARCH Models
Chou (1988) notes that there are some problems with the Poterba-Summ
ical estimation of the time varying variance 0 He notes that the Poterb
.
:
measure of the variance remains constant within a given horizon (i.e. a mo
then assumed to vary over longer horizons (i.e. the AR model of equation
It
02). is therefore not a correct measure of the conditional variance. Chou
Poterba-Summers analysis using an explicit model of conditional variances
GARCH(1,l) model which is explained briefly below and more fully in Chap
also that the Poterba- Summers estimation technique being a two-step proce
inconsistent estimates of the parameters (see Pagan and Ullah (1988)). Chou, a
and Summers, assumes expected returns on the market portfolio are given by
(plus RE):
(&+1 - rr) = h[E,o;+J+ E f + l

where R,+l = return on the market portfolio, r, = risk-free rate. Taking exp
(17.9) it is easy to see that the best forecast of the excess return depends
forecast of the conditional variance



The conditional forecast error is therefore

- ErRr+l = &f+l
&+l

and the conditional variance of the forecast error is



-
We assume Er+1 N ( 0 , U:+,) and hence has a time varying variance. Accor
CAPM the expected excess return varies directly with the time varying variance
errors: large forecast errors (i.e. more risk) require compensation in the form
expected returns. It remains to describe the time path of the conditional vari
+
assumes a GARCH(1,l) model in which the conditional variance at t 1 is
average of last periods conditional variance o and the forecast error (square
:



The GARCH(1,l) model is a form of adaptive expectations in the second mo
distribution. The best guess of o + at time t (i.e. Erof+l)is given by the RHS
:,
+
The expected value of the variance for time t 1 is:
The ai are constrained to be non-negative so that the conditional variance is a
+
negative. If (a1 a2) = 1 then a change in the current variance of has a
+
effect on all future expectations. If a1 a2 < 1 then the influence of o on :
+
away exponentially. Thus (a1 a2) measures the degree of persistence in the
+
+
1 012 2 1 then the unconditional variance ao/[l - (a1 ay^)] is
variance. If a
and we have a non-stationary (explosive) series in the conditional variance.
(17.9) + (17.13) taken together are often referred to as a 'GARCH in mean' or
model. Chou (1988) estimates these two equations simultaneously, using the
likelihood method. His data is for weekly returns (Tuesday-Tuesday closing pr
NYSE value weighted stock price index (with dividends assumed to be reinv
the period 1962- 1985 (1225 observations). The price index and weekly returns
in Figures 17.1 and 17.2. The crashes of 1974 and 1982 are clearly visible in
and periods of tranquillity and turbulence in returns are noticeable in Figure
Chou finds that the estimate of the market price of risk (or index of r
aversion) A over various subperiods is not well determined statistically and
being statistically insignificant. However, it has plausible point estimates in the
(Poterba and Summers obtain a value of 3.5 and Merton (1973) finds a value o
+
value of a 1 a is very stable over subperiods and is around 0.98 indicating
2




1000
l2O01
J
800




400


200


of I I
I I I
1 I
1

1962 1965 1968 1971 1974 1977 1980 1983 1985

Figure 17.1 NYSE: Stock Index. Source: Chou (1988). Reproduced by permission o
and Sons Ltd
Figure 17.2 NYSE: Weekly Stock Returns. Source: Chou (1988). Reproduced by p
John Wiley and Sons Ltd




Figure 17.3 NYSE: Variance of Stock Returns. Source: Chou (1988). Reproduced by
of John Wiley and Sons Ltd

persistence in the conditional variance (Figure 17.3). It follows from our previ
sion that observed sharp falls in stock prices can now be explained using the R
+
Poterba-Summers framework. Indeed, when a1 a2 = 1 (which is found to
acceptable on statistical grounds by Chou) then stock prices move tremendou
elasticity d(ln P,)/d(ln a;) can be as high as (minus) 60.
+ +
values of N . He then estimates o&+l= a a1q& Vr+l for these various v
0
He finds that a1,the degree of persistence, varies tremendously increasing from
for N = 5 (working) days to a = 0.6 for N = 20 days (i.e. one month) and t
1
for N = 250 days (i.e. one year). This suggests that the Poterba-Summers m
not have correctly captured the true degree of persistence. Note, however, t
experiment should really have used a closer measure of o2 to that used by P
Summers, namely
U , = I=(P N ) 2 / N
-R
;

For daily data over one month as used by Poterba and Summers, E will be cl
but over longer horizons (i.e. N increases), is likely to be non-zero because
bear markets. Hence the above measure is more representative of the Poterba
approach and may not yield such sensitivity in the estimate of a1, as found
experiment. However, Chou™s use of a GARCH model is preferable on U prio
as it correctly estimates a measure of conditional variance.
As a counterweight to the above result by Chou consider a slight mod
Chou™s model as used by Lamoureux and Lastrapes (1990). They assume that
volatility is influenced both by past forecast errors (GARCH) and by the volum
(VOL) (i.e. number of buyhell orders undertaken during the day):



They model dairy returns (price changes) and hence feel it is realistic to assum
expected returns p :




+
They estimate (17.17) (17.18) for 20 actively traded companies using abo
of daily data (for 1981 or 1982). When y is set to zero they generally find a si
+
to Chou, namely strong, persistent GARCH effects (i.e. a 0.8-0.95).
1a 2
the residuals Et+l are non-normal and hence strictly these results are statistic
given the assumption (17.19). When VOL, is added they find that a = a2 = 0
1
and the residuals are now normally distributed. Hence conditional volatility
determined by past forecast errors but by the volume of trading (i.e. the pe
VOL accounts for the persistence in okl). They interpret VOL as measuring
of new information and therefore conjecture that in general GARCH effec
studies are really measuring the persistence in the arrival of new informatio
this data set, the Chou model is shown to be very sensitive to the specificat
of introducing VOLr into the GARCH process. Note that volatility (risk) i
varying but its degree of persistence is determined by the persistence in tradi
There are some caveats to add to the results of Lamoureux and Lastrapes. In
as their model uses data on individual firms a correct formulation of the CA
include the covariance between asset i and the market return. This makes th
17.2.3 CAPM, Noise lkaders and Volatility
The empirical analysis above has highlighted the potential sensitivity of
equations to assumptions about the equilibrium model determining returns
precise parameterisation of any time varying conditional variances. Below we
explore the ability of the CAPM to explain equilibrium asset returns but use
specifications of the GARCH process. We then examine a model in which n
have an influence on equilibrium returns.

The CAPM and Dividends
The study by Attanasio and Wadhwani (1990) starts with the empirical obser
from previous work, it is known that the expected excess return on an aggr
market index (which is assumed to proxy the market portfolio) depends on t
period™s dividend price ratio. The latter violates the EMH under constant expe
returns. Previous work in this area often assumed a constant risk premiu
sometimes interpreted the presence of the dividend yield as indicative of a ti
risk premium. Attanasio and Wadhwani suggest that if we explicitly mod
varying risk premium then we may find that the dividend yield (= Z,) does n
expected returns. If so, this would support the CAPM version of the EMH. If
the CAPM is the correct equilibrium pricing model:



then we expect h > 0 and 6 = 0. The time varying conditional variance of
return is given by the variance of E r + l which is assumed to be determined by th
GARCH( 1,2) model (with the dividend yield added)

+ W O ; + a24 + + nz,
2
a, = a0
,,
where ai and TC are constrained to be non-negative. The GARCH model imp
+
expected variance for period t 1, that is E,o:+l, depends on a weighted ave
periods of the variance 0 and the two most recent forecast errors squared (
:
addition, equation (17.21) explicitly allows the dividend yield to affect the
variance - this is not a violation of the CAPM and the EMH. Using monthly d
1953-November 1988 on an aggregate US stock price series, a representati
(Attanasio and Wahwani 1990, Table 2, page 10):

+ 0.552, - 4.05rf + 22.30:+,
[ R c l - Y , ] = - 0.035
(0.025) (0.39) (1.05) (11.3)
53(1)-88(11),R2 = 0.059, (.) = standard error
+ 1.5(10-2)˜f+ 2.2(10-2)˜;-1 + 0.870; + 5.310-22
= ao
2
Or+,
(2.9. 10-2) (4.0.10-2) (0.06) (2.4. 1Ov2)
equilibrium returns (under the assumption of RE).
In the GARCH equation (17.23), the dividend yield Z , has a statistically
effect on the conditional variance and this would explain why previous resea
assumed a constant risk premium found Z, significant in the CAPM retur
(17.22). Note that there is also considerable persistence in the conditional var
++ a = 0.91.
3
a1
y

Noise Traders, Risk and Serial Correlation in Stock Returns
Noise traders are now introduced into the market who (by definition) do no
asset decisions on fundamental value. Positive feedback traders buy after a pr
sell after a price fall (e.g. use of ˜stop-loss™ orders or ˜portfolio insurers™). Th
to positive serial correlation in returns, since price rises are followed by an
demand and further price rises next period. Negative feedback traders pursue t
strategy, they ˜buy low™ and ˜sell high™. Hence a price fall would be followed
rise if these traders dominated the market. (The latter would also be true fo
who assign a constant share of market value wealth to each asset, since a p
asset i will lead to a fall in its ˜value share™ in the portfolio and hence lead to
purchases and a subsequent price rise.) The demand for stocks by noise trade
proportion of the total market value of stocks) may be represented

N, = YR,-1
with y > 0, indicating positive feedback traders and y < 0,indicating negativ
traders and R,-1 is the holding period return in the previous period. Let us
the demand for shares by the smart money is determined by a (simple) me
model (Section 3.2).
s, = [Et& -a]/@,
where S , = proportion of stock held by smart money, a = expected rate o
which demand by the smart money is zero, p, = measure of the perceived
shares. We assume p f is a positive function of the conditional variance 0; of
(i.e. p = ˜ ( 0 )Thus the smart money holds more stock the higher the expe
˜).
and the smaller the riskiness of stocks. If the smart money holds all the s
S, = 1 and rearranging (17.25) we have the CAPM for the market portfolio:
return (E,R, - a ) and depends on a risk premium, which is proportional to
tional variance of stock prices, p r = m2.Equilibrium in the market require
to be held:
S,+N,=l

Substituting (17.24) and (17.25) in (17.26) rearranging and using the RE
+
R, = E,R, E, we have:
+ 802 + ( ˜ + yi02)Rt-i +
o
R, = a Et

The direct impact of feedback traders at a constant level of risk is given by the
However, suppose yo is positive (i.e. positive serial correlation in R,) but y1
+
Then as risk 0; increases the coefficient on R,-1, namely yo no:, could
and the serial correlation in stock returns would move from positive to n
risk increases. This would suggest that as volatility increases the market bec
dominated by positive feedback traders, who interact in the market with the sm
resulting in overall negative serial correlation in returns.
Sentana and Wadhwani (1992) estimate the above model using US
1855-1988, together with a complex GARCH model of the time varying
variance. Their GARCH model allows the number of non-trading days t
conditional variance (French and Roll, 1986) although in practice this is not
statistically significant. The conditional variance is found to be influenced d
by positive and negative forecast errors. Ceteris paribus, a unit negative fo
leads to a larger change in conditional variance than does a positive forecast
The switch point for the change from positive serial correlation in returns
serial correlation is q? > (-yo/y1) and they find yo = 0.09, y1 = -0.01 and
point is a > 5.8. Hence when volatility is low stock returns at very sho
:
(i.e. daily) exhibit positive serial correlation but when volatility is high retu
negative autocorrelation. This model therefore provides some statistical supp
view that the relative influence of positive and negative feedback traders ma
the degree of risk but it doesn™t explain why this might happen. As is becom
in such studies of aggregate stock price returns, Sentana and Wadhwani also f
conditional variance exhibits substantial persistence (with the sum of coeffici
GARCH parameters being close to unity). In the empirical results 8 is not
different from zero, so that the influence of volatility on the mean return on
works through the non-linear interaction variable yla;Rt-l. Thus the empirica
not in complete conformity with the theoretical model.


17.3 SUMMARY
In the past 10 years there has been substantial growth in the number of empir
examining the volatility of stock returns, particularly those which use ARCH an
processes to model conditional variances and covariances. Illustrative exam
work have been provided and in general terms the main conclusions are:

Only a small part of the conditional volatility in stock prices is explai
0

volatility in economic fundamentals or by other economic variables (such
of gearing).
There is considerable support for the view that the conditional variance of t
0

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