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working paper
department
of economics
THE PEESISTEKCE OF VOLATILITI MI
STOCK MAEKET rLUCTDATIOMS
James M. Poterba
and
Lawrence H. Summers
Number 553
September 1984
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
THE PEESISTEKCE OF VOLATILITT AHD
STOCI MAEKET FLUCTDATIOKS
James K. Poterba
and
Lawrence H. Summers
number 353
September 19B4
The Persistence of Volatility and Stock Market Fluctuations
James M. Ppterba
MIT and NBER
and
Lawrence H. Summers
Harvard University and HBER
Revised
September I98U
We are indebted to Carla Ponn and especially to Ignacio Mas for research
assistance, to Fischer Black and Robert Me rt on for helpful discussions, and to
the KBER and the HSF for financial support. This research is part of the NBER
Program in Financial Markets and Monetary Economics.
Seotember
IQiS'-
The Persistence of Volatility and Stock Market Fluctuations
ABSTRAirr
This paper examines the potential influence of changing volatility in stock
market prices on the level of stock market prices.
It demonstrates that vola-
tility is only weakly serially correlated, implying that shocks to volatility do
not persist.
These shocks can therefore have only a small impact on stock
market prices, since changes in volatility aJfect expected required rates of
return for relatively short intervals.
These findings lead us to be skeptical
of recent claims that the stock market's poor performance during the 1970 's can
he explained
"by
volatility-induced increases in risk premia.
James M. Poterba
Department of Economics
Massachusetts Institute of Technology
Cambridge, MA 02139
(617) 253-6673
Lawrence H. Summers
Department of Economics
harvard University
Cambridge, MA 02138
(617) k9^-2kk-l
ThiB paper examlncB the potential Influence of changing stock marKet vola-
tility on the level of stock market prices.
It QemonEtrates that volatility is
only weakly serially correlated, implying that shocks to volatility do not persist.
These shocks can therefore have only a small ingjact on stock market pri-
ces, since changes in volatility affect expected required rates of return for
only short intervals.
These findings lead us to be skeptical of recent claims
that the stock market's poor performance during the 1970 's can be explained by
volatility-induced increases in risk premia, as suggested by MalXiel (1979) and
Pindyck
(l98'*)«
They also lead us to doubt that fluctuations in risk premia
associated vith changing return volatility can account for nuch of the observed
variation in stock prices.
The finding that volatility
is
not highly serially
correlated is puzzling in light of Black's (1976) observation that stock market
returns and changes in volatility are negatively correlated.
The paper is divided into five sections.
The first clarifies the theoreti-
cal relationship betveen return volatility, the level of share prices,
required rates of return.
ajid
The second section examines the time series proper-
ties of stock market volatility as measured using both monthly and daily data.
The results suggest that although volatility is serially correlated, changes in
current volatility should have only a negligible impact on volatility forecasts
over intervals as short as one or tvo years.
The third and fourth sections use
data on the iniplied volatilities in option premia to re-examine the persistence
question, and again find evidence of only weeik serial correlation.
The conclu-
sion discusses the implications of our results for alternative explanations of
recent stock market movements and for our understanding of the sources of asset
price fluctuations more generally.
1,
Volatility. Required Returns, and Stock Price Fluctuations
This section discuBBes the relationship between changes in volatility and
changes in the level of stock market prices.
Por simplicity we assume that
firms are not levered and that expected dividends grow at a constant rate.
The
former assumption allows us to ignore Black's (1976) important observation that
the level of share prices, by affecting the degree of leverage, should have a
direct impact on volatility.
The latter assiimption is maintained for con-
venience and could be relaxed easily.
Because of the nature of the volatility
estimates used in our empirical vork, ve use a discrete time formulation.
We
assiune that share
E (P
t
)
- P
t+1
D
t ^
t
^ ^
/,
is the risk-free interest rate,
dividend paid in period t.
= ^^ *
^.1
^
t
t
where r
prices satisfy the standard requirement that
^
*
a
is the risk premium,
Equivalently, equation
\K
-
\
*
(l)
and
D.
can be written as
is
the
:
^2)
\^t
where
=
E
(P
t
,
t+1
J
- E iP
)/P
t t+1
t
(3)
is a random disturbance assumed to be uncorrelated with any information avail-
able at time t.
It reflects the impact of revisions in expectations about
future values of D, a, and r which take place between periods t and t+1.
Equation
(2)
is a difference equation for P
,
and it can be solved forward
subject to an appropriate transversality condition to yield
P
=
t
Ell
t
J
{
II
j^Q \^Q
T
(1
+ a
t+i
+
r_)-^}
t+i
]'.
D
t+J
{k)
-•^-
ABBuming that the risk free rate Ib constant over time, this expression may be
linearized around the mean value of a, a, to obtain:
EjD
-
K
-
J,
I
-
1
+
dP^
I
7r-^(E.ia_.] -a)
•
E
(5)
where
dP
^
'^Vj
Equation
,
» - (l*r^a)->l. I
k=0
(6)
.
(l+r+a)^
Assuming that expected dividends grow at a constant rate
future risk premia.
Id.
.
,
t+j
dP
)
-^^-^^^
expreBBes current stock prices as a linear function of expected
(5)
BO that E
t
(D
^
t
1
= (l+g)
t
-D^d+g)''
t
_
^Vj
J^+t
g,
the derivative in (6) can be sinrplified as:
»
-D^d+g)''
,^^ xk
U+gJ
-^^^^^
r
I
t
=
_,.. _
(l+r+a ) J ( rHQ-g )
.
(l+r+a)J''^ k=0 (1+r+a)^
It is natural to postulate that a
.
depends on
2
,
C7)
.
*
the variance of
e
.
We
assiune for Eiirolicity that
a^ = -ro;
(B)
vhere T is a constant of proportionality that depends on investors' levels of
Merton (19T3,1980) derives a similar relationship between a^
risk aversion.
and the variance of returns in a continuous -time model.
.To stuify
the effect of changes in volatility on P
adopt some assumption about the evolution of
2
.
it is necessary to
We assume that o
2
follows
an AR(1) process:
+11.
0=p+DO
t
t-1
^0
^1
t
(o)
^^'
Evidence to support the AR(l) assumption is presented in subsequent sections.
From
and (9), it immediately follows tnat a
(8)
\
vhere v
' ^'O *
yw
\
^^°^
The mean value of a
•
t
t
*
^iVl
also follovs an AR(l) process:
t
is therefore
TP^/d-P^
u
1
),
and the deviat-
and a obeys
ion between a
e
- a
a
P
-
(a
a) + v
(ll)
.
Equation (ll) enables us to simplify (5) substantially, since E (a
Substituting this relationship into
p''(a -"a).
"
P
.
I
K^K^.^
-1-^
J=0
(1+r+a)''
•
-
=
/
_
—
(12)
J=0 (l+r+a)'^(r+a-g)
D (l+r+a)
t
and using (T) yields
D^(l+g)'^p.^(a -a)
^
_^
_
I
(5)
- a) =
-
I
r + a - g
D
l+r+a
—
r
1+r+a-p (l+g)
,
t
J*
I
—t— j(a^
,
,
—V
- a)
r+a-g
The last expression shows the effect of risk premia shocks on share prices;
XX
the second term is (dP /da
)
•
- a).
(a
This may be rewritten in terms of
X
volatility shocks, using (8), as
t
- Y (l+r+a")
_
dcr
X
,
^^^^
1—1—1
ll+r+'a-p (l+g)]
r+"a-g
X
or
^^°S\
2
dlogo^
where
2
TO
X
t
= a
-^°t\
~
Z
'
1
Il+r+-^
1+r+a-p^ (l+g)]
1
r+a-g]
is the dividend yield, D /P .
t t
and X^ = r^+a^-g,.
(ll»)
Z
*
The numerator simplifies since
_
Evaluating both expressions at a yields
>^.
2"
dlogcT
~
^^^^
•
ll+r+a-p tl+g)]
Hotice tbat the absolute value of the derivative of share prices vith respect to
current volatility rises vith p^.
This result
is
intuitively natural.
If
increases in volatility are expected to persist, they vill have a greater impact
on the discount factors applied to future cash flows, and therefore on share
prices.
In order to examine possible relationships between volatility and the level
of share prices, it is useful to insert some plausible parameter values into
(15).
The mean annual return on common, stocks for the period 19^*6-1983 was
11.6 percent. 1
The mean nominal return on Treasury bills was
^^.6
percent per
year over the same period, iaiplying an average value of 7-0 percent for
average real return on Treasury bills, which we use to estimate r, was
cent.
The
a.
,k
per-
The estimated variance of the market return, expressed at annual rates,
ranged from 26.83 in I96U to 638.57 in 197^, averaging 238.3.
The last sta-
tistic in conjunction with the mean estimate for a implies a value of .029 for
Tf.
Merton (19BO) estimated this parameter to be .032 for the period 1952-1978,
The growth rate of nominal dividends on the S&P 500 during the 19^8-1983 period
vas 5.2 percent annual-ly.
Combining this with our inflation rate of k,2 percent
yields an average growth rate for real dividends,
g,
of .01.
The effect of changes in volatility on the level of share prices
sensitive to the level of Pi,
is
very
The derivative in (15 ) equals -.070 when P1=0,
.131 when Pl=.5, and -.1*09 when Pl=.9.
We have defined Pl as the serial corre-
-t>-
lation in annual volatility; an annual value of Pi = .90 iurplies a monthly auto-
correlation of more
thaji
.99.
Stated another way, a 50 percent increase in
market volatility from itB average level would reduce the value of the market by
3.5 percent if Pi = 0, by 6.5 percent of Pi
«=
.5,
and by 20 percent if Pi=.9.
It is clear that if fluctuationB in volatility are to play a significant role in
explaining market fluctuations, then Pi must be quite large.
The next two sec-
tions examine the serial correlation properties of several measures of volatility.
2.
Serial Correlation in Market Volatility
Ab enrphaBized in Merton (ipBO), a great deal of
variance estimation.
vork,
remains to be done on
In this section, we use crude estimators for the variance
of market returns to study the serial correlation properties of volatility. ^
"2
"2
use tvo estimators of market variance, c and o , computed respectively from
monthly and daily returns data.
We
While daily data are preferable for variance
estimation, they were available to us only for the 195B-8^ period.
Monthly
returns data were available from 1926 to 1983.
Volatility Estimation Using Monthly Data
2.1
Our first variance estimator, based on monthly data, vas calculated as:
^2
^2
%
vhere
=
J^^^it
2
- ^it) /12
(16)
denotes the annualized return on common stocks in month i of year t,
s
it
and r
IbbotBon
is the Treasury bill rate. 3
Our monthly returns data were obtained from
(I981t).
~2
In Table 1, ve report summary statistics on
and 19^8-83.
o
for two periods, I926-I9B3
It is immediately clear from the autocorrelogram and partial auto-
correlogram that there is no substantial positive serial correlation in market
volatility.
For the 19^*8-1983 period, the first order autocorrelation coef-
ficient is only
.lli<;
for the longer 1926-83 period, it is .675.
The high
autocorrelation coefficient for the whole period is sensitive to the inclusion
of the Depression years in the data sample.
The first order autocorrelation for
the sample period 1935-1983 is .50, and for the 19^0-1983 period, the estimate
declines to .05.
Besults similar to those for the postwar period were obtained
- 8 -
Table
1:
Autocorrelation in Annual Stoel: Market VolatllltieB
I9UB-I983 Sample
1926-1983 Sample
Autocorrelation
Partial
Autocorrelation
0.6T5 (.131)
0.675 (.131)
Lag Length
(Years
)
0.269 (.131)
-0.31*5
(.131)
Autocorrelation
0.1ll»
(.167)
Partial
Au t ocorrelat ion
O.lll*
(.167)
-0.115 (.167)
-0.129 (.166)
-0.221*
(.167)
-0.200 (.167)
3
0.110 (.131)
0.206 (.131)
k
0.077 (.131)
-0.058 (.131)
0.233 (.166)
0.287 (.167)
5
0.11»5
(.131)
0.217 (.131)
o.ooB (.160)
-0.122 (.167)
6
0.215 (.131)
0.002 (.125)
0,197 (.167)
0.255 (.167)
Source:
Annual volatility estimates vere calculated as the average of twelve
squared monthly values of the return on common stocks minus the return
Treasury "bills. These data were dravn from Ibbotson (198I1) for the
period I926-I983, a total of 696 observations. The second sample, for
the I9I+B-I983 period, contains 1*32 observations. Standard errors are
shown in parentheses. See text for further details.
-
9
-
using data for only the I96O-I983 period, vhen the estimated first order serial
correlation coefficient was .131.
Tne hypothesis that annual volatility was a
vhite noise process -could be rejected at standard levels in the 19'<B-19B3 or
I96O-I9BI4 periods.^
As a further check on the autocorrelation properties of our volatility
"2
cBtimates, we estimated some sinrple autoregressive nodels for o^.
The results,
which are presented in Table 2, corroborate the conclusions reached above.
They
suggest no great persistence in volatility, and indicate that the simple first
order autoregressive model used in the preceding section's theoretical development fits the data quite well. 5
Higher order models did not yield appreciably
smaller sums of squared residuals for either sample period under consideration.
The point estimates of Pi for each sample period are substantially less
than unity.
They imply that volatility shocks do not persist for long periods.
For the I926-I9B3 sample perod, where we find the greatest amount of persistence, ninety percent of a volatility shock will have dissipated
after the shock.
required.
tsf
six years
In the AR(2) case for this period, only four years are
The half life of the shock, the time required to move half way back
to the long-run equilibrium, is two years for both processes.
period, the in5>lied haJ.f life is much shorter
—
In the postwar
less than one year. 6 These
results confirm Schmalensee and Trippi's (19T8) findings of relatively little
persistence in volatility for individual firms.
The estimated autocorrelations all suggest little persistence in volatility, and conventional t-tests reject the hypothesis that volatility is a random
walk vPi c 1).
However, recent work on the estimation of time series models
with unit roots, such as Dickey and Fuller (198I), has shown that the actual
- 10 -
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Blze of these testB nay be substantially different from their nominal size.
Dickey and Fuller (198I) present tables of adjusted critical t-values for
The last column in
various saniple sizes to test the hypothesis of a unit root.
Table 2 showB that t-statistic against the hypothesis Pi =
1
along vith the
appropriate critical t-value for the 95% confidence interval. T
in each case we
are able to reject the null hypothesis of a unit root.
Our analysis so far has assumed that market volatility can be modelled as a
Btationaiy time series.
may trend over time.
Some research, however, has suggested that volatility
If so, this would in^jly that our estimated autocorrela-
tions are biased upwards.
The equations in rows 3,^,7, and 8 of Table 2 present
simple tests for the existence of trends over various sample periods.
They
suggest a positive trend for the time period 19^8-1983, and a negative trend for
1926-1983.
However, the findings also show that the null hJTOthesis of no trend
cannot be rejected in the 19^8-1983 period cannot be rejected.
These results contradict Pindyck's
(l9Bli)
claim that there has been a clear
upward trend in market volatility over the last thirty years.
monthly volatility estimates
"ty
He pre-smooths
computing twelve-month moving averages before
looking for trends, and this makes his conclusions difficult to evaluate.
Even
if there were trends, however, it is important to recognize that only unexpected
deviations from trend should affect asset returns.
If the trend rate of volati-
lity growth rises abruptly, that should lead to a one-time adjustment in share
prices, bat it cannot account for a low rate of return over an extended period.
2.2
Volatility Estimates Based on Daily Data
Our second volatility estimator was based on daily data; it follows closely
on Merton's (198O) estimator. 8
Using daily returns on the Standard and Poor's
-12-
500 Stock Index, we conrputed
^2
o;
^
vhere
b.
xt
2^ -2
=
r
1=1
B
" /21
(IT)
adjusted for non-trading days.
1b the daily return
In daily data,
measuring the returns around the risk-free rate would have virtually no effect
on the estimated volatilities.
The twenty-one trading day intervals which we
use correspond roughly to months of calendar time.
The autocorrelogram and partial autocorrelogram for this volatility esti-
mator are shown in Table 3.
In this case, the data clearly exhibit positive
serial correlation. Again, the persistence of volatility is relatively unim-
portant from an economic perspective.
The first order autocorrelation coef-
ficient obtained using monthly data is .59^, which is equivalent to an
autocorrelation coefficient of only (.596)12, or .002, in annual data.
This is
not inconsistent with the estimates obtained using the post-war monthly volati-
lity data in the last section, since we could not reject the null hypothesis
that there was no serial correlation in volatility.
Table
k
shows estimated auto regressive models for the volatility series
calculated from daily data.
Once again, the first or second order autore-
gressive process provides an adequate description of the data.
The hypothesis
that the coefficients on all variables lagged more than two periods equalled
zero could never be rejected.
Higher lagged terms have small, as well as sta-
tistically insignificant, coefficients.
exceeded .10 in absolute value.
Kone of the estimates of P3 or
pit
The tests against the null hypothesis of
again clearly reject the hypothesis that volatility follows a random walk.
half life for a shock in the AR(l) model is less than three nxanths.
ever
P.,=l
The
-a.
-13-
Table
3:
Autocorrelation
In Monthly Stock Market
Partial
Autocorrelation
Lag Length
(Months)
•
Autocorrelation
0.596
0.1U2
0.027
-0.025
0.059
0.056
0.596 (.071)
1
2
3
k
0.1*U6
0.330
0.225
0.196
0.1B6
0.169
5
6
7
8
0.15i*
0.172
0.192
O.16B
0.050
0.021
0.009
0.021
-0.003
0.013
-0.036
-0.097
-0.113
-0.107
-0.115
-0.085
-0.062
9
10
11
12
13
Ik
15
16
17
IB
19
20
21
22
23
2k
Source;
.
(.071)
[.071)
(.071)
(.071)
(.071)
..071)
(.071)
(.071)
..071)
(.071)
:.07l)
..072)
..071)
.071)
.075)
.069)
.070)
.071)
.071)
.071)
{ .071)
.071)
( .071)
Volatility
(.071)
(.071)
(.072)
(.072)
(.071)
(.072)
(.071)
(.072)
..071)
(.071)
0.021*
(
0.016
0.067
0.069
-0.009 ..071*)
-0.161 ..071)
-0.009
.069)
0.023 .070)
0.031+
.072)
-0.063 .071)
0.020 .073)
-0.070 .071)
-0.097 .072)
(
-0.01*7
(
(
(
1
(
(
-
(
(
1
(
(
(
0.017
-0.003
(
(
(
0.01*5
(
0.002
(
.071)
.072)
.070)
.072)
.091)
The table shows the estimated autocorrelogram for monthly estimates of
market volatility. Each month's estimate is based on the average of
squared daily returns on the S&P 500 Index for the period 1968:001 to
19Bl*:l80, A total of 197 monthly observations, based on 1*137 daily
observations, are \ised. Standard errors are shown in parentheses.
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Substantlve economic concluBlons about the importance of volatility Bhocks
are unaffected by the choices between the different autoregresBive modelB.
All
of our resultB suggest very little perslEtence.^
Moreover, the results again
question the iE53ortance of trends in volatility.
None of the estimated time
trend coefficients is statist ically significant.
In both the AR(l) and AR{2)
cases, the estimate trend coefficients have t-Etatistics of less than unity and
the addition of trend variables has little effect on the other coefficients in
these equations.
-J6-
3.
Market Volatilities Implied
"by
Option Prenla
The estimates presented in the last section suggest that volatility shockB
are short-lived.
However, the estinated aerial correlation parameters might be
biased downward by measurement error, and they may also fall to reflect market
participants' beliefs about volatility persistence.
To address these problems,
we analyzed the persistence in volatilities inferred from option premia.
Unfortunately, options on stock market indices such as the SiP 500 have been
traded for too short a period to make analyzing them informative.
However, the
Chicago Board Options Exchange (CBOE) has computed an index of the price of a
standardized stock option on every Thursday since January 8, 1976.10
...the CBOE Call Option Index is an average of percent
option premiums; for each CBOE ;inderlying stock, a narket premimn is estimated for a hypothetical six-month,
at the money option using the market premiums of existing option series. This estimated market premium is
expressed as a percentage of the stock price. The CBOE
Call Option Index for a given day is the arithmetic average of all such percent premiums on CBOE underlying
stocks on that day. CBOE (1979) , p. l)
1
These data are now available for a period of eight and one half years and it is
possible to analyze the persistence of volatility expectations using them.
The CBOE Index does not correspond to the option premium of any traded
security.
for
trtiich
It is a measure of this option premiiim on the "representative share"
options are traded on the CBOE.
As such, the implied volatility
should be substantially higher than the volatility of the market, since the
market is a weighted average of many imperfectly correlated shares.
While our
estimates of the implied volatility on a representative share are not directly
comparable to the volatilities estimated in the last section, our assumption is
that their serial correlation properties should be reasonably similar. 11
11-
To estimate the volatility of the "representative Btock" linplied by the
CBOE index, ve assumed that the dividend yield on this share equalled that on
the SiP 500.^^
We followed Black's (1976) suggestion for -dividend adjustment and
subtracted the present value of dividend payments over the life of the option
from the price of the stock.
Vie
assumed that the option on the representative
stock vas priced according" to the Black-Scholes (19T3) formula, and applied a
numerical search algorithm to determine the variance of returns vhich was consistent vith the observed option price, risk free rate, and market dividend
yield.
The CBOE Index is standardized to apply to an option on a stock vith a
current price of $U0.O0, and since the index applies to at the money options
the strike price is $U0.00 as veil.
The veekly data on the CBOE Index, as veil
as our estimates of the injjlied volatilities, are shown in the Data Appendix.
The movements our infilled volatilities vere compared vith those of sixmonth ex post volatilities estimated from dally returns on the StP 500; the tvo
series cohere reasonably veil.
Figure 1 shows the movments in these tvo series,
each divided by its mean, for the 19T6-19Bi+ period.
A positive association bet-
Both series rise throughout the late
veen the series is readily apparent.
1970s, and decline during the 1982-1981* period. The CBOE implied volatility does
not rise as dramatically as the ex poste volatility series during the I9BI stock
market rally, although it does increase.
Table
5
shovs the estimated autocorrelogram and partial autocorrelogram
for the implied volatility series.
These are veekly data, and so the estimated
autocorrelations are higher than those in the earlier sections.
autocorrelation, for exanple, is .971.
The first order
Hovever, these results confirm the
-18-
CD
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1
1
NU3W 01 3AIIU13y AIIIIIUIOA
IN-
LU
T-
CD
;i;
-19-
"Pable 5:
Autocorrelation in Volatility Forecasts IciDlled by Option Premia
Lag Len^rth
(veeks)
1
2
3
U
5
6
7
B
9
10
11
12
13
Ik
15
16
U..
17
IB
19
20
21
22
23
2k
25
26
Autocorrelation
0.971
0.939
0.903
0.869
0.835
0.802
0.767
[.OI47)
O.73I*
[.IBI4)
0.703
0.673
0.650
0.629
0.612
0.589
0.566
..195)
..195)
..195)
.195)
>.195)
.195)
.195)
.195)
.195)
.195)
.195)
( .195)
.195)
( .195)
( .195)
( .195)
( .195)
( .195)
0.5i«5
0.525
0.507
0.k93
0.ii82
0.1*73
O.I466
0.1;58
0.1;50
0.H3
0.1;39
[.082)
[.106)
1.125)
l.ikz)
[.157)
[.171)
(
<
(
Partial
Autocorrelation
0.971
-0.056
-0.087
0.029
-0.028
-0.003
(.0147)
-0.0143
[.OI47)
(.OI47)
(.0I47)
(.OI47)
(.OI47)
[.0I4B)
-0.003 [.OI43)
0.032 [.OI47)
-0.003 [.056)
0.092 [.OI47)
-0.002 ..0l;l4)
0.062 [.0147)
-0.130 >.Ol47)
-0.026 ..OI47)
(
0.0614
(
(
-O.Oll*
(
(
(
(
0.012
0.059
0.053
0.025
0.028
-0.032
-O.OOB
(
(
.OI47)
.0146)
.OI46)
.0147)
.OI47)
(
.OI47)
(
.OI48)
(
.Oli7)
O.OOl;
(
.OI48)
.0I46)
0.027
{
.01*8)
(
Source ; Estimates of volatility forecasts were determined try inverting the
Black-Scholes option valuation formula to obtain the volatility implied
tjy CBOE option premia indices. These data were available for the period
1976:1 to 19814:26, for a total of l*l47 weekly observations. See appendix
for further details and data description. Standard errors are shown in
parentheses.
-20-
earlier contluBions that volatility changes are not perslBtent.
One year after
a shock to volatility, expected volatility vill exceed its mean by only (.9Tl)52
»,22
or twenty two percect, of the i-nitial shock.
The partial autocorrelogram
again suggests that a first order autoregressive representation is appropriate
for this series.
The statistical insignificance of partial autocorrelations at
lags of more than one week is indicative of an AR(l) structure in these data.
Table 6 reports estimates of several time series models for these data.
Equations for our entire data period, comprising hWj weeks, are reported in
the first four rows of the tahle.
The hypothesis that the residuals from the
AR(1) model are white noise is nearly rejected at standard levels, as shown ty
the reported Q-statistics.
ficulty.
The AR(2) results do not suffer from this dif-
The higher order (third and fourth order) autocorrelation parameters
are never statistically significant.
The implied responses to a volatility
shock are similar in all of the estimated models.
They suggest that the
hEilf
life for a volatility shock is about six months. 13 Although the estimated weekly
autocorrelation is near unity, the last column of the table shows that the
hypothesis of a unit root is still rejected in each case.
Because each weekly observation on the CBOE Index depends on forecasts of
volatility for each of the next twenty six weeks,
two consecutive observations
on the iJ35)lied volatility vill have twenty five weeks of forecast volatilities
in common.
This may bias our estimated autocorrelations.
Ve therefore esti-
mated autoregressive models using non-overlapping data periods, corresponding to
every twenty-sixth observation in our data set.
The estimated AR(l) ani AR(2)
models are reported in the last two rows of the table.
The estimated six-month
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-22-
autocorrelation coelTicient from these data
than the Bix-month autocorrelation iniplied
is
toy
.UZ, vhicb is only slightly lower
cur weekly estimateB.
The resultB
agai^ suggest that over half of a volatility shock vanishes within six months.
-23-
U.
The Term Structure of Icrplled Volatilities
The CBOE
CeJ.1
Option Index data provide strong support for the transient
character of volatility shocks.
However, they do not permit us to directly
investigate how long-term expectations of volatility respond to changing short-
A second source of option data can illuminate
term volatility expectations.
this issue.
Since 1979* Valueline has computed indices of option premia at
We inverted these option premia indices using
three and six month maturities.
the sane procedure vhich ve applied to the CBOE data.l^
The availability of two different maturity option indices provides an
opportunity for additional tests of the persistence hypothesis.
The implied
volatility for the six month options was assumed to equal the average of the
expected three month volatilities for the next three months as well as the three
months following them:
In this notation,
,
&
,i^
o
2
t
is the volatility expected to prevail, as of time t, over
the k months beginning in week
s.
The assumption in (iB) allows us to solve for
an estimate of the inplied forward volatility
vhich
is
expected to prevail for
the three month period beginning three months from the current week:
*p
^P
t+13,3^t
'^
-
•^p
2t,6°t " t,3°t
(19)
We can use the estimated forward volatilities to study the change in the
implied forward volatility vhich occurs when the current three-month "spot"
Implied volatility changes.
The results of this estimation are shown below:
-2i-
^^t -t+l?
**T.
-^"Ll
t-1
t+12.3
t+13.3
=
-°233
(.0769)
L 7°. ^ - t^"'»^
1 ^^
^^T
(.050) ^»^ ^
+ 0.511
1
^20)
These resultB indicate that when current volatility expectations change,
expected volatility in future periods also changes.
However, they also consti-
tute further evidence for our contention that volatility shocks are not persistent.
One year after a volatility shock, these estimates imply that only
seven percent of a shock vill still persist.
"
The half life of a volatility
shock according to these data is Just over three months.
One year after a ten
percent shock to volatility, the three-month forecast of volatility vould only
be 1.3 percent greater than in the initial period.
'
-25-
5. ConcluBJonp and InrplicationB
Our findings suggest that Bhocks to stock market volatility are not perThis is illuBtTated in Figure 2, vhich showB the impulse response
sistent.
functions corresponding to several of the estimated time series models for vola•tllity.
These functions, vhich show the moving average representation of each
process, depict the evolution of volatility following a "shock" equal to ten
percent of the steady-state value of volatility.
shocl: dissipates varies
two years.
While the speed with which the
across modeD^, the half life of the shock never exceeds
For the equations corresponding to the CBOE and daily volatility
estimators, the half-life of a volatility shock is less than half a year.
Our engiirical results suggest that changes in volatility should affect
expected required returns for periods not substantially greater than two years.
This means that they can only have a very limited impact on the level of share
prices.
A doubling of volatility would reduce the level of the market by only
Since actual volatility fluctuations ere usually smaller
about nine percent, 15
than this, we doubt that changing volatility accounts for any large fraction of
market fluctuations.
This observation applies both to the problem of explaining
recent events and to the deeper problem of explaining the sources of stock price
fluctuations.
Our work deepens the puzzle of explaining the strong negative correlation,
observed
"ty
Black (1976) and Schmalensee and Trippi {19TB), between stock market
returns and volatility.
Black (1976) showed that the inverse correlation
between volatility and returns was so strong that a positive one percent return
on a stock in5)lied more than a one percent reduction in volatility;
this
-26-
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vo
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2
•^
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0^
.-<
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(^
tx nv
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-27-
inplies that raising the share price actually reduces the dollar volatility of
the stock.
The finding that volatility is not highly perBistent Buggests that
autonomous changes In ^volatility should have only a relatively small effect on
share prices.
If Black's (1976) "leverage effect" explanation of the rela-
tionship between returns and volatility were correct, then one would expect to
observe that volatility, like prices, would follow a random walk.
One possible explanation for the observed data is that the same events
which make the returns to capital more uncertain also reduce their expected
value.
This coincidence in the arrival of stochastic shocks would lead to on
apparent relationship between volatility changes and share prices, although in
^"'
:_:
,
fact no such causal link exists.
Another possibility is that when adversity
strikes, firms are expected to respond with new strategies.
Between the time
the market anticipates that some new policy will be chosen and the time this
,™ policy is actually announced, uncertainty and therefore volatility may
increase.
-26-
Endnotes
1. These data are
based on Ibbotson
(198I4).
2. To the extent that there is measurement error in our estimates of volatility, the estimated serial correlation coefficients nay be biased downward. In
the next section we use estimates of volatility expectations which are less
susceptible to these difficulties.
3. Merton (I98O) observes that although estimators of the variance vhich
center the estimated returns around a point vhich is not their Ban53le mean will
be biased, these biases are trivial. He estimates variances using uncentered
returns; Pincfy'ck (19S^) estimates variances for twelve month periods hy computing second moments centered around the overall mean return in his sample.
li. We used Box-Jenkins
(19T0) Q-statistics to test the null hypothesis of no
serial correlation of up to fifth order. The values of the test statistics were
32.9 for the I926-19B3 period, i+.70 for 19l*B-1983, and k.kl for I96O-I983. The
sriticELl v&lue for rejecting the null hypothesis of at the .05 level is 12.59.
5. The finding that lew-order autoregressive processes provide an adequate
description of the variance of market returns suggest that it may be possible to
apply the econometric techniques for time- varying heteroscedasticity, suggested
by Engle{l9B2}, vhen studying the behavior of security returns.
6. The same conclusion emerged vhen we examined changes in volatility.
is showE by *be folloving equation, estimated for the 19^B-19B3 period:
This
i^
' cl . = 1083.3
- .i*3B lof . - of -1
^"^ (3079.1)
^^
^^
(.155)
The coefficient on the lagged volatility change is negative and has a t-value of
2. 83, clearly different from zero.
T. Critical values of the unit root tests are drawn from Dickey and Fuller
(1981). Tables I and II.
8. Merton (198O, Appendix A) discusses tvo adjustments to estimated volatility series. The first, for nontrading days, is implemented in our study. The
second, vhich corrects for nontrading shares in the stock index, multiplies the
estimated volatility by a constant. Since our stuc^ is concerned vith the autocovariance and not the level of the volatility series, and the former is unaffected 'by multiplication by a constant, we did not make the correction.
9« The results of estimating a model in changes were:
\
'
Vl
= ^-^^^10"^ -
-315
•
I
Vi
-
^-2^
•
(3.1*2x10-^)
(.068)
Again, this suggests negative serial correlation.
10, A further description of this series may be found in CBOE(l979).
-29-
11. Latane and Rendelnan (19T6) coiaputed implied standard deviations for a
series of pptions over a thirty nine week period and discovered tnat these
lumlied standard deviations tended to move together. Schmalensee and Trippi
(1978) found Bimilar results. Tnese coincident movements in volatility are the
.market-vide volatility shifts we hope to capture.
12. We assumed that dividends were paid as a continuing flow at rate X per
year, where A is the current yield on the StP 500.
13. We also considered the serial correlation of changes for the non-
overlapping differences
«2
-I,t
"2
I.t-2b
"2
-^'^^^
(.OliU)
(0.00013)
where the subscript I denotes an implied volatility.
"2
^'^^^
Ik, The Value Line data were available for the period 19BO:l6 to 19Bl4:26;
this constitutes a total of 220 weeks. However, there were 17 weeks of missing
dAta. This precluded calculating the long autocorrelograms which are reported
for the other volatility series. However, the first order autocorrelations for
these series, .88 for the three month implied volatility and .87 for the six
month, were roughly consistent with earlier findings.
15. This calculation is based on Pi = .21, the estimate from the CBOE
inrolied volatilities, and the other parameter values described in Section 1.
i3n.
-30-
Ref rrences
Black, Fischer, 19T5, Fact and fantasy in the use of options, Financial Analysts
Journal 31, 36-1^1, 6l-72.
•
Black, Fischer, 1976, Studies of stock price volatility changes, Proceedinps of
the 19T6 Meetings of the American Statistical ABSOCiation, business and
Economic Statistics Section . 177-ltSl.
Black, Fischer and ftyron Scholes, 1973, The pricing of options and corporate
liabillticE, Journal of Political Econony 8l, 637-659.
Box, G.E.P. and G, Jenkins, 1970, Time Series Analysis: Forecasting and Control
(Bolden Day: San Francisco).
Chicago Board Options Exchange, 1979, The CBOE call option index; Methodology
and technical considerations (Research I>epartment of the CBOE, Chicago, XL).
Dickey, David A. and Wayne A. Fuller, 19Bl, Likelihood ratio statistics for
autoregressive time series vith a unit root, Econometrica ^9, 1057-1072.
Engle, Robert F., 1982, Autoregressive conditional heteroscedasticlty vith estimates of the variance of United Kingdom inflations, Econometrica 50,
987-lOOB.
Ibbotson, Roger G., 1981*, Stocks, bonds, bills, and inflation;
(R.G. Ibbotson Associates, Chicago, IL;.
19B1<
yearbook ,
Latane, E. and R. Rendleman, 1976, Standard deviations of stock price ratios
ingjlied in option prices. Journal of Finance 31, 369-381.
Malkiel, Burton G,, 1979, The capital formation problem in the United States,
Journal of Finance 3^*, 291-306.
Merton, Robert
C,
1973, An intertemporal capital asset pricing model,
Econonietrica ^1, 867-887.
Merton, Robert C, 1980, On estimating the expected return on the market.
Journal of Financial Economics 8, 323-361.
Pindyck, Robert S. , 198U, Risk, inflation, and the stock market, American
Economic Review 7^, 335-351-
Roll, Richard, 1977, An analytical valuation formula for unprotected American
call options on stocks with known dividends. Journal of Financial Economies
5, 251-258.
Schmalensee, Richard and R. Trippi, 1978, Common stock volatility expectations
implied by option premia. Journal of Finance 33, 129-^7.
^50^1^
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