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Department of Economics
Working Paper Series
INSTITUTIONAL INVESTORS AND
STOCK MARKET VOLATILITY
Xavier Gabaix
Parameswaran Gopikrishnan
Vasiliki Plerou
H. Eugene Stanley
Working Paper 03-30
October 2, 2005
Revised: May 12, 2010
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INSTITUTIONAL INVESTORS AND STOCK
MARKET
VOLATILITY*
Xavier Gabaix
Parameswaran Gopikrishnan
Plerou
Vasiliki
H.
Eugene Stanley
October
2,
2005
Abstract
We
present a theory of excess stock market volatility, in which market
movements are due
to
trades by very large institutional investors in relatively illiquid markets. Such trades generate
significant spikes in returns
mentals.
We
and volume, even
in the
absence of important news about funda-
derive the optimal trading behavior of these investors, which allows us to provide a
unified explanation for apparently disconnected empirical regularities in returns, trading
and investor
size.
I.
Ever since
market
Shiller [1981],
prices,
volume
Introduction
economists have sought to understand the origins of volatility in stock
which appears to exceed the predictions of simple models with rational expectations
editor and referees for very helpful comments. We also thank Tobias Adrian, Nicholas BarBlanchard, Jean-Philippe Bouchaud, John Campbell, Victor Chernozhukov, Emanuel Derman,
Peter Diamond, Alex Edmans, Edward Glaeser, Joel Hasbrouck, David Hirshleifer, Harrison Hong, Soeren
Hvidjkaer, Ivana Komunjer, David Laibson, Augustin Landier, Ananth Madhavan, Andrew Metrick, Lasse
Pedersen, Thomas Philippon, Marc Potters, Jon Reuter, Bryan Routledge, Gideon Saar, Andrei Shleifer,
Didier Sornette, Dimitri Vayanos, Jessica Wachter, Jiang Wang, Jeffrey Wurgler, and seminar participants
at University of California, Berkeley, Departement et Laboratoire d'Economie Theorique et Appliquee, Duke
University, Harvard University, John Hopkins University, Kellogg School of Management, Massachusetts Institute of Technology, National Bureau of Economic Research, New York University, Princeton University,
Stanford University, Wharton School of Business, the Econophysics conferences in Indonesia and Japan, the
Western Finance Association and Econometric Society meetings. We thank Fernando Duarte, Tal Fishman,
and David Kang for their research assistance. Gopikrishnan's contribution was part of his Ph.D. thesis. We
thank Morgan Stanley, the National Science Foundation and the Russell Sage Foundation for support.
*We thank our
beris, Olivier
and constant discounting. 1 Even
after the fact,
using only observable news [Cutler, Poterba and
We present
a model in which volatility
is
it is
hard to explain changes in the stock market
Summers
caused by the trades of large institutions. Institutional
investors appear to be important for the low-frequency
Gompers and Metrick
sheds light on
2004; Cohen,
many
[2001].
issues,
1989; Fair 2002; Roll 1988].
movements
of equity prices, as
shown by
Understanding better the behavior of institutional investors also
such as
momentum and
Gompers and Vuolteenaho
2002; Choe,
positive feedback trading [Chae
Kho and
and Lewellen
Stulz 1999; Hvidkjaer 2005], bubbles
[Brunnermeier and Nagel 2004], liquidity provision [Campbell, Ramadorai and Vuolteenaho 2005],
and the importance of indexing [Goetzman and Massa 2003]
how
trading by individual large investors
may
create price
.
We
further this research by analyzing
movements that
are hard to explain by
fundamental news.
In our theory, spikes in trading volume and returns are created by a combination of news and
the trades by large investors.
Suppose news or proprietary analysis induces a large investor to
trade a particular stock. Since his desired trading volume
is
then a significant proportion of daily
turnover, he will moderate his actual trading volume to avoid paying too
The optimal volume
will
much
in price
impact. 2
nonetheless remain large enough to induce a significant price change.
Traditional measures, such as variances and correlations, are of limited use in analyzing spikes
market
in
activity.
Many
empirical
moments
are infinite; moreover, their theoretical analysis
is
typically untractable. 3 Instead, a natural object of analysis turns out to be the tail exponent of the
distribution, for
which some convenient analytical techniques apply. Furthermore, there
empirical evidence on the
tails of
is
much
the distributions, which appear to be well approximated by power
laws. For example, the distribution of returns r over daily or weekly horizons decays according to
P(\r\
>
x)
behavior
is
~
x~^r where £r
useful to guide
is
the
tail or
Pareto exponent. 4
and constrain any theory of the impact of large
our theory unifies the following stylized
(ii)
~
3;
the power law distribution of trading volume, with exponent £g
~
1.5;
(iii)
the power law distribution of price impact;
(iv)
the power law distribution of the size of large investors, with exponent £5
~
1.
'See also Campbell and Shiller [1988], French and Roll [1986], LeRoy and Porter [1981].
See Section II. C.
2
3
The
variance of volume and the kurtosis of returns are infinite. Section
Appendix
1
reviews the relevant techniques.
II
tail
investors. Specifically,
facts,
the power law distribution of returns, with exponent £r
(i)
This accumulated evidence on
provides more details.
Existing models have difficulty in explaining facts
behavior in general, but also the specific exponents. For example,
on news
move stock
to
prices
~
3.
However, there
hypothesis that justifies this assumption. Similarly,
to be fine-tuned to replicate the exponent of
We
rely
efficient
markets theories rely
and thus can explain the empirical finding only
distributed with an exponent £r
on previous research to explain
3.
is
power law
together, not only the
(i)-(iv)
if
news
nothing a priori in the
is
power law
efficient
GARCH models generate power laws,
markets
but need
5
(iv),
and develop a trading model
to explain
We
(iii).
use these facts together to derive the optimal trading behavior of large institutions in relatively
illiquid
The
markets.
fat-tailed distribution of investor sizes generates a fat-tailed distribution of
Whe we
volumes and returns.
derive the optimal trading behavior of large institutions,
to replicate the specific values for the
power law exponents found
In addition to explaining the above facts, an analysis of
tail
in stylized facts
behavior
(i)
may have
we are able
and
a
(ii).
6
number
of
wider applications in option pricing,' risk management, and the debate on the importance of large
returns for the equity
premium
Our paper draws on
Weitzman
[Barro 2005; Rietz 1988; Routledge and Zin 2004;
several literatures.
The behavioral
2005].
finance literature [Barberis and Thaler
2003; Hirshleifer 2001; Shleifer 2000] describes mechanisms by which large returns obtain without
significant changes in fundamentals.
We
idiosyncratic trades of institutions.
The microstructure
O'Hara
1995]
and Spatt 2005;
literature [Biais, Glosten
shows that order flow can explain a large fraction of exchange rate movements [Evans
and Lyons 2002] and stock
and Seppi
propose that these extreme returns often result from large
price
movements, including the covariance between stocks [Hasbrouck
Previous papers combine these behavioral, microstructure and asset pricing
2001].
el-
ements to explain the impact of limited liquidity and demand pressures on asset prices [Acharya
and Pedersen 2005; Gompers and Metrick 2001; Pritsker 2005;
ravskaya 2002].
We
complement
this research
by focusing on
Shleifer 1986;
tail
Wurgler and Zhu-
behavior, partially in the hope
that understanding extreme events allows us to understand standard market behavior better.
This article
to study
is
also part of a broader
economic
issues,
movement
utilizing concepts
and methods from physics
a literature sometimes referred to as "econophysics"
8
.
Econophysics
GARCH models are silent about the economic origins of the tails, and about trading volume.
This includes the relative fatness documented by facts (i), (ii) and (iv) (note that a higher exponent means a
thinner tail). Since large traders moderate their trading volumes, the distribution of volumes is less fat-tailed than
°Also,
6
that of investor
volume
7
sizes. In turn,
a concave price impact function leads to return distributions being
less fat-tailed
than
distributions.
Our theory
indicates that trading
volume should help forecast the probability
of large returns.
Marsh and Wagner
[2004] provides evidence consistent with that view.
Antecedents are Simon [1955] and Mandelbrot
[1963].
More
recent research includes Bak, Chen, Scheinkman,
is
similar in spirit to behavioral economics in that
behavior, and explores their implications.
psychological microfoundations, and
Section
II
tains our baseline
V
Section
Appendix
1 is
postulates simple plausible rules of agent
it
differs
by putting
results of the interactions
emphasis on the
less
among
agents.
behavior of financial variables. Section
tail
model that connects together power
concludes.
II.
However,
more on the
presents stylized facts on the
it
laws. Section
IV
then con-
III
discusses various extensions.
a primer on power law mathematics.
The Empirical Findings That Motivate Our Theory
This section presents the empirical facts that motivate our theory, and provides a self-contained
tour of the empirical literature on power laws.
~
II. A.
The Power Law Distribution
The
distribution of returns has been analyzed in a series of studies that uses an ever increasing
tail
number
of Price Fluctuations: (r
3
and de Vries 1991; Lux 1996; Gopikrishnan, Plerou, Amaral, Meyer,
of data points [Jansen
and Stanley 1999; Plerou, Gopikrishnan, Amaral, Meyer, and Stanley
The
logarithmic return over a time interval At.
1999].
Let r t denote the
distribution function of returns for the 1,000
i
largest U.S. stocks
and
several
major international indices has been found to
P{\r\>x)
(1)
Here,
~
positive
9
^-withCr-3.
denotes asymptotic equality up to numerical constants. 10
and negative returns separately and
be:
is
This relationship holds for
best illustrated in Figure
I.
It
plots the cumulative
probability distribution of the population of normalized absolute returns, with In a; on the horizontal
axis
and lnP(|r| > x) on the
vertical axis. It
lnP(|r|
(2)
and Woodford
[1993],
Bouchaud and Potters
>
x)
[2003],
shows that
= — Cr hix +
Gabaix
constant
[1999, 2005], Plerou, Gopikrishnan,
Amaral, Meyer, and
Stanley 1999], Levy, Levy, and Solomon [2000], Lux and Sornette [2002], Mantegna and Stanley [1995, 2000]. See
also Arthur, LeBaron, Holland, Palmer, and Tayler [1997], Blume and Durlauf [2005], Brock and Hommes [1998],
[1993], Jackson and Rogers [2005] for work in a related vein.
To compare quantities across different stocks, we normalize variables such as r and q by the second moments if
they exist, otherwise by the first moments. For instance, for a stock i, we consider the returns r\ t = {tu — n) /ov.i,
where r is the mean of the r« and ay,i is their standard deviation. For volume, which has an infinite standard
deviation, we use the normalization q'it = qn/qi, where qu is the raw volume, and gT is the absolute deviation:
Durlauf
t
<?i
=
\qn
-qitY
Formally, f(x)
~ g (x)
means /
(i) j g (x) tends
toward a positive constant (not necessarily
1)
as
x
—
>
oo.
yields a
—3.1
good
± 0.1,
should be
-3: in
1
for
\r\
(1). It is
i.e.,
Equation
to
fit
as
between 2 and 80 standard deviations.
estimation yields
not automatic that this graph should be a straight
a Gaussian world
u
OLS
it
line, or
that the slope
would be a concave parabola. In the following, we
the cubic law of returns"
— Cr =
shall refer
.
Insert Figure
I
here
Insert Figure II here
Furthermore, the 1929 and 1987 "crashes" do not appear to be outliers to the power law
distribution of daily returns [Gabaix, Gopikrishnan, Plerou,
and Stanley
2005].
Thus there may
not be a need for a special theory of "crashes": extreme realizations are fully consistent with a
fat-tailed distribution.
Equation
1999].
1
12
appears to hold internationally [Gopikrishnan, Plerou, Amaral, Meyer, and Stanley
For example, Figure
indices are very similar.
III,
shows that the distribution of returns
for three different
country
13
Insert Figure III here
Having checked the robustness of the (r
~
3 finding across different stock markets, Plerou,
Gopikrishnan, Amaral, Meyer, and Stanley [1999] examine firms of different
have higher
also
volatility
than large firms, as
shows similar slopes
for the
is
graphs of
distribution of each size quantile by
its
verified in Figure IVa.
all
four distributions. 10
sizes.
14
Small firms
Moreover, the same diagram
Figure IVb normalizes the
standard deviation, so that the normalized distributions
all
The particular value £ r ~ 3 is consistent with a finite variance, but moments higher than 3 are unbounded. ( r ~ 3
contradicts the "stable Paretian hypothesis" of Mandelbrot [1963], which proposes that financial returns follow a Levy
1
'
A Levy distribution has an exponent £r < 2, which is inconsistent with the empirical evidence
[Fama 1963; McCulloch 1996; Rachev and Mittnick 2000].
'"Section IV.D reports quotes from the Brady report, which repeatedly marvels at how concentrated trading was
on Monday, October 19, 1987.
The empirical literature has proposed other distributions. We are more confident about our findings as they rely
on a much larger number of data points, and hence quantify the tails more reliably. We can also explain previous
findings in light of ours. Andersen, Bollerslev, Diebold, and Ebens [2001] show that the bulk of the distribution of
realized volatility is lognormal. In independent work, Liu, Gopikrishnan, Cizeau, Meyer, Peng, and Stanley [1999]
show that while this is true, the tails seem to be power law.
H Some studies quantify the power law exponent of foreign exchange fluctuations. The most comprehensive is
probably Guillaume, Dacorogna, Dave, Miiller, Olsen, and Pictet [1997], who calculate the exponent £r of the price
movements between the major currencies. At the shortest frequency At = 10 minutes, they find exponents with
average Q r = 3.44, and a standard deviation 0.30. This is tantalizingly close to the stock market findings, though the
standard error is too high to draw sharp conclusions.
15
There is some dispersion in the measured exponent across individual stocks [Plerou, Gopikrishnan, Amaral,
Meyer, and Stanley 1999]. This is expected, as least because measured exponents are noisy. Proposition 5 makes
stable distribution.
1
predictions about the determinants of a possible heterogeneity in the exponents.
have a standard deviation of
to (r
~
1.
The
plots collapse
on the same curve, and
all
have exponents close
3.
Insert Figure
The above
IV here
results hold for relatively short time horizons
- a day or
less.
16
Longer-horizon return
sum
distributions are shaped by two opposite forces.
One
law distributed variables with exponent £
power law distributed, with the same exponent
Thus one expects the
The second
force
is
tails of
is
also
is
that a finite
the central limit theorem, which says that
becomes more Gaussian, while the
smaller probability, so that they
33-35] for
of independent
power
17
£.
monthly and even quarterly returns to remain power law distributed.
of the distribution converges to Gaussian. In sum, as
[2003, p.
force
tails
may
if
T
returns are aggregated, the bulk
we aggregate over
T
returns, the central part
remain a power law with exponent
but have an ever
£,
not even be detectable in practice. See Bouchaud and Potters
an example. In practice, the convergence to the Gaussian
returns were independently and identically distributed
(i.i.d.),
and one
still
horizons [Plerou, Gopikrishnan, Amaral, Meyer, and Stanley 1999, Figure
is
slower than
if
sees fat tails at yearly
9].
A
likely
explanation
the autocorrelation of volatility and trading activity [Plerou, Gopikrishnan, Amaral, Gabaix, and
is
Stanley 2000]. 18 »A useful extension of the present model would allow the desire to trade (or signal
occurrences) to be autocorrelated, and might generate the right calibration of autocorrelation of
volatility
and slow convergence to a Gaussian.
In conclusion, the existing literature shows that while high frequencies offer the best statistical
resolution to investigate the
horizons, such as a
II. B.
To
month
tails,
power laws
still
appear relevant
for the tails of returns at longer
or even a year.
The Power Law
Distribution of Trading Volume: (q
better constrain a theory of large returns,
it
is
~
3/2
helpful to understand the structure of large
trading volumes. Gopikrishnan, Plerou, Gabaix, and Stanley [2000] find that trading volumes for
Our analysis does not require exact power laws. It is enough that an important part of the tail distribution is well
approximated by a power law. For instance, lognormal distributions with high variance are often well approximated
by Pareto distributions. The exponent is then interpreted as a local exponent, i.e. ( (x) = —xp' (x) /p(x) — 1, rather
than a global exponent:
This is one of the aggregation properties of power laws reviewed in Appendix 1.
Aggregation issues may also be important to understand the dispersion of exponents [Plerou, Gopikrishnan,
Amaral, and Stanley 1999].
'" Dembo, Deuschel and Duffie [2004], Ibragimov [2005] and Kou and Kou [2004] develop further the importance of
1
1
'
1
fat tails in finance.
20
the 1,000 largest U.S. stocks are also power law distributed:
P (q >
(3)
The
P
(<?)
~
precise value estimated
9"" 2 5
'
ii
and Mills
e -i (3)-
is
The exponent
[2001] likewise find ( q
=
Qq
x)
=
j-
1.53
±
Figure
.07.
V
1.4
±
0.1 for the
test the robustness of this result,
the density satisfies
illustrates:
of the distribution of individual trades
is
close to 1.5.
Maslov
volume of market orders.
Insert Figure
To
C ^ 3/2.
with
V
here
we examine 30
large stocks of the Paris Bourse
from
1995-1999, which contain approximately 35 million records, and 250 stocks of the London Stock
Exchange
in 2001.
As shown
in
Figure V, we find
=
C,
q
1.5
±
0.1 for
each of the three stock
markets. The exponent appears essentially identical in the three stock markets, which
is
suggestive
of universality.
The low exponent
trading
is
(g
~
3/2 indicates that the distribution of volumes
very concentrated.
Indeed, the
1
is
very fat failed, and
percent largest trades represent 28.5 percent (±0.6
percent) of the total volume traded. 21
The power law
of individual trades continues to hold for volumes that are aggregated (for a
given stock) at the horizon At
15 minutes [Gopikrishnan, Plerou, Gabaix, and Stanley 2000]:
P (Q >
(4)
We
be
=
refer to
x)
~
-J- with Cq
m
Equation 3-4 as the "half-cubic law of trading volume".
It is
intriguing that the exponent of returns should be 3
1.5.
To
see
3/2.
if
there
is
and the exponent
of volumes should
an economic connection between those values, we turn to the relation
between return and volume.
II. C.
The Power Law
The microstructure
We
define
~ V1
[1993, 1995] estimate a range of 0.3 to 1 percent;
volume as the number of shares traded. The dollar value traded
a given security,
"'The
r
literature generally confirms that substantial trades can have a large impact.
Chan and Lakonishok
"
of Price Impact:
it is
Keim and Madhavan
[1996]
yields very similar results, since, for
number of shares traded.
±0.3 percent of the total volume
essentially proportional to the
0.1 percent largest trades represent 9.6
on the 100 largest stocks of the Trades and Quotes database
in the
traded.
period 1994-5.
We
computed the
statistics
find 4 percent for smaller stocks.
prices: see
A
Brady
There are also many anecdotal examples of large investors affecting
[1988], Corsetti, Pesenti,
simple calculation illustrates
significantly.
The
Wang
hence daily turnover
2001]:
sell its entire
=
0.5/250
is
e.g.,
is
Coyne and Witter
that a large fund can
[2002].
move
22
the market
50 percent of the shares outstanding [Lo and
0.2 percent
based on 250 trading days per year.
the 30th largest fund. At the end of 2000, such a fund held
market and hence, on average,
0.1 percent of the
[2002],
why one can expect
typical yearly turnover of a stock
Consider a moderately large fund,
To
and Roubini
0.1 percent of the capitalization of a given stock.
holding, the fund will have to absorb 0.1/0.2 or half of the daily turnover.
23
This
supports the idea that large funds are indeed large compared to the liquidity of the market, and
that price impact will therefore be an important consideration.
We
r
(5)
with
fc
V
next present evidence that the price impact r of a trade of size
>
0,
< 7<
1,
~
scales as:
k\P,
which yields a concave price impact function [Hasbrouck 1991, Hasbrouck
=
and Seppi 2001; Plerou, Gopikrishnan, Gabaix, and Stanley 2002]. The parameterization 7
is
often used,
Kahn
[1999],
by Barra
e.g.,
Hasbrouck and Seppi
Equation 5 implies (r
the value 7
[1997],
=
1/2
is
=
1/2
Gabaix, Gopikrishnan, Plerou, and Stanley [2003], Grinold and
[2001].
(v/l by
rule (42) in
Appendix
Hence, given (r
1.
a particularly plausible null hypothesis.
From
=
3
and (y
this relationship,
=
we
3/2,
see a
natural connection between the power laws of returns and volumes.
The exact value
7
=
1/2.
We
start
volumes Vi,...,Vn
Q=
(6)
,
of
7
is
a topic of active research.
from the benchmark where,
of independent signs £j
Y^i=\ ^' an d aggregate return
We report
here evidence on the null hypothesis
in a given time interval,
= ±1
n blocks
with equal probability.
are traded, with
Aggregate volume
is
is:
r
= u + k^T,e V 1/2
i
i
"See also Chiyachantana, Jain, Jiang, and Wood [2004] for international evidence, and Jones and Lipson [2001]
and Werner [2003] for recent U.S. evidence.
"3
It had $19 billion in- assets under management. The total market capitalization of The New York Stock Exchange,
the Nasdaq and the American Stock Exchange was $18 trillion.
where u
is
some other orthogonal source
E
\r
2
I
Ql
=
+
el
k
2
of price
movement. 24 Then,
E
o*
E
(7)
[r
2
|
Q]
=
a u2
VI reveals an
results of Figure
2
k Q.
VI here
Insert Figure
Our
+
+ k 2 Q + 0.
affine relation predicted
rather than any clear sign of concavity or convexity.
A
by Equation 7
for large
volumes Q,
formal test that we detail in Appendix 3
confirms this relation.
Measuring price impact and
its
following reasons. First, order flow
all
dependence on order
size
is
a complex problem due to the
and returns are jointly endogenous. To our knowledge, virtually
empirical studies including ours, suffer from this lack of exogeneity in order flow. 25
Second, the unsplit size of orders
size of individual trades q,
is
unobservable in most liquid markets.
not the size of the desired block V.
If
One
observes the
one does not pay attention
to aggregation, different exponents of price impact are measured, depending on the time horizon
chosen [Plerou, Gopikrishnan, Gabaix, and Stanley 2002, 2004; Farmer and Lillo 2004]. 26
Third, order flow
Potters,
is
autocorrelated [Froot, O'Connell and Seasholes, 2001; Bouchaud, Gefen,
and Wyart 2004;
of different traders.
It is
Lillo
and Farmer
also predicted
Chriss 2000; Berstimas and
Lo
2004]. This autocorrelation could
come from the actions
by models of optimal execution of trades [Almgren and
1998; Gabaix, Gopikrishnan, Plerou, and Stanley 2003], as large
transactions are split into smaller pieces. 27
Although the empirical evidence we gathered
impact more accurately
flow.
and
In particular,
split of
it
will require better
is
suggestive, measuring the curvature
7 of price
data and a technique to address the endogeneity of order
would require knowing desired trading volumes, magnitude of price impact
trades for a set of large market participants. In the meantime,
"4
we consider evidence such
Via Equation 6, a model such as ours provides a foundation for stochastic clock representations of the type
proposed by Clark [1973].
An exception is Loeb [1983], who collected bids on different size blocks of stock. Barra [1997] and Grinold and
Kahn [1999, p. 453] report that the best fit of the Loeb data is a square root price impact.
' In a related way, part of the linearity of Equation 7 can arise because in some simple models total volume and
squared returns depend linearly on the number of trades [Plerou, Gopikrishnan, Amaral, Gabaix, and Stanley 2000].
27
If the trades are executed in the same time window, Equation 7 still holds. If they do not, the estimate of 7 is
typically biased
downward
[Plerou, Gopikrishnan, Gabaix,
and Stanley
2004],
VI
as Figure
possible that the true relationship
it is
why we
II. D.
between volume and squared return. However,
as supportive of a linear relationship
is
different, or
may
vary from market to market. This
is
present a theory with a general curvature 7.
The Power Law Distribution
of the Size of Large Investors: £s
~
1
highly probable that substantial trades are generated by very large investors. This motivates
It is
us to investigate the size distribution of market participants.
A
power law formulation:
1
often yields a good
fit.
The exponent £g
both
~
1,
often called Zipf's law,
is
of firms in general follows Zipf's law,
management
it is
[Axtell 2001;
(20 percent cutoff) via
across years of 0.08. 30
deviation of 0.07.
OLS.
The
We
and Aref
money management
is
Hence we conclude
If
the distribution
technique gives a
that, to a
money
[2004] find this is the
under management) of
find an average coefficient
Hill estimator
true for
the core business: mutual funds. 28
we estimate the power law exponent
t,
is
management.
to obtain the size (dollar value of assets
from 1961-1999. For each year
relation
Okuyama, Takayasu,
and Souma 2004].
firms in particular follows Zipf's law. Indeed, Pushkin
investigate firms for which
CRSP
use
Gallegati,
common. This
plausible to hypothesize that the distribution of
case for U.S. baAk sizes, measured by assets under
We
particularly
Gabaix and Ioannides 2004] and firms
cities [Zipf 1949;
and Takayasu 1999; Fujiwara, Di Guilmi, Aoyama,
'
^_L
P(5>x)
(8)
Q =
mean
1.10,
all
We
mutual funds
29
£ of the tail distribution
with a standard deviation
estimate
Q=
0.93
and a standard
good approximation, mutual fund
sizes follow a
power law distribution with exponent:
(s
(9)
*
1.
For this paper, we can take this distribution of the sizes of mutual funds as a given.
fact,
not difficult to explain.
One can apply
the explanations given for
cities
It is, in
[Simon 1955; Gabaix
the main findings. Gabaix, Ramalho, and Reuter [2005] present much more detail.
count as x different funds, not as one big "Fidelity" fund.
The estimates
'"'We cannot conclude that the standard deviation on our mean estimate is 0.08 (1999 — 1961 + 1)~
"
"9
Here we sketch
The x funds of
Fidelity, for instance,
are not independent across years, because of the persistence in mutual fund sizes.
10
1999; Gabaix and Ioannides 2004] to mutual funds. Suppose that the relative size
fund
i
follows a
random growth process Su = S^t-i
minor element of
follows Zipf's law with
=
C,s
£it),
with en
of a
and mean
i.i.d.
0.
mutual
Add
a
a steady state distribution; for instance, very
friction to small funds to ensure
small funds are terminated and are replaced by
+
(1
Su
new
Then,
funds.
this steady state distribution
1.
Gabaix, Ramalho and Reuter [2005] develop this idea and show that these assumptions are
verified empirically. This
means that the random growth
distribution to satisfy Zipf's law,
i^g
=
of
mutual funds generically lead their
size
1.
Insert Figure VII here
It is
only in the past 30 years that mutual funds have
ketplace.
It
would be interesting
follow Zipf's law, as the
The evidence we
small
number
have evidence on the
to
became important. For
before mutual funds
number
is
very
size distribution of financial institutions
instance, pension funds of corporations are likely to
necessarily tentative. Estimating a
is
and
difficult,
all
participants in the U.S. market.
their
found to describe the
to describe well the
as a
we had
access to only a subset of the
institutions. It
would be useful to weight the funds by
annual turnover. Nevertheless, given that Zipf's law (Equation 9) has been
size of
upper
many
tail
other entities, such as banks and firms in general, and appears
of the empirical distribution of mutual funds,
we view Equation
9
good benchmark.
Summary and Paradoxes
II. E.
The
relatively
Other important participants are hedge funds, pension funds,
and proprietary trading desks, and foreign
and
power law with a
estimators require somewhat arbitrary parameters
[Embrechts, Kluppelberg, and Mikosch 1997]. Furthermore,
their leverage
to represent a large part of the mar-
of employees in firms follow Zipf's law.
present here
of points
come
facts
summarized
difficulties in
GARCH
'it
important challenges.
may be
First,
economic theories have
explaining the power law distribution of returns, as the efficient market theory, and
models, need to be fine tuned to explain
exponent of
satisfies
in this section present
why
the distribution of returns would have an
3.
useful to give a short proof.
the forward Kolmogorov equation
and a cumulative distribution
P (S >
x)
=
Suppose the process
=
dtp
k/x,
=
i.e.,
\-j§i {c
2
Zipf's law.
11
is:
dSt
=
StadBt- The steady state density p(S)
S 2 p(S)). This
implies p(S)
=
k/S 2
for a constant k,
Second,
it is
institution size
and trading
surprising that the Pareto exponent of trading volume
is
£g
policies,
~
In models with frictionless trading,
1.
except that they are scaled by the size
S
is
(q
~
1.5,
while that of
agents have identical portfolios
all
of the agents (which corresponds to
wealth). Hence frictionless trading predicts that the distribution of trading volume of a given stock
should
Cg
>
reflect the distribution of
33
A
Cs-
likely
cause
is
institutions, because price
Finally, the basic price
its
impact
is
(q
=
Cs
—
monotonically increasing in trade
=
=
£q
.
To explain why
I-
32
However, we find that
size.
Cg/Cr
is
close to 1/2,
we require a model with
1/2.
present a model that attempts to resolve the above paradoxes. 34
The Model
III.
for
i.e.
impact model [Kyle 1985] predicts a linear relation between returns and
curvature of price impact 7
We
investors,
the cost of trading; large institutions trade more prudently than small
volume, which would imply (T
We now
the size of
We
consider a large fund in a relatively illiquid market.
the price impact of
its
trades. Next,
the core contribution of this paper.
One
we
link the various
first
describe a rudimentary
power law exponents;
model
this represents
could employ different microfoundations for price impact
without changing our conclusions.
III. A.
A
Simple Model to Generate a Power
Law
Before presenting, in Section III.B, the core of the model, we
for the
The
square root price impact.
Subsequent models, such as Seppi
[1996], generate
basic
[1990],
model
first
present a simple microfoundation
of Kyle [1985] predicts a linear price impact.
Barclay and Warner [1993], and
Keim and Madhavan
a concave impact in general. Zhang [1999] and Gabaix, Gopikrishnan, Plerou, and
Stanley [2003] produce a square root function in particular. 30
''"Solomon and
Price Impact
Richmond
[2001] have
proposed a model that
relies
are sympathetic to this approach that links wealth to volumes.
of institutions, rather
The model used
in this section
on a scaling exponent of wealth £s
In the present study
than individual wealth, because most very large trades are
rather than by private individuals. Also, the Pareto exponent of wealth and income
we use the
likely to
is
=
3/2.
We
size distribution
be done by institutions
[e.g., Davies and
quite variable
Shorrocks 2000; Piketty and Saez 2003].
The lower the power law exponent, the fatter the tails of the variable. See Appendix 1.
ii
Gabaix, Gopikrishnan, Plerou, and Stanley [2003] presents a reduced form of some elements of the present
N
io
is
article.
= V 1 ^"
Gabaix, Gopikrishnan, Plerou, and Stanley [2003] predicts that a trade of size V will be traded into
smaller chunks. This has the advantage of generating a power law distributions of the number of trade with exponent
Cn = 2£v = 3, which is close to the empirical value [Plerou, Gopikrishnan, Amaral, Gabaix, and Stanley 2000]. We
did not keep
it in
the current model, because we wanted to streamline the microfoundation of price impact.
12
a formalized version of a useful heuristic argument, sometimes called the "Barra model" of Torre
and
Ferrari [Barra 1997].
We
consider a single risky security in fixed supply, with a price p
("he") buys or
The timing
the security from a liquidity supplier ("she").
sell
at time
(t)
The
t.
of the
large fund
model
as
is
follows:
At time
is
t
=
0,
the fund receives a signal
M about mispricing. M <
is
a
a buy signal. Without loss of generality, we study a buy signal. The analysis
sell signal,
M>
symmetric
is
for a
sell signal.
At
=
t
1
—
2e (e
a small positive number), the fund negotiates a price with the liquidity
For simplicity, we assume liquidity provision
supplier.
bargaining power.
where p
is
The
2e)
the transaction
At
t
= 1 — e,
At
t
=
between
difference
the price before impact and
is
7r
and
R
is
= p + n (V),
t
=
1
is
the price concession, or
where n (V)
a standard
Brownian motion with
of the stock at price p(t): she
is
liquidity supplier continues to
time
The
is
full
p + R,
price impact.
the permanent price impact.
=R—
tt.
Equilibrium
price concession
36
will
The
determine
R(V).
+ TT{V) + aB(t),
B (1) =
0.
Also, at
replenishing her inventory. She continuously meets sellers
after a
of shares, at a price
public.
n(V) and the
P {t)=p
B
V
full
onwards, the price follows a random walk with volatility a:
(10)
where
is
the temporary price impact t
the value of the permanent price impact
From
R
announced to the
the price jumps to p (1)
1,
competitive so that the fund has
liquidity supplier sells to the fund the quantity
= p (1 —
is
is
T=
V/V. The
who
t
=
1,
the liquidity supplier starts
a price taker as she can credibly assure that she
buy shares
until her inventory
Vdt
are willing to sell her a quantity
is
is
not informed.
fully replenished,
The
which happens
price continues to evolve according to (10).
liquidity provider benefits
from the temporary price impact
tainty as she replenishes her inventory [Grossman
and
Miller 1988].
r,
but then faces price uncer-
To evaluate
these effects,
we
To keep the mathematics simple, the impact is additive, and the price otherwise follows a random walk. It is easy,
though cumbersome, to make the price impact proportional and the log price follow a random walk. Our conclusions
about the power law exponents would not change.
7
We wish to add two comments about the timing of the model. In our model, the large fund trades in one block
(at time t = 1), and the liquidity provider trades in many smaller chunks (at time ( £ (1,1 + T]). Alternative timing
assumptions would leave the scaling relations unchanged (12), with the same 7. Also, the exchange between the large
fund and the liquidity supplier is an "upstairs" block trade. In an upstairs trade, the initiator typically commits not
to repeat the trade too soon in the future. This prevents many market manipulation strategies that might otherwise
be possible with a non-linear price impact, such as those analyzed by Huberman and Stanlz [2004].
13
assume that the
amount
liquidity provider has the following
U = E[W] -
risk of
>
and
>
5
The
0.
standard deviation
preferences, 5
=
2.
In
=
l.
X[var (W)]
liquidity supplier requires
i.e.
cr,
many
which corresponds to 5
One
variance utility function on the total
W of money earned during the trade
(11)
with A
mean
,
compensation equal
to
Xa s
to bear a
has "6-th order risk aversion." 38 With standard mean- variance
cases, a better description of
what behavior
is
first-order risk aversion,
comes from psychology.
Prospect theory [Kah-
39
justification for first-order risk aversion
neman and Tversky
s/2
1979] presents psychological evidence for this behavior,
which has also been
formalized in disappointment aversion [Gul 1991; Backus, Routledge, and Zin 2005]. Second,
order risk aversion
[1990]
is
frequently needed to calibrate quantitative models, such as Epstein
and Barberis, Huang, and Santos
reflect a value at risk penalty,
where A
[2001].
Another institutional
rule to accept trades
if
if
their
third justification
is
justification
Sharpe ratio
is
and Zin
institutional, as (11)
the size of the penalty, and var
is
to the value at risk.
and only
A
(W)
via the Sharpe ratio.
can
proportional
'
is
If
a trader uses a
greater than A, then he will behave as
is
first-
if
he
exhibits first-order risk aversion.
Proposition
,
1
The setup
of this section generates the temporary price impact function:
r{V)
(12)
with
H = \a
S/2
5
/
(3V)
= HV'
r
and
7=y-l.
For future reference,
Proposition 2
If
it is
useful to state separately our central case.
the liquidity provider
is first
order risk averse, then the price impact increases
with the square root of traded volume:
7=1/2
Essentially all non-expected utility theories need a postulate on how different gambles are intergrated. Here, we
assume that the liquidity provider evaluates individually the amount
earned in the trade.
3g
The model also generates a square root price impact with a different specification that generates first order risk
aversion, for instance the loss averse utility function: U = E [max (W, 0)] + A£ [min (W, 0)] with A > 1.
W
14
and
t(V)
(13)
a
is
V
\o(^
/
=
\
1/2
the daily volatility of the stock, and A the risk aversion of the liquidity provider.
V
In practice,
is
likely to
be proportional to the daily trading volume.
Hence the scaling
predictions of Equation 13 can be almost directly examined.
The proof
is
in
V
buy back the
to
Appendix
shares.
2.
The
intuition
that the liquidity provider needs a time
is
During that time, the price
diffuses at a rate a.
provider faces a price uncertainty with standard deviation
is first
r
order risk averse, the price concession r
~ ay/V
To
,
i.e.,
Equation
close the model,
is
ayT ~ uyV
.
If
T = V/V
Hence the
liquidity
the liquidity provider
proportional to the standard deviation, hence
13.
we need
to determine both the
permanent and the
full
price impact.
The
de-
termination of these two variables typically depends on the fine details of the information structure
processed by the other market participants.
We
use a somewhat indirect route, which drastically
simplifies the analysis.
Assumption
We
1
assume that the market uses a
R{V) = Bt(V)
(14)
some
for
linear rule to determine the full price impact,
B >
0.
Section IV. A presents conditions under which the linear rule (14)
optimal.
Assumption
1
closes the price
impact part of the model.
Proposition 3 The above setup generates the
R(V) = hVi
(15)
where h
price concession function
= BH,
and
H
and 7 are determined
in Propositions
15
1
and
2.
is
actually
The Core Model: Behavior
III.B.
Fund
of a Large
We now
lay out the core of our model.
tunities,
which indicate that the excess risk-adjusted return on the asset
are independent. s t
Mt
mispricing.
Assumption
The model
is
in fact real.
mispricing
is
If
We
assume that
M
misspecification risk
t
not too fat-tailed:
is
is
C
is
st
M C.
t
st
,M
t
and
C
the expected absolute value of the
E [M 1+1
to be not too fat-tailed.
/7
]
<
oo.
captures uncertainty over whether the perceived mispricing
For example, the fund's predictive regressions
may have
C=
M
the sign of the mispricing.
since been arbitraged away.
signals the fund perceives are pure noise,
is 0.
periodically receives signals about trading oppor-
drawn from a distribution / (M), which we assume
is
2
= ±1
The fund
C*, the mispricings are real.
C
may
from data mining, or the
result
can take two values,
and C*
.
If
C=
0,
the
and the true average return on the perceived mispricings
We
specify
E C
1,
so that
M represents the expected
value of the mispricing.
The fund has S
R (Vt),
dollars in assets. If
the total return of
its
portfolio
rt
(16)
where
If
ut is
mean
the model
to
buys a volume
is
price concession
= V (CM -R(V )+u )/S
t
t
wrong, expected returns
t
t
are:
t
assume that the manager has a concern
—A
be below some value
percent
if
Equation 18 can be
is
t
t
He does
for robustness.
|
C=
model
not want his expected return
wrong. Formally, this means:
is
> -A.
0J
One
is
a psychological attitude towards model
and Schmeidler
[1989]
and Hansen and Sargent
[2005].
a useful rule of thumb, that can be applied without requiring detailed infor-
fine details of
and Vishny
t
justified in several ways.
uncertainty, developed in depth by Gilboa
Second, Equation 18
Q\=-V R{V )/S.
his trading
E\r
(18)
[Shleifer
and pays a
is:
E\r \C =
mation about the
Vj of the asset,
zero noise.
(17)
We
it
model uncertainty.
1997]. If trader ability
is
A
third explanation
uncertain, investors
16
may
is
delegated
management
wish to impose a constraint
such as Equation 18 to prevent excessive trading.
To
simplify the algebra,
we assume
wants to maximize the expected value of
that, subject to the robustness constraint, the
his excess returns -E^r].
40
We now
summarize
manager
this.
Definition 1 Suppose that the fund has S dollars under management. The fund's optimal policy
a function
is
V(M,S)
mispricing of size
M.
V
that specifies the quantity of shares
It
E [r
maximizes the expected returns
t
traded when the fund perceives a
subject to the robustness constraint
]
(18):
max E\r
(19)
t]
E
subject to
rt
I
C=
> -A
V(M,S)
III.C.
We
Optimal Strategy and Resulting Power Law Exponents
can now derive the large fund's strategy. Given Equation
1
max -
f°°
/
16, Definition 1 is equivalent to:
V{M,S)(M -R{V(M,S)))f(M)dM
V(M,S) b Jo
s.t.
Appendix 2
~
V(M,S)R {V (M, S)) f (M) dM >
I
J Jo
-A.
establishes the following Proposition.
Proposition 4
If
constraint (18) binds, the optimal policy
to trade a volume:
is
V (M, S) = vM ^S ll{l+l)
l
(20)
The
price change after the trade
is:
(21)
R(M,S) = hv^MS l/{ l+l)
for a positive constant v, defined in
Equation
-
Equation 21 means that price movements
M
,
40
u.
.
and the
One might
size of the
fund S.
48,
which
reflect
is
prefer the formulation raaxv(M.s)
E [u (r)]
subject to
many
decreasing in
h.
both the intensity of the perceived mispricing
Concretely, a large price
Fortunately, this does not change the conclusions in
A and
increasing in
movement can come from an extreme
E [u (r) C =
|
instances, such as
0]
> u — Ft), with a concave utility
= — e~ ar a > 0. On the other
u (r)
{
,
hand, with a non-linear function u the derivations are more complex, as they rely on asymptotic equalities, rather
than exact equalities. To keep things simple, we use the linear representation (19).
17
signal or the trade of a large fund [Easley
In the remaining analysis,
Assumption
we assume
and O'Hara
1987].
that (18) holds over the support of S.
3 The robustness constraint (18) binds for
funds in the market above a certain
all
size.
A
simple calibration presented in Appendix 2 shows that Assumption 3 holds for funds that
manage
less
than S*
Section IV. C shows a
We
=
Assumption
$21 trillion dollars.
way
to ensure
3
not very stringent.
is
Assumption 3 without any
finite size effects.
Alternatively,
41
next derive the distribution of volume and price changes.
Proposition 5 The traded volume and the
price changes follow
power law distributions with
respective exponents:
Cv
(22)
Ge
(23)
= min[(l+7)Cs,7CMl
= min
1
+-
JCs.Cm
Equation 21 implies that the distribution of price movements
coming from proprietary
analysis), as reflected in
on the news. Equation 23
two
tails of signals
and
M
sizes that matters.
of the agents that act
it is
the fatter of the
Mathematically, this comes from the properties (38)
exponent of the product of two independent random variables X\ and
tail
is
equal to the
exponent of the more fat-tailed variable,
X\ and X%- Economically,
is
S
illustrates the resulting exponent. In equilibrium,
power laws: the
trading volume,
both the "news" (perhaps
as well as the size
,
of
tail
reflects
this
means that the polar
captured when
Cm >
(
1
+~
)
(24)
Cv
=
(l
(25)
Cr
=
(l
)
the lower of the exponents of
affect the tail of
get:
Cs
+ -\
4
is
where large investors
Then, we
Cs-
+7
case,
i.e.,
X2
Cs
'Such cutoffs are generally present when handling power laws, and are sometimes called "border" or "finite size"
The cutoff affects only very little predictions. For instance, it affects the power law exponent of returns only
-3
3
by a factor 10
if a large fund has a size S = 10~ 5", which is a plausible empirical order of magnitude.
effects.
18
Equation 23 then means that, when there
is
the actions of a very large institution (the
of
news (the
M term).
Cutler, Poterba,
S
movement,
it is
and Summers
more
likely to
come from
term), rather than an objectively important piece
This potential importance of a large institution
Long Term Capital Management
the
a very large
may
explain why, during
the October 1987 crash, and the events studied by
crisis,
[1989], prices
moved
absence of significant news items.
in the
In
the context of our theory, the extreme returns occurred because some large institutions wished to
make
substantial trades in a short time period.
Proposition 5 says that
and returns are
also
when the
more
distribution of the size of institutions
fat-tailed.
when the
more
fat-tailed,
However, when the curvature 7 of price impact
returns are less fat-tailed, but volumes are
trade more moderately
is
more
price impact
fat-tailed.
is
The reason
We now
steeper.
is
is
volume
smaller,
that large institutions
apply Proposition 5 to our
baseline values.
Proposition 6 With a square root
—
(Qs
l)i
=
price impact (7
1/2)
and Zipf 's law
for financial institutions
volumes and returns foUow power law distributions, with respective exponents of 3/2 and
3.
(26)
Cv
=
3/2
(27)
Cfi
=
3.
These exponents are the empirical values of the distribution of volume and returns.
Proposition 6 captures our explanation of the origins of the cubic law of returns, and the halfcubic law of volumes.
(s
=
1-
The model
Random growth
of Section
III.
A
of
mutual funds leads
to Zipf 's
law of financial institutions,
leads to a power law price impact with curvature
7
=
1/2.
As
large funds wish to lessen their price impacts, their trading volumes are less than proportional to
their size. This generates a
power law distribution
the size distribution of mutual funds.
The
resulting exponent
value. Trades of large funds create large returns,
with exponent (r
=
of the size of trades that
is
£y
=
is less
3/2, which
fat-tailed
is
than
the empirical
and indeed the power law distribution of returns
3.
19
Robustness and Extensions
IV.
IV.A.
So
far
Permanent versus Transitory Price Impact
we have analyzed the
transitory
component
r:
full
R=
permanent and the
full
must come from an
inference,
n
+
We
r.
is
the
sum
of a
permanent component
which from Proposition 4
=E
(V)
?r
conditional expectation (28)
is
is
M
I
see
It is difficult to
hypothesize that
it is
know
<
£r.
The
case where they believe
is
equal to the exponent (n of the
&-=
the updaters believe
is
Cm <
a fraction b
£m <
full
(j?-
Then, the exponent (^
price impact,
and
is
given by
=
min
1
+-
a constant
is
'
)
.
>
b
s.t.,
for large
volumes, the permanent
is:
= E[M\
n{V)
updaters performing (28) believe
function that varies
Cr
0?> there
(30)
If
(28) believe
5,
(29)
price impact
particularly
close to the distribution of returns. This motivates the following Proposition.
permanent price impact
Proposition
1S
when
the distribution of mispricings perceived by other agents, one might
Proposition 7 Suppose that updaters performing
of the
Qm = Cr
how
M, and
not a directly observable quantity. However, these difficulties vanish in a class of cases -
plausible. If one does not
If
impact
hV 1 = hv^M^S l/{l+l)
complicated and can be non-linear.
agents use (28) with the belief that (,m
and
is:
agents would apply Bayes' rule to compute (28), which requires knowing the distribution of
M
ir
provide a sufficient condition that will ensure that the
price impact are proportional. In a Bayesian framework, the price
(28)
The
price impact R, which
Qm =
Cr, then
V]
~ 6V 7
n (V)
more slowly than any polynomial
Proposition 7 presents sufficient conditions for
it
=
(see
Appendix
is
a "slowly varying"
1).
(V) to preserve the power law price impact
under Bayesian updating, and thus to justify Assumption
20
VL (V), where L
1.
IV.B.
Multiple Stocks
The model can
= hjV 1
Ri (V)
C
risk
is
,
easily
that the signal
common
trading over
all
Suppose stock
be extended to multiple stocks.
M's
across stocks. 4
and the model misspecification
are independent across stocks,
The
trader's
program
maximize the expected
to
is
has a power law impact
i
profit
from
stocks,
f
1
maX
Vi(Mi,S),i=l...n
00
V
/
JC J2
*—r Jo
i
(
M» S M
i
) (
~
R V
i (
i (
M5
'
))) /*
(
A
subject to the robustness constraint that he does not lose more than
M
i)
dM
>
percent in price impact
costs:
f°°
1
l
l
<A
(V(M))f (M )dM
-J2j V (M )R
l
l
l
l
5 )7
=
power law exponents derived
in
Following the proof of the main Proposition, one can show that the solution
KM S1 ^ l+1 \
where
l
K
does not depend on
i
and
S. Hence, the
1
is
hiVi (Mi,
Propositions 5-6 follow.
Different Quality of Signals Across Firms
IV. C.
We now
our
is
allow the quality of signal
We
results.
assume that fund /
M to
differ across funds,
receives signals distributed according to
K=
(A//i)
4,
Xf m
min [(1+7)
-
Cs,
The average
iCm] and
</j
i
i
M-iS^-y
K
j—
M
= K-
i±3
>
where Xf
S
is still,
for a
constant
SU-oWt) and
for a
=
fund
C
1
!
1+7
i
Hence, one
still
obtains (y
=
Cv/l-
satisfies
the quality of signals
easy to verify that
1+2
quality Xf °f the signals disappears.
according to a production function x (F)
optimal investment
i
m->Si+->
E m
-r
In general, one expects larger firms to have a higher x-
'It is
M = x/ m
:
E
=
does not affect
the same across funds. Following
the optimal trading quantity of a fund of size
V (m, S) =
M
is
this
1/(1+7)
I
as
m
the quality of the fund's signals, and the distribution of
the proof of Proposition
and show that
is
x
= FK
where
F
for 6
if
signals are generated
denotes investment in research, then the
maxp CF K S 1 ^ 1+
~ Se
F° r instance,
'
)
^
—
F, for a constant C.
Hence
= jk/ [(1 - k) (1 + 7)].
could be also specific to each stock, or to each one of different classes of stocks.
21
F ~
—
This framework allows us to provide a microfoundation for Assumption 3 without any upper
The proof
cutoff.
of Proposition 4 shows that
(1+1/7)
g[M"-^][(l+7)]—
Dp <.
—
Assumption 3 holds
E[mW][(l+ 7 )}-W»
J
h^A
which holds
if
k > 1/2 and S
is
Is it
^
O
1
~
.
O
1-K
h^A
large enough.
production function of market research
IV.D.
if:
Thus Assumption
rises faster
than \
=F
1
3
is
automatically verified
if
the
^-
Discussion and Questions for Future Research
reasonable to believe that there are institutions large enough to cause the power law distribution
of returns? In view of the empirical facts,
we
believe so.
The
large volumes in Figure V,
which
can be 1,000 times bigger than the median trades, must come from very large traders. They are
also associated
with extreme price movements (Figure VT). However, a natural analysis would be
to investigate directly whether extreme
1989] are caused
by a small number
movements without news
may
same way.
versus correlated
movements
initial
is
of beliefs of
most traders would be
A
We
tions.
and our baseline model of
second example
October
...
market.
is
19, 1987, "this
Sell
...
many
traders will
is
43
Long Term Capital Man-
Our
a model of the initial impulse - the form and the power law distribution of the
disruption by a large trade.
present,
interesting.
of a large fund disrupting the market
leave to future research the important task of modeling the
specifics of the cascade that followed the initial impulse.
we
that, subsequently,
created a volatility spike that did not subside for several months.
Its collapse
contribution
availability of
Quantifying the importance of idiosyncratic movements of large trades
One prominent example
agement.
The growing
allow such a study to be conducted in the near future.
Note that the existence of prime movers does not preclude
in the
and Summers
of large institutional investors.
databases that track individual trades
move
[Cutler, Poterba,
initial
44
We
speculate that the empirical facts
impulses, will be useful for this future research.
the Brady [1988] report on the 1987 crash.
On
the crash day of Monday,
trading activity was concentrated in the hands of surprisingly few institu-
programs by three portfolio insurers accounted
Block sales by a few mutual funds accounted
for
for just
under $2
billion in the stock
about $900 million of stock
sales,"
on
Gabaix [2005] finds that the idiosyncratic movements of large firms explain a substantial fraction of macroeconomic activity, and Canals, Gabaix, Vilarrubia, and Weinstein [2005] find that idiosyncratic shocks explain a large
'
fraction of international trade.
4,1
[Abreu and Brunnermeier 2003; Bernardo and Welch 2004; Gennotte and Leland 1900; Romer 1993; Greenwald
and Stein 1991] also present elements for a theory of crashes.
22
a total of $21 billion traded
In the
first
group"
(p.
and "One portfolio insurer alone sold $1.3
v)
half hour of trading, "roughly 25 percent of the
30).
The
report concludes that
hands of surprisingly few
sharpness of the decline"
traders,
(p.
institutions.
(p. 41).
Of
A
"much
volume
...
billion" (p. 111-22).
came from one mutual fund
of the selling pressure
was concentrated in the
handful of large investors provided the impetus for the
course,
some
of the investors in the
Brady report are program
which amplify existing movements, rather than cause them. Also, our model
limited to allow the rich
from the report
is
dynamic analysis suggested by the Brady
too
is still
report. Nonetheless, the evidence
strongly suggestive of the hypothesis that a few traders
,
move a
relatively illiquid
market.
Our theory suggests a number
of research angles.
First,
it
would be desirable to study
dynamic extensions of the model. The analysis becomes much more
difficult [see e.g.,
fully
Vayanos 2001,
Gabaix, Gopikrishnan, Plerou, and Stanley 2003], but the simplicity of the empirical distributions
may be
suggests that a simple dynamic theory of large events
Second,
it
within reach. 45
would be interesting to study the distribution of fund
by leverage) across
classes of stocks.
distribution of traders, the
this prediction directly
more
"effective" size (assets multiplied
Proposition 5 predicts that the more fat-tailed the size
fat-tailed the distributions of
volume and returns. Investigating
might explain a cross sectional dispersion of .power law exponents.
Third, our model predicts that the total price impact cost paid by a fund of size
S
will
be
proportional to S, and that the total volume traded with sizable price impact will be proportional
to
SVU+i). Testing
this proposition directly
would be
useful.
Fourth, the model suggests a particularly useful functional form for "illiquidity"
sponds more closely to the prefactor of Proposition
H
(31)
3:
,
which corre-
46
\n\
\,V?)
S,^cov(\r
varVt
t
(32)
=
Again, the evidence, and some models, suggests 7
1/2, but other values
may
prove better suited.
Expressions 31 and 32 are likely to be more stable than other measures. Indeed, volume
tailed variable
(it
has infinite variance), so using a square root of volume
45
is
is
likely to yield
a
fat-
a more
Engle and Russell [1998] and Liesenfeld [2001] present interesting empirical investigations of the dynamic relations
between trading and returns.
4
One could even calculate the two expressions only for volumes above a certain threshold, e.g., the mean volume.
23
stable measure than
volume
itself.
proportional to cr/M 1 / 2 where
,
a
is
Furthermore, the model also suggests that
M
the volatility of the stock and
is its
H
and H'
will
market capitalization.
approach suggests a way to estimate the power law exponent of price impact,
Fifth, our
be
7,
and
the power law exponent of the distribution of financial institutions, (s, f° r instance across markets.
One
first
estimates separately the power law exponents of volumes and returns, £9 and £r
.
Then
one defines the estimators 7 and Cs by:
c9
7=
(33)
'
Cr
1
1
1
(34)
Proposition 5 indicates that these are consistent estimates of 7 and (s
Cm>Cs(1+7)/7-
[the
sum
the polar case where
47
Finally, the theory
volume
m
makes predictions about the comovements
of volumes traded on a price increase
in returns,
volume, and signed
minus volume traded on a price decrease).
Gabaix, Gopikrishnan, Plerou, and Stanley [2003] adds predictions in the number of
Its variant in
trades and signed
in Figure 3 of
number
of trades.
The
results
show non-linear
patterns,
and the
results,
reported
Gabaix, Gopikrishnan, Plerou, and Stanley [2003], show a quite encouraging
fit
between theory and data.
Conclusion
V.
This paper proposes a theory in which large investors generate significant spikes in returns and
volume.
We
posit that the specific structure of large
sizable institutional investors, stimulated by news.
large traders' moderation of their trading volumes
movements
The
is
due to the desire to trade of
distribution of fund sizes, coupled with
and a concave
price impact function, generates
the Pareto exponents 3 and 3/2 for the distribution of returns and volumes.
We
introduce some
new
questions that finance theories should answer.
Matching, as we do,
the quantitative empirical regularities established here (in particular explaining the exponents of
3
and 3/2 from
first
principles rather than by assumption) should be a sine
the admissibility of a model of volume and
will constrain
'it is
and guide future
tempting to
call
theories.
volatility.
Given
its
We
hope that the
regularities
criterion for
we
establish
simple structure, the present model might be a
Equation 34 a "reciprocity law" that holds irrespective of
24
qua non
7.
useful point of departure for thinking about these issues.
Appendix
1:
Some Power Law Mathematics
A. Definitions
We present
properties
how
here some basic facts about power law mathematics, and show
make them
how
their aggregation
and empirical work. They also show
especially interesting for both theoretical
our predictions are robust to other sources of noise.
A
random
variable
X
has power law behavior
if
there
is
a (x
>
such that the probability
density p(x) follows:
!
p(x)
(35)
for
x
—
oo,
>
and a constant C. This implies
xCa-+i
Resnick, 1987, p. 17] that the "counter-cumulative"
[e.g.
distribution function follows:
P{ x>x)~-^.
(36)
A
p(x)
more general
definition
~ L (x) /x^x+1
£x
is
that there
is
so that the tail follows a
,
a "slowly varying" 48 function
power law up to slowly varying
the (cumulative) power law exponent of
is
X
£y implies that
has fatter
tails
L
A
X.
(x)
and a (x
corrections.
lower exponent means fatter
than Y, hence the large X's are
s.t.
(infinitely, at
tails: Ca'
the limit)
<
more
frequent than large Y's.
If
a
is
a constant,
E \\X\ a =
oo for
]
returns have power law exponents £r
infinite.
is
Of
4S
=
49
moments
If all
are finite
=
E [|^"| a <
a > £x> an d
3, their
(e.g., for
]
kurtosis
< a<
oo for
is infinite,
and
(x- For instance,
if
their skewness borderline
a Gaussian distribution), the formal power law exponent
co.
Embrechts, Kluppelberg, and Mikosch 1997, p. 564] if for all t >
L = a and L (x) = a\nx for a non-zero constant a.
4!,
This makes the use of the kurtosis invalid. As the theoretical kurtosis is infinite, empirical measures of it are
essentially meaningless. As a symptom, according Levy's theorem, the median sample kurtosis of T i.i.d. demeaned
0,
L(x)
is
lim I _ 00
L
said to be slowly varying
(tx) J L (x)
variables n,...,rT with
,
=
k.
1.
t
[e.g.,
Prototypical examples are
= (eLi
r
V T ) I (Y,I=i r ?/ T )
>
increases to
should be banished from use with fat-tailed distributions. As
+oo
like
T 1/3
is
positive
if
tails are fatter
than predicted by a Gaussian.
25
=
Cr
3.
The
use of kurtosis
a simple diagnostic for having "fatter tail
normality", we would recommend, rather than the kurtosis, quantile measures such as
which
if
P (\(r
—
r) /ov|
>
than from
1.96) /.05
— 1,
B. Transformation Rules
Power laws have
to a
power law
is
The property of being distributed according
excellent aggregation properties.
conserved under addition, multiplication, polynomial transformation, min, and
max. The general rule
is
that,
when we combine two power law
one with the smallest exponent) power law dominates." Indeed,
variables,
variables, "the fattest
for
X\,
the
independent random
and a a positive constant, we have the following formulas:
Cx1 +...+xn
(37)
Cx
(38)
1
,..-A-„
= min (C^,..., CO
= min(Cxi,-,Cx„)
(39)
CmaxfA'!,...,^)
=min(Cx ,-,Cx n
(40)
(m in(X u ...,X n
)
=
Cxi
(41)
CaX
=
CX
(42)
Cx«
= ^.
For instance,
if
X
is
1
+-+
)
Cx n
a
a power law variable for £y
<
Y
and
oo,
is
power law variable with an
exponent £y > (x, or even normal, lognormal or exponential variable
X + Y,X
normal
Xn
...,
(i.e.,
Y, max(X, V) are
still
(so that
£y
=
oo),
then
power laws with the same exponent (x- Hence multiplying by
variables, adding non-fat tail noise, or
summing over
i.i.d.
variables preserves the exponent.
This makes theorizing with power law very streamlined. Also, this gives the empiricist hope that
those power laws can be measured, even
such as variances,
"essence" of the
it
will not affect the
if
the data
is
phenomenon: smaller order
and
b
b are
other
have thinner
random
tails
than r
with dominating power law),
counterpart
Proof.
?
factors not
(£a
its
,
£{,
>
will affect statistics
power law exponent. Power law exponents carry over the
effects
do not
For example, our theory gives a mechanism by which (r
where a and
although noise
noisy:
modeled
3). If
affect the
=
3.
in the theory.
power law exponent.
we
In reality,
observe:
7=
We will still have (r =
the theory of r captures the
i
first
(r
ar
=
3
+ b,
if
a
order effects (those
predictions for the power law exponents of the noisy empirical
will hold.
See Breiman [1965] and Gnedenko and Kolmogorov [1968] for rigorous proofs, and
Sornette [2000]
for.
heuristic derivations.
Here we just indicate the proofs
26
for the simplest cases.
r
By
induction
it is
enough to prove the properties
P (max {X,Y)>
x)
=
n
for
2 variables.
= l- P (max (X,Y)<x) = l-P {X <
Y<
x and
x)
= l-P(X<x)P(Y<x) = l-(l-^-](l-^-}
X^X
X^
J \
where C"
=C
if
Cx < Cy,
P (min (X, Y) >
Finally,
if
P {X >
x)
<?"
x)
~
C
if
Cx > Cy, and
= P (X >
Ca;
P (X Q >
C. Estimating
=
-
x)
^,
x and
y>
C= C+C
= P (X >
x)
if
x
Cx
)
=
i"nin((x,Cy)'
)
Cy-
P (Y >
CC
x)
x Cx+Cy
then
= P (X >
x
l'
a
\
~C
(i 1/Q )
_CX
~ Cx-^' /Q
.
Power Law Exponents
There are two basic methodologies
power law exponents.
for estimating
with the example of absolute returns. In both methods, one
orders the observations above this cutoff as
rm >
•
•
>
•
first selects
There
T( n y
is
We
illustrate
them
a cutoff of returns, and
yet no consensus
on how
to pick the optimal cutoff, as systematic procedures require the econometrician to estimate further
parameters [Embrechts, Kluppelberg, and Mikosch 1997]. Often, the most reliable procedure
use a simple rule, such as choosing
The
first
coefficient
method
is
on rn\ in the regression of log of the rank
]ni:=A- ( OLS In
with standard error £ OLS
and
yields a visual
Figure
1.
(n/2)~
goodness of
The second method
i
on the log
(i)
+
(44)
C"
is
i=\
i
which has a standard error t^ l "n
-1 / 2
.
27
(i)
is
the simplest,
the approach used, for instance, in
estimator
= (n-l)/£(lnr
OLS
size:
71-1
"
estimated as the the
noise
power law. This
for the
is Hill's
l
is
[Gabaix and Ioannides 2004], This method
'
fit
to
the observations in the top 5 percent.
all
a "log rank log size regression", where (
(43)
is
-lnr (n)
)
Both methods have
345]
pitfalls,
discussed in Embrechts, Kluppelberg, and Mikosch [1997, pp. 330-
and Gabaix and Ioannides
[2004].
vations. In reality, trading activity
is
One
large pitfall
is
the assumption of independent obser-
autocorrelated which causes standard errors to be underesti-
mated; however, point estimates remain unbiased. In a future paper we plan to propose a method
of estimating the standard errors. In any case, the stability of the estimates across different periods,
countries and classes of assets gives us confidence that the empirical estimates
we report here
are
robust.
With the samples
one can
of millions of points available in finance, standard errors are so small that
any null hypothesis. Hence, researchers estimating power laws typically
reject essentially
do not use
tests to see
many data points,
if
a distribution with more parameters would offer a better
statistical tests
in
fit
would be minimal. Rather £
optimal one-parameter approximation of the
may be
best
left for
is
tail
best interpreted as the
is
by a Pareto family. Explaining the value of this
Explaining the higher order terms
already a difficult challenge.
future decades of research.
Appendix
Proof of Proposition
2.
We
use
T = V/V
and
2:
Proofs
B (1) =
to calculate:
T
l+T
B (t) dt
var
var
(I dB (1 + u)
I
]
ds
=
/
r
T
ds\
var
.JO
T
Jo
liquidity provider sells
inventory during
+ T]
[1, 1
at
V
'
du
a total cost
K=^
T3
3
J
shares to the fund at a price p
+
dB{u+
1)
\Ju
[ (T-u)dB(u+l) = f (T
Jo
var
The
it
+
r,
V3
W
6
and replenishes her
p (t) Vdt. Her net income from the transaction
is:
W = {p +
r\+T
fl+T
-k
so
would always justify a higher-dimensional parameterization, even
though economically, the improvement
one-parameter approximation
With
fit.
+
V-
t)
/
p(t)Vdt
=
(p
+
rl+T
= tV -o-V
/
B{t)dt.
28
tt
+ t)V -
/
{p
+ n + aB (£)) Vdt
=
Her
utility
is:
_,
/
U = E [W] -
A (varW)
S/2
= tV - A
a
I
2
V
a/2
rl+T
r
var
= tV -
B (tt)dt
/
a2y3
—
A
^/2
31/
The fund has
U=
0.
full
bargaining power, and so leaves the liquidity supplier with a reservation utility
This implies:
Economically, the liquidity provider purchases the stock back at an average price p
which has expected value p
compensation
,
ir
J
It is sufficient to
(Af
)
-
optimize on
=
„T
(1
I
we use the notation
+ fi) hV (Af
V (Af
)
ffL = __^_
SV(M)
(45)
.
Thus, using Equation
M-
y (M) =
.
The temporary impact r
V (Af)
V (M, S).
rather than
[y
+ /x)
(1
+
)
dAf
/ (Af ) dAf.
)
(M
)
M-
+ n) (1 + 7) hV
[(1
V (Af) i? (V (Af )) / (Af
(
7) h]~
(1
:
+M
)
W
(Af )
1+7
1
/ (Af)
Af ) 7
lh
M ^.
1
17,
n c=o =£
I
Constraint (18) binds
(1
7
)
- n f
separately for each Af
<9V(Af)
=
iff fi
>
Jj
p (t)
is
the
I
V{M){M -R{V (M))) / (M) dM
= /V (Af
(46)
I
is:
£ =
-25
(v\
-=
l
/
v \ 1/2
a r=
J
In this proof
4-
,
and standard deviation a
for this price risk of
Proof of Proposition
Lagrangian
+
= T~
l/2
/iV(Af)
1+7
/S =
/i£
0, i.e.
5<5*
29
,1/
1+1/7
[(i+
A
o(i+7)/r (1+1/7)
/s.
The
dt,
with:
.
(4?)
If
—E
the constraint binds,
\rt
f?[M^/7]
s* =
C=
\
=
+ 7 )]-(H-l/7)
l
A. This implies:
+M )(l+ 7 )/ ]-<
[(l
[(
^=
1+1
l
VV
ttS
[M 1+1 /7]
i
and going back
to Equation 45,
we
V (M) = uM
get
1
/tS' 1 /( 1+ t) with:
\ 1/(1+7)
/
'
w=
(48)
'
JlE
The
expression for
To
E [M 3
to
R
]
=
10 percent (which
be conservative),
A=
motivated by Sections
=
'.
following parameters, which
is less
C and
R(V) =
III. A:
A2 =
Act (tj)
A=
l/2<
effects,
D=
daily market turnover. Using the 1999
which means that up to
Proof of Proposition
5.
Cfi
A2
=
s
3
D=
=
=
=
We
1/2,
likely
take a price impact,
daily market volatility
market capitalization
1/2 x $18 trillion/250
=
A
of a total equity
21.
=
$36
billion.
So
m
$21trillion.
We
apply the rules in Appendix
1
to derive:
(MS-y/^+f) b y applying (41)
Csr/cn-r)]
Cm,
]
o,
2
from Equation
= min [Cm,
mm
cj
(3/2)
QhviMSi/^+-,)
50
where a
,
number
E\M 3
D
=
We start
7
25 percent of the market fluctuations are due to our
and a 50 percent annual turnover,
S*
as simply indicative:
impact costs paid annually.
0.01,
of $18 trillion,
we view
than the annual standard deviation of the market, hence
2 percent of price
II.
l
l
R = hV 1
comes from
we use the
calibrate S*,
[M + h] J
by applying (38)
by applying
Cs
(42)
7
which proves the Proposition. One derives (y
Proof of Proposition
^f the fund gets
F
6.
Assumption
signals per year, 5"
is
in the
2 implies
divided by
F
1
30
same way.
Cm >
/2
,
as
M
1
is
+
1/7
=
Then, Proposition 5 gives
divided by
3.
F
1
^2
and
A
by F.
(R
= min (3, Cm) =
Proof of Proposition
fatter tails
XY
Lemma
Assume Cx <
x)
=
3/2.
will start
E [X XY =
\
z]
X
Y
and
(i/x*)" Cl
are independent
E {X XY =
varying function.
If
for
power distribu-
variables, with exact
x > x* and y > y*
.
Define z*
=
x*y*.
Cx < Cy,
^W
~
1Jw=*J ~ y*lnz
y*ln(z/z*)
Z
L
(52)
(z)
'
Proof of
k.
that an
Cy,
v
(j
is
has
= L(z)z
&*<*>=
5i )
=
The reason
z.
X
LM= *[f*"W-]
(z) is a slowly
Cx
z]
|
'
If
random
P(Y >y) = (y/y*r (Y
,
(50)
(
proportional to z for large
is
if
Cv- Then:
(49)
L
with the following Lemma, which means that
probably comes from an extreme value of X.
8 Suppose that
P{X >
tions:
=
-y( R
We
7.
than Y, then
extreme value of
=
and (v
3
Lemma
the densities of
X
By
8.
and
.
normalization,
By
Y".
Z
}
it is
Bayes' rule,
for z -» oo.
enough to study the case x*
p(X =
x
\
=
=
y*
Calling / and
1.
XY = z) = kf (x) g (z/x) jx for
a constant
So:
rIy vv =
E[X\XY
,
,
z}
x )9(z/x)/x
J f(
= J
rf/
,
J
=
z
^/yr ix
l
z
I ^/y)~
~1
Cy,
When (x =
l
=
z
E [Y^1 y<2
]
-»
2
J f(z/y)g(y)/y dy by the change
1
dy
J f{z/y)g{y)/y
J
2
9(y)/y dy
ix ~ 1
1
When Cx <
dx
.
.
f{x)g(z/x)/x dx
J
Z
.
E
= J'y^-'gjy)
z
]
<
=
_
E[Y^-
1
1
Y<Z
e[y^i y<z
I y^g(y)dy
g(y)/ydy
[Y Cy
dy
.
of variable x
1
oo
Cy,
= J'y^^gjy)
fiy
dy
ix 9{y)dy
=
ff^-iy-Cr-i
dy
ffytxy-Cy- 1 dy
D
31
2
.
= fiy-
dy
l
fcy- dy
=
1
- z~
\az
l
]
]
=
= zL
z y
For the proof of Proposition
j
Q^
3
Given the hypothesis Cm
Lemma
^
when they
1
S 1 ^ l+1 \ We
call
calculate the conditional expectation
Equation 23 gives (^
Cfti
M and Y = hv
X=
8 with
J
<
<
(r
(l
+ ^)Cs =
Cy- So using
8,
M\R = /i^M5 7/(1+7)
= XY
L
£w
E
a slowly varying function
(R) of R. In the case
J
RL{R)
jE
l
Appendix
3:
Y*>m
a constant.
,
Finally, given
+
<
(
tt
= RL(R),
/subj
= E Y^m
b'
we use Lemma
the exponent of the distribution agents use
(28).
for
7,
1
^
Csi
)
we
and L
When
Confidence Intervals and Tests
get: lim/j^oo
is
L
(R)
slowly varying,
Has
a Variable
Infinite Variance
A. Construction of the Confidence Intervals for Figure VI
Q=Q
In a given bin conditioned by
is
the sample
mean
of the r? which
we
z
,
with k elements
call
r
2
2
,-..,r
,
the point estimate of
m. Getting a confidence
interval for
has infinite variance, so the standard approach relying on asymptotic normality
m
is
E [r
mean and a
ratio
=
t
1 '2
fc
is
(m —
we simulate draws
fj.)
/a follows a non-degenerate distribution
=
a T i k~ l
>
2
,
for large k.
following a power law with exponent 1.5, which
finite
— t, which we
variance value, which would be
To construct 95 percent confidence
Arf
2
But the
invalid.
if fi is
the
-
from their
differ
Qi\
empirical standard deviation of the r 2 in the bin with k observations, then the
tabulate 2.5 percent and 97.5 percent quantiles of
They
Q=
delicate, as r
is
theory of self-ndrmalizing sums of Logan, Mallows, Rice, and Shepp [1973] shows that
true
|
we can
intervals,
first
x~
is
call
=
X
+
By Monte
Carlo analysis
the exponent of r
,
and we
— x~ = —1.1 and x + —
=
1-95.
5.5.
51
calculate the empirical standard error
the sample standard deviation of the observations divided by the square root of
the number of observations.
A
95 percent confidence interval
is
[m^
— x~Ar?,mi +
"^ 2
1
x"
]-
We
should stress that we make the simplifying assumption of idenpendent and identically distributed
draws. Given that the data are likely to be autocorrelated, our confidence intervals are likely to be
too narrow.
B. Test of Relation (7):
"
law
When
k
P (r 2 >
there
-3 / 2
is finite,
i)
=
x
for
is
x
E
[r
2
|
Q]
= a + (5Q
some sensitivity of x~ an d X +
1, and k = 200, to reflect our
>
32
We take a pure power
extreme values.
to the underlying distribution.
typical
sample
size in bins of
.
For each bin Qi of Q, we set r 2
i.
By
least squares
we
fit
an
=E
[r
2
|
Q=
affine relationship
5 (Q)
=
E
The standard
errors are in parentheses
[r
0.07
(0.59)
and the
and Ar 2 sample standard
Qi],
2
|
Q]
=
g (Q), with
+ 0.60Q
(0.013).
R2 =
0.90.
We
find that for all values Qi
predicted value g (Q,) belongs to the 95 percent confidence interval: g (Qi) £
We
conclude that, at the 95 percent confidence
error in interval
level,
we cannot
\r
2
— x~Ar 2
reject the linear
form
E
,
r
[r
>
2
2
3,
the
+ x + Arf]
Q] =
|
g(Q)forQ>3.
Massachusetts Institute of Technology, Economics Department, and National
Bureau of Economic Research
Boston University, Physics Department, Center for Polymer Studies
Boston University, Physics Department, Center for Polymer Studies
Boston University, Physics Department, Center for Polymer Studies
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MA:
Addison-
x (Units of standard deviation)
Figure
Empirical cumulative distribution of the absolute values of the normalized 15 minute
I:
returns of the 1,000 largest companies in the Trades
And Quotes
database
for the 2-year
period
1994-1995 (12 million observations). We normalize the returns of each stock so that the normalized
returns have a mean of
and a standard deviation of 1. For instance, for a stock i, we consider the
returns
r'
it
=
Cr
=
large
3.1
±
— r;) /oy^, where T{
< x < 80 we find an
(ja
In the region 2
0.1.
x between
is
the
mean
of the r^'s
ordinary least squares
and aT)
fit
i
their standard deviation.
is
lnP(|r|
>
x)
= — (Vinz + b, with
> x) ~ x~^ for
This means that returns are distributed with a power law P(\r\
2
and 80 standard deviations
of returns. Source:
44
Gabaix
et al. [2003].
r
10
10"'
•a
&o"
5
So"
7
"9
10
-60
-40
20
-20
Normalized returns
Probability density function of the returns normalized 5 minute returns of the 1,000
Figure
II:
largest
companies
in the
Trades
And Quotes
in the center of the distribution arise
database for the 2-yr period 1994-1995. The values
from the discreteness in stock
fractions of U.S. dollars, usually 1/8, 1/16, or 1/32.
The
solid curve
prices,
is
which are
a power-law
2 < x < 80. We find £ = 3.1 ± 0.03 for the positive tail, and £ = 2.84 ± 0.12
The dotted line represents a Gaussian density. Source: Plerou et al. [1999].
45
for
set in units of
fit
in the region
the negative
tail.
10"
r^^^tt^aCfli^.
•1
10"'
"
•.
^^^i«.
.-5
%
10~ 2
"S
J5.
c
I
;
s
io-
>
;
o
,0 Nikkei -84-97
m Hang-Seng '80-'</7
o
SAP 500
°*V
i
'62- 'V6
10" 5
.
.
1
10
1
Normalized daily returns
Figure
III:
Empirical cumulative distribution function of the absolute value of the daily return of the
Nikkei (1984-97), the
behavior in the
Hang Seng
tails is
(1980-97), and the
S&P
500 (1962-96).
The apparent power-law
characterized by the exponents (r = 3.05 ± 0.16 (Nikkei), £r
3.34 ± 0.12 (S&P 500). The fits are performed in the region
(Hang-Seng), and £r =
and 10 standard deviations of returns. Source: Gopikrishnan
46
et al.
[1999].
=
\r\
3.03
±
0.16
between
1
i
*
^i™"— "
"-""
•
»
10"
a
J
10
1
Ot
\
5 HT*
\
fc
a
V
10"
<*
\
%i
10**
at
-
6
io-
10"
10"'
10
So
-
f
-i
10"
Normalized daily returns
Figure IV: Cumulative distribution of the conditional probability P{\r\
italization
K,
CRSP
K
We
database, 1962-1998.
define uniformly spaced bins
returns for in each bin:
e [10
5
6
10*
to
icr
Daily returns
of companies in the
\
™
s
K
on a logarithmic
6
x) of the daily returns
7
and show the distribution of
scale,
K
(o),
]
,
10
]
6 [10
7
8
K
8
9
10 ](D),
G [10 10 (a).
(),
is measured in 1962 constant dollars, (a) Unnormalized returns. Each cumulative distribution
corresponds to a bin of sizes. Small stocks are to the right, because they are more volatile, (b)
,
10
6 [10
>
consider the starting values of market cap-
,
,
]
K
Returns normalized by the average
distribution, with £r
=
2.70
± .10
volatility
ax
of each bin.
for the negative tail,
and £r
=
The
plots collapsed to an identical
2.96
± .09
for the positive tail.
The
horizontal axis displays returns that are as high as 100 standard deviations. Source: Plerou et
[1999].
47
al.
!
—
10'
—
-i
c
i
— —miTTi
inn
——
i
i
1
1
rt
r
n
—
1
it
i
ttttt
f
10
cr 10
i
-1
rm
10"
r^>
10"
C/}
10
£
>-»
10"
10
5
10"
-o
10
jo
10
*-
1+^ =2.5
-4
7
8
NYSE
(1000 stocks)
• Paris Bourse (30 stocks)
10
LSE (250
-11
stocks)
10
•12
Mill
10
I
I
L-L
lllti
II ll
L
10
10
10
10
Trade
L_
;
\
10
size
q
10"
10'
Figure V: Probability density of normalized individual transaction sizes q for three stock markets (i)
NYSE for 1994-5 (ii) the London Stock Exchange for 2001 and (iii) the Paris Bourse for 1995-1999.
= — (1+C?) lnx+constant
for Q q = 1.5 ±0.1. This means a probability density
and a countercumulative distribution function P (q > x) ~ x~^ q The
three stock markets appear to have a common distribution of volume, with a power law exponent
of 1.5 ± 0.1. The horizontal axis shows invidividual volumes that are up to 10 4 times larger than
OLS
fit
yields
function p(x)
lnp (x)
~ x~
1 +^«)
(
the absolute deviation,
.
)
\q
—
q\.
48
100
a
__
LU
10
+|
0)
km
•a
o
i_
(0
3
1
or
'
^^lULu^-;^
(A
0)
D)
n
u
0)
£
0.1
j
0.001
i
i
imiii
i
i_u
i
0.01
_i
0.1
i
i
i
i
1
1
ii
i
the aggregate volume
Q
in At.
r
E
is
i
i
i
1
1
1|
10
1
Aggregate volume
Figure VI: Conditional expectation
i
i
i
H
100
Q
2
Q] of the squared return r in At = 15 minutes, given
in units of standard deviation, and Q in units of absolute
[r
2
|
— Q\. The results are averaged over the largest 100 stocks in the New York Stock
Exchange market capitalization on January 1, 1994. The data spans the 2-year period 1994-95
and is obtained from the Trades and Quotes database, which records all transactions for all listed
securities in the NYSE, AMEX and NASDAQ. Formal tests reported in Appendix 3 show that one
cannot reject £[r 2 |Q] = a + (3Q large enough (Q > 3). This is consistent with a square root price
impact of large trades. Appendix 3 reports the procedure used to compute the 95% confidence
deviation, \Q
intervals.
49
Lag
P lize
A
I
leg x
Figure VII: Cumulative distribution of the size (assets under managements) of the top mutual funds
in 1999. Source:
Center
for
Research on Security Prices.
50