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orKmg raper series
Noisy Trade Disclosure and Liquidity
Subu Venkataraman
Working Papers Series
Issues in Financial Regulation
Research Department
Federal Reserve Bank of Chicago
Septem ber 1995 (W P -95-18)
FEDERAL RESERVE BANK
OF CHICAGO
Noisy Trade Disclosure and Liquidity
by
S.
Venkataraman
Research Department
Federal Reserve Bank of Chicago
230 S. LaSalle Avenue
Chicago, IL. 60604-1413
Current Version
Septem ber, 1995
The conclusions of this paper are strictly my own, and not necessarily those of the Federal Reserve Bank of
Chicago or the Federal Reserve Board of Governors. This paper was begun while I was at the University of
Florida. I would like to the faculty of the Department of Finance, Insurance, and Real Estate, and especially M.
Nim alendran, for their many useful comments. All errors remain my own, of course.
Noisy Trade Disclosure and Liquidity
Abstract
This paper examines the implications of noisy trade disclosure on trading and liquidity
in a multiperiod security market where some traders receive private information before others.
We find that an increase in the precision of trade disclosure creates a natural incentive for
traders to either (i) trade aggressively in early rounds, implicitly giving up the opportunity to
trade later, or, (ii) scale down trading and effectively withdraw from the market in the early
rounds in order to protect their opportunity to trade later. Not surprisingly, the consequences
of improved disclosure for the liquidity of the market depends on which of these two
situations arises. While improved disclosure could lead to a worsening in liquidity in the
early rounds of trading in the former case, it leads a worsening in liquidity in later rounds
under the latter case. Traders who transact in later rounds are ambiguous in their views on
improved disclosure as well, since they trade-off the corroborative effect of trade disclosure
for their private information against the additional information conveyed to the market-maker
who sets prices on this basis. These tradeoffs lead to liquidity being a non-monotonic function
of the precision of trade disclosure. These implications for market equilibrium, which reflect
the response of traders to the changes in the disclosure environment, should be incorporated
into the current debate over the optimal level of transparency in security markets.
Noisy Trade Disclosure and Liquidity
1. Introduction
The internationalization of security markets has led to increased attention being paid to
the appropriate form of security exchanges in a global trading , and on the need for regulatory
harmonization to achieve these goals.
This is reflected, for example, in the threats posed to
the New York Stock Exchange by competition from both domestic and foreign exchanges
over the last decade. There has been concern about the movement of trading off the Big
Board, and the extent to which this is caused by differences in trading and regulatory
structures. Similarly, bourses in Europe are also increasingly aware of the importance of
market structure in attracting trading volume to the exchange. One aspect of the trading
environment that has been especially contentious is the transparency of a security market, and
the extent to which increased transparency enhances welfare.1-2
This paper examines the implications of one specific dimension of transparency: the
extent to which the form and content of reported order flow reflects the motivations for trade.3
The disclosure of the details of trades after they occur does convey useful information about
the extent to which these trades were motivated by liquidity needs as opposed to private
information about the security itself. One can therefore view the details of trade disclosure as
See, for exam ple, the views of the SEC on the regulation of markets in the recently released M arket
2000 study (especially study IV).
2
The importance o f this dim ension of transparency is illustrated by the movement of order flow to
London. Ninety percent o f the cross-border trading in European securities occurs in London's SEAQ
International (W all Street & Technology (1993)) which is of concern to continental exchanges like Paris. One
reason for this is thought to be the Paris Bourse requirem ent that trade details (like price, size, and the identities
o f the buyer and seller) be published im m ediately, while London's SEAQ International does not require the
publication of such details until the next day, and even then only as a part of the aggregate volume of a
particular stock. (Barron's (1991), Econom ist (1994)). In contrast, for NASDAQ-NM S securities, trade details
are disclosed within 90 seconds o f the trade occuring (Franks and Schaefer (1992)).
3 There are other dim ensions of transparency that have been considered in the literature as well. For
example, some authors have exam ined the impact of the ex ante transparency of the stock market, as captured
by the extent to which quotes, the specialist's limit order book, or details about the order flow are observed
before a trade occurs (exam ples of formal models that examine the im plications in such an ex ante framework
are Admati and Pfleiderer (1991), Forster and George (1992), and Pagano and Roell (1993)). W hile these
dim ensions of transparency are im portant as well, we restrict the analysis of this paper to the implications of the
transparency of the trade reporting process for the liquidity of the security market. There is also a large literature
that deals with the generic im plications of public disclosure. See for exam ple Alles and Lundholm (1993) (and
the references therein).
2
having implications for the
ex p o s t
transparency of the market. The regulatory perspective on
transparency is reflected in the following quote from Mary L. Schapiro (Commissioner at the
Securities and Exchange Commission (SEC) at the time) in the Wall Street Computer Review
(1991):
"A s i g n i f i c a n t p e r c e n ta g e o f th e v o l u m e in N e w Y o r k S t o c k E x c h a n g e - l i s t e d s e c u r i t i e s
is n o w b e i n g d o n e in m a r k e t s th a t a r e l e s s tr a n s p a r e n t.
A n d b eca u se fir m s d o
s o m e t i m e s p r e f e r a l e s s tr a n s p a r e n t m a r k e tp la c e , it d o e s b e c o m e a c o m p e t i t i v e is s u e
f o r U .S . e x c h a n g e s .
I f th e s ta n d a r d is l e s s { r i g o r o u s } in o t h e r c o u n tr ie s w h e r e th e r e
a r e v i a b l e m a r k e ts , th e n th e U .S . m a y s e e m a r k e t s h a r e m o v e o f f s h o r e ."
The SEC's approval of extended trading hours on the NYSE, during which the trade reporting
requirements are much lower, can be viewed as a tacit recognition that stringent reporting
requirements might lead to a movement of order flow off the exchange.4
Before discussing the structure and results of this paper, it would be useful to
summarize the typical perspective that is taken on the benefits and costs of enhanced
transparency. A good definition of market transparency is that it a measure of the ability of
market participants to observe the information contained in the trading process (O'Hara
(1995)). Obviously, while some aspects of transparency can be associated with the trading
process itself, other aspects are discretionary, and represent policy tools under the control of
regulators.5 Furthermore, the recent literature on transparency also distinguishes between
factors that influence the
e x a n te
versus
ex p o s t
transparency of markets (Madhavan (1995)).
While regulators view increased transparency as being extremely desirable, presumably
because it improves liquidity and reduces the losses suffered by uninformed traders, increased
transparency also reduces the incentives of traders to invest resources in the collection of
4 Hasbrouck, Sofianos and Sosebee (1993) provide a useful overview of the operating procedures of the
New York Stock Exchange.
5
For exam ple, M adhavan (1992) and Pagano and Roell (1993) highlight the differences in the levels of
transparency betw een trade and order driven systems, and also between continuous versus batch auctions. The
tim ing and content o f trade disclosure, however, is a regulatory choice variable.
3
information (Mulherin (1993)). Some authors have also questioned the benefits of increasing
market liquidity, since it could reduce the effectiveness of the market for corporate control
(Bhide (1993).
Two papers that formally examine the trade reporting dimension of market
transparency are Fishman and Hagerty (1991) and Madhavan (1995). These papers view trade
disclosure as either being nonexistent or perfect, and examine the consequences for liquidity,
fragmentation and the regulation of markets (see also, Harris (1992)). This paper, on the
other hand, is motivated by the observation that trade disclosure as a signal of the private
information available to a trader is inherently a noisy proxy (as is market transparency).
Delays in reporting, the aggregation of trades, the anonymity of the market, all contribute to
the noisiness of the trade reporting process.6 Consequently, examining the implications of the
entire spectrum of disclosure regimes for market equilibrium might provide insights that are
not available by examining the two polar extremes (i.e nonexistent versus perfect disclosure).
The analysis of the consequences of the increased precision of trade disclosure is
conducted in a multiperiod model of trading in a single asset. The structure of private
information available to traders is such that traders are distinguished from each other not just
on the basis of whether or not they possess private information, but also based on when they
obtain this information.7 Specifically, we examine the situation where one trader obtains
noisy information about the asset's payoff before the rest of the market. However, he realizes
that another trader will also be receiving private information in the next period, and will be
utilizing both this private information, and any information conveyed by the first round of
trade, in deciding his trading strategy. Some of this information is revealed through prices,
while additional information is revealed through a noisy trade reporting system that has been
6 M assimb and Phelps (1994) suggest, for example, that one of the differences between the open outcry
system and a typical electronic trading system is that the former provides more details about com pleted
transactions than the latter. They concede, however, that such disclosure is not perfect under the open outcry
system, since participants m ight only be able to infer the customer's trading house, but not the identity of the
custom er itself.
7 This structure is sim ilar to that utilized in a recent paper by Hirshleifer, Subrahmanyam and Titman
(1995), and is also discussed in Froot, Scharfstein, and Stein (1992). The sim ilarities are discussed when the
model is form ally developed in Section 2.
4
put in place to enhance the transparency of the market.8 Consequently, in formulating his
trading strategy, this trader takes into account the fact that the noisy disclosure of his current
trades reveals information to the traders with whom he will transact in later rounds.9 This
information arrival process seems to be consistent with the concerns that traders have about
the implications of trade disclosure on their activity.
We find that an increase in the precision of the trade reporting process creates several
conflicting incentives for both the trader who receives private information early and the one
who receives it late. The trader who receives information early recognizes that his trades in
the early rounds convey information to both the market maker and the informed trader who
will be competing with him in later rounds. If he views trading in later rounds as being very
lucrative, the trader responds by rationally reducing the extent to which he trades early, in
order to control the amount of his private information that is revealed through trades. This
behavior leads to an improvement in liquidity in the early rounds. If trading in the early
rounds is viewed as being more lucrative, however, an increase in the precision of trade
disclosure causes the trader to effectively forego profitable trading in later rounds by trading
very aggressively in the early rounds. This leads to a worsening in liquidity in the early
rounds when disclosure is made more precise.
The trader who transacts in the later rounds recognizes that increased disclosure has
two distinct effects that are relevant to him. The information that is provided by trade
disclosure makes his private signal more informative, which he clearly finds useful. However,
the information in trade disclosure is also available to the market maker who then sets prices
on this basis, thereby reducing the profits available to the trader. We show that while there
8 The assumption that the trader who is informed early has no com petition in the first round of trade
leads to a much more tractable analysis o f the costs and benefits of the trading strategies adopted and the
m anner in which they are affected by more precise disclosure. Allowing for com petition would result in
informed traders transacting more aggressively in early rounds, causing inform ation to be reflected earlier in
prices (Holden and Subrahm anyam (1992)).
9 Benabou and Laroque (1992) argue that insiders can attem pt to m anipulate markets through "distorted"
announcem ents. In our model, the attem pts of the insider to control his trading in early rounds is m otivated by
the recognition that this influences the "trade report" that is announced at the end of the first round. In this
sense, the incentive effects in the two papers are quite similar.
5
are circumstances where increased precision in trade disclosure increases competition in later
rounds, there are also circumstances where the opposite occurs. The net impact of these
conflicting effects is that an increase in the precision of trade disclosure need not be
unambiguously preferred by the trader who transacts late.
The results of the paper highlight a general point. The impact of increased
transparency on trading incentives and liquidity is complicated by the endogenous adjustment
in the trading strategies of the market participants. In fact, in fairly reasonable circumstances,
increased transparency can have very different effects than have been recognized so far in the
literature. These insights should provide a useful starting point in the attempts of policy
makers to better understand the implications of transparency in a global marketplace.
The remainder of the paper proceeds as follows. The model of trade is developed in
Section 2, and the equilibrium for this model is developed in Section 3. Section 4 examines
the relationship between the precision of trade disclosure and market equilibrium, while
Section 5 discusses the implications and some possible extensions. Section 5 concludes.
2. A multiperiod model of trading with differential timing in receiving information
In this section, we develop the economic setting for the multiperiod model of trading,
which is a simplified version of that developed in Holden and Subrahmanyam (1992),
modified to incorporate the noisy disclosure of trades. Specifically, the economy consists of
three time periods indexed by t=0,l,2. Trading takes place in a single asset that only pays off
at t=2. The payoff on the asset is an uncertain quantity v which is distributed N(0,ay)Trade in this asset occurs at both t=l and t=2. There are two traders in this market who
6
invest resources to gather information about the firm.101 The nature of the information
collection process is such that information about the terminal (t=2) asset payoff is first
revealed to one of the traders before the first round of trade. In the next round, the other
trader also receives some information about the final asset payoff.
Without loss of generality we refer to the trader who receives the information early as
trader #1. At the beginning of the trading game, trader #1 receives a noisy signal 0 ( = v +e,
about the terminal asset payoff. The noise in the signal, e, is distributed N(0,c£) • The
second informed trader (#2) receives no signal in the first round of trade, but does observe a
signal 02= v +e2>but only after the first round of trading is over. Consequently, this trader
does not participate until the second round of trading.
In addition to these informed traders, there is a random order, u„ placed by noise
traders in the first round, where u, ~N(0,a;;1)- Similarly, in the second round, noise traders
place a random order, u2, where u2~N(0,a^2)-U 1° each round of trader, the orders are
submitted to a market maker, who sets a price based on the cumulative order flow that he
observes, to ensure that he breaks even. All participants in the economy are risk neutral, and
all the random variables are assumed to be independent of one another.
Market transparency is modelled as a signal that is observed by all players at the end
of the first round of trade. This signal, which is related to the order placed by informed
trader #1 in the first round, is denoted by S=x(+r|. where x, is the order submitted, and
T)
is
10 W e assume, for sim plicity, that both traders have incentives to invest in the inform ation collection
technologies. W hile an exam ination of the profits of the traders will provide some insights into the im pact of
better disclosure on the endogenous incentives to collect inform ation, we defer a form al analysis of this issue to
future research.
11 The assum ption that the liquidity traders are constrained to transact in a given round is obviously
restrictive. W hile the model can be extended to allow for some of these traders to have the discretion on which
round to trade, it is still necessary to have some amount of non-discretionary trading in each round (Admati and
Pfleiderer (1988)). A lternately, one could relax the assumption of risk neutrality, in which case the hedging
m otives of the uninform ed would result in their transacting in both rounds (see Speigel and Subrahm anyam
(1992)).
7
the noisiness in the signal, which is distributed N(0,a^) • There are several motivations for
modelling transparency in this manner. Much of the debate in markets centers around the
reporting of previous transactions (The Economist (1992, 1994)). Moreover, different market
structures also impact the ability of traders to identify the motives of trades (information
versus liquidity). For example, Benveniste (1992) suggests that one of the benefits of the
specialist structure is that it provides the market maker with an efficient way of inferring the
motives for trade
e x a n te .
Similarly, Admati and Pfleiderer (1991) suggest that to the extent
that some liquidity traders are able to convey information that their trades are not motivated
by private information (i.e engage in sunshine trading), this information in conjunction with
aggregate order flows also provide noisy information about the trading strategies of the
informed trader. In any trading system, therefore, a combination of the trading methods and
disclosure requirements provide participants with (potentially imperfect) information about the
motives for trade. In this paper, instead of endogenously developing the noisiness in the trade
disclosure process, we model this noise exogenously, and examine the implications for market
equilibrium.
The assumptions on the arrival of information, and the resultant trading strategies of
the two informed traders require some elaboration as well. The assumption that the process
of collecting information provides some parties with early success relative to others seems a
reasonable one. It has been utilized, for example, by Hirshleifer, Titman and Subrahmanyam
(1994), to explain the incentives of investors to focus on only a subset of available securities,
and certain empirical trading patterns.12 This information structure is also consistent with that
utilized by Froot, Scharfstein and Stein (1992) in their examination of the impact of short
term speculation on trading patterns.
Moreover, the "first mover" advantage that it provides to the trader who receives
information early is at the heart of the debate on transparency. While a regulator would like
this information to be revealed to the market as soon as possible (i.e. improve informational
A few similarities and differences are worth noting. W hile the arrival of information is similar, HST
focus on the case where the informed traders are risk averse (leading to a hedging motive), have the signals of
the traders perfectly correlated, and utilize a competitive (rational expectations approach) equilibrium. Utilize
their own description of FSS to characterize why it is a less that satisfactory set of assumptions.
8
efficiency), the trader would like to maximize the rents associated with his private
information. Any trading strategy that he adopts will reflect both the information and the
disclosure environment in which he trades. By assuming that this trader faces no competition
in the first round of trade, we can highlight the manner in which he adjusts his trading
strategies in response to changes in disclosure. In the more general environment, where he
faces competition in the first round as well, the adjustment in trading strategies would also
reflect his perception of the manner in which the "competition" reacts to the change in
disclosure requirements. Insights into the implications of disclosure in the absence of the
threat of competition can also be easily obtained from this model, since this would just be the
special case where a 2 approaches infinity. An extension of this modelling framework to allow
multiple "early" and "late" informed traders will increase notational complexity considerably,
without changing the basic insights of the paper.
The development of equilibrium in this model consists of identifying the trading
strategies for traders #1 and #2, and price setting mechanisms for the market maker. We
assume that both traders adopt trading strategies to maximize expected profits, while the
market maker sets prices in each round in order to break even (i.e earn zero profits). In this
economy, we conjecture the following linear equilibrium.13 In the first round, trader #1
submits the order
(1)
while the market maker sets prices according to
pi= *'[*,+“,]
(2)
In the second round, the two informed traders trade according to
x2 = a 2[0,-Pi]
(3)
and
13
While the existence of non-linear equilibria is not precluded in this model, our focus
on linear equilibria is consistent with the focus of much of the literature in the area (see, for
example, Madhavan (1995), who argues that the merits of restricting attention to linear
equilibria are that these impose the fewest computational burdens on agents, exhibit stability,
and yield closed-form solutions).
9
yi ~ Pa[®2
(4)
^2]
while the market maker's pricing rule in this round is given by
P2 = Pv + r [jc2 + y2 + M2]
where |iv, (i, and
represent the conditional expectations of v, 0, and 02 based on the public
information available at t=l, i.e. the market clearing price in the previous round, P„ and the
signal about past trades by informed trader #1, S. Equilibrium is therefore characterized by
the parameters (<x„ X, o^, (32, T) that are consistent with this conjectured equilibrium.
3. Market Equilibrium
The second round of trade
We begin by considering the restrictions characterizing equilibrium in the second
round of trading. At this stage, all the players have observed the market clearing price in the
previous round, P,, and the signal S. Given the joint normality of all the variables, it follows
that the distribution of (v, 0,, 02) conditional on (P,, S) is also multivariate normal (Anderson
(1984), Theorem 2.5.1). The parameters of this distribution are given in the next proposition.
Proposition I: The distribution of (v, 0,, 02) conditional on (P,, S) is multivariate normal with
conditional means
Pv
p, =¥. ’^1 = 1 a »(g v+ a »)
-s \ A
Xa2
ul
a i(g y + g i)
a*
[ 5
a .<*v
(7)
and conditional covariance matrix
10
[l + <Daf]o*
_2
(8)
where
(9)
and
(10)
Proof: All proofs are in the Appendix.
■
The elements of the 4* matrix will be referred to as \yy, i=v,l,2 and j=P,S, and the terms in the
conditional covariance matrix, denoted Q, will be denoted by CDy (ij=v,l,2), for the subsequent
discussion.
Based on this posterior joint density, the problem faced by the three players can be
formulated. Specifically, while the two traders maximize their expected profits, the marketmaker sets prices in order to break even. These strategies impose the following restrictions
on the equilibrium
Proposition 2: A comparison of terms between the conjectured equilibrium, and the
individually rational strategies, leads to the following parameter restrictions being imposed in
the second round of trading.
= J_ Ovi-TMiz
2r
(on
(11. a)
11
q)v2-ra2q)12
to22
(H.b)
and
r
T
<*2<0vl+Mv2
(11. c)
[a2con + p 2C022 + 2a 2p2a),2+ a 2u2
Proof: Appendix.
Notice that the preceding results characterize the manner in which the equilibrium depends on
any distribution that is multivariate normal. To better understand the manner in which the
strategy on the trader in the preceding round affects the equilibrium here, it is useful to
explicitly characterize the solution both in terms of the level of disclosure and the trading
intensity in the preceding round. The following result restates the restrictions implied by (8.ad) in terms that makes this comparison easier.
Proposition 2': The equilibrium in the second round of trade can be characterized by the
following parameter restrictions.
a2
i-rp2
2r
oj_
P2
^
(12.a)
ol+o*'
P q ? + ( l - r a 2)
2 r[(l + Oo?)(o| +o*)+4»a*o*
(12.b)
[(q2+ p2)+ a>qf]q;
r =
[(a2+ p2)2cj+c^af
+ V l G 2 +<y2
u2] + <t>[$2
2o 2
l ( o l + o 2) + o 2
vc 2 + ( o l + o 2
l )o2
u2]
Proof: Follows from substituting for the variance and covariance terms from (7) into the
equilibrium conditions in Proposition 2.
(12.c)
12
As can be seen from restating the equilibrium conditions this way, the parameter <I> captures
all the information that is being brought from the first round into the second round. This
information is based on (i) how aggressively trader #1 transacted in the previous round (a,),
(ii) the extent to which uninformed order flow allows trader #1 to hide his inform ation^^),
and (iii) the extent to which noisiness in the trade disclosure process allows him to hide his
information (crp- It is also easy to show that as O approaches zero, which can be interpreted
as the case where no information is conveyed by the previous round of trade, the solution
characterized by (12 a-c) is nothing but the equilibrium in a single round of trade with two
informed traders. Similarly, an environment of perfect trade disclosure would be one where
0*_*o, or equivalently, one where < j > _ •
Since O fully captures the effect of the first round on equilibrium in the second round
of trade, it is useful to examine the relationship between it and equilibrium in the second
round. Notice that the term <I> is not present in the restriction on ctj directly, reflecting the fact
that in the second round, trader #1 is primarily concerned about the strategies of the other
players alone, and not about <I> itself, since he has already observed 0,, about which P, and S
are providing information .
For trader #2 (the trader who trades in the second round only), the increased precision
of the revelation of information about the asset through the price P, and through trade
disclosure (loosely interpreted as an increase in <I>) has conflicting effects. First, to the extent
that this revelation provides information about the signal observed by the first trader (0,), it is
useful in "corroborating" the information contained in trader #2's own signal, 02, and also
provides this trader with information on the nature of competition in the second round.
However, the disclosure announcement also conveys valuable information to the market
maker, who bases his price on this information. This leads to a reduction in the value of
trader #2's private information. While the corroboration effect would cause trader #2 to trade
more aggressively, the revelation of the information to the market maker creates the opposite
effect, suggesting that trader #2's strategy could be non-monotonic in O. The fact that this
can happen is illustrated by the following example.
13
Example:
Consider the case where the model is characterized by the following set of parameters.
ctv 5 ,, O 2
1* a u,
3, ^ u2
^
The solution values of P2 (the intensity of trading by trader #2) are computed for different
levels of the precision of trader #l's signal 0,, i.e. over the range a, = 0 to 0.35, and for
different values of the parameter 0=0 to 5.14
These solutions are plotted in Figure 1 (all figures appear after the Conclusions),
which shows that the trading intensity of the second trader can be nonmonotonic in O. This
nonmonotonicity is most pronounced when the signal being received by trader #1 is quite
precise. When the signal being received by trader #1 becomes less precise, the trading
intensity of trader #2 becomes a monotonic increasing function of O.
These findings are consistent with the trade-off faced by trader #2 between the benefits
of having knowledge about the competition and the costs of having information revealed to
the market maker.
Not surprisingly, trader #2 is most concerned about the revelation of
information to the market maker when trader #1 receives a very precise signal, i.e. C, is low.
In this case, increases in <1> (loosely interpreted as an increase in the precision of trade
disclosure) quickly lead to situations where any benefit (in terms of corroboration) to trader
#2 are offset by the fact that this information is also revealed to the market maker.
Consequently, beyond a certain point, trader #2 is forced to respond to increases in O by
decreasing his intensity of trading (the conventional impact of disclosure).
However, as the precision of trader #l's information is lowered (i.e. a, is increased)
trader #2 is less concerned about the information revealed to the market maker, and is more
interested in the manner in which the disclosure corroborates the information in his own
signal. Since, in this case, the corroboration effect is the dominant one, an increase in <f>
leads to an unambiguous increase in P2 (at least over the range of computation).
It is important to emphasize that any insights provided by this example are only within
14
The numerical com putations were performed in M athCad (version 5). All solutions are at a precision
of 1*10A-10. Details are available from the author upon request.
14
the context of a partial equilibrium, in the sense that, in equilibrium d> depends endogenously
on a, and a,. The example will be extended in Section 4 to include these aspects, and the
insights provided here will be useful in explaining the factors that affect the overall
equilibrium.
The First Round of Trade
In order to characterize equilibrium in the first round of trade, we need to know the
expected profits from the second round. The expected profits to the two traders in the second
round are given next.
Proposition 3: The expected profits to the two traders in the second round are given by
Jtjz = TjCj = rotj^i — Pi] ,and n 22 = T) ,2 = 1'P2[®2 — P 2 ]
where
(13)
denotes the profits by trader i in round j.
Proof: See Appendix.
■
In selecting his optimal trading strategy in round #1, informed trader #1 solves
max£:{x1[? -X 1(x1+w1)]l01} + £o[7i12(xI)|01]
(14)
*1
where the first term in (14) reflects the expected profits to trader #1 from the first round of
trade, while the second term reflects the implications for the expected profits from the second
round of trade. Knowing the expected profits for trader #1 in the second round of trade from
Proposition 3, we can now solve for the additional conditions imposed.
Proposition 4: The equilibrium restrictions imposed by the trading strategy of the informed
trader in the first round and the market-maker's pricing strategy are
15
(14)
where
and
X =
a
2
(15)
Proof: Appendix.
■
Notice that when \|/1P and \|/IS are zero, the equilibrium is identical to the single period
equilibrium in Kyle (1985). This is not surprising, since this corresponds to the case where
trader #l's current trades do not reveal any information, and consequently have no
implications for the second round of trade.
This completely characterizes equilibrium in the model. The complete solution
requires solving equations (9 a.-c.), (14), and (15) for the parameters (a,,
X,
0^, fi2, O- In
order to understand the complex, non-linear system characterizing equilibrium, we begin by
considering equilibrium under the extreme cases of no disclosure (a^= °°) and perfect disclosure
(an=0). This will make it easier to understand the consequences of noisy disclosure regimes
on equilibrium.
4. Analysis of Equilibrium
If there is no disclosure of trades in the economic environment under consideration, i.e
the noise in the disclosure process (an) is infinite, the resulting equilibrium is fairly obvious,
16
given the results of the previous section.15 Specifically, while obvious changes are made to
both the conditional means and the conditional covariance matrix in Proposition 1, the
restrictions characterizing equilibrium remain unchanged. The same is true for as Gn
approaches zero, i.e. the disclosure of trades becomes perfect. However, an examination of
this special case will provide some useful insights into the solution under generic levels of
disclosure.
When disclosure is perfect, trader #1 has to decide whether to trade in the first or the
second round only. Any trading in the first round immediately renders his information
worthless for the subsequent round of trade since his trade (and therefore his private
information 0,) is disclosed before the next round of trade. Alternately, trader #1 could
forego trading in the first round altogether, and trade in the second round alone.
Consequently, in this extreme case, trader# 1 will trade either in the first round or in the
second round only. The benefit of trading in the first round is that the trader has no
competition to worry about. However, if there are a large number of noise traders in the
second round (i.e. o ^ o ^ j ), the trader might be willing to forego trading in the first round,
and trade exclusively in the second round instead, despite the presence of competition from
trader #2. His willingness to do so, of course, will also depend on the relative precision of
his signal relative to the precision of the signal received by trader #2. Obviously, his choice
of which round to trade i is influenced by his desire to maximize expected profits. It is well
known from the existing literature that the expected profits to trader #1 in the first round are
increasing in Gv and <*„i. while decreasing in a,. Similarly, the expected profits that he
generates in the second round are increasing in ctv, gu2, and g 2, while they decrease in a,. As a
consequence, trader 1 is more likely to trade in the first round when Gul is high, while he is
more likely to trade in the second round if g u2 and/or g 2 are high. The implications of higher
Gv or a, are less clear, since they affect profits in either round similarly.
To illustrate the types of outcomes that can occur under the cases of no disclosure and
perfect disclosure, consider the following two cases.15
15
The equilibrium in this case is can be shown to be a variant of H olden and Subrahm anyam (1992),
m odified to allow the num ber o f informed traders to change over time.
17
Case 1: o = 3 , c^O.5, o2=l, aul=l, au2=l
Case 2: av=3, a,=0.5, a2=l, cul=l, au2=l
It should now be clear why case 1 and 2 have been parameterized the way that they
have. Specifically, Case 1 corresponds to the situation where trader 1 would rather transact in
the first versus the second round if there is perfect disclosure. The pool of noise traders is
similar in both rounds (i.e. crul=au2) and transacting in the second round exposes trader 1 to
some amount of competition, thereby reducing his expected profits. In Case 2, on the other
hand, the pool of noise traders in the second round is much "larger", though trader 1 does
face increased competition by transacting in this round. One would expect the benefit of
having a larger amount of noise trading to dominate the "cost" of increased competition,
resulting in an equilibrium where trader 1 can make superior profits by transacting in the
second round alone. The equilibrium along with the expected profits of the two traders are
summarized in Table 1.
18
Table 1: Equilibrium Under "No Disclosure" and "Full Disclosure"
(A Numerical Example)
No disclosure (an= °°)
Full disclosure (Gn= 0)
Case 1 (aul=l)
Case 2 (au2=2)
Case 1 (<7Uj=l)
Case 2 (ou2=2)
a,
0.26365
0.20900
0.329798
0
X
1.44423
1.33969
1.47959
0
<*2
0.32876
0.59946
******
0.48928
P2
0.25629
0.48447
0.81284
0.42256
r
1.08263
0.58596
0.12035
0.70013
n„
1.44423
1.33964
1.47959
0
^12
0.65878
1.38721
0
1.55038
1^22
0.42099
0.95983
0.00058
1.25014
n,
2.10302
2.72690
1.47959
1.55038
^total
2.5240
3.68673
1.48017
2.80052
19
Having understood these extreme cases will make the cases of interim levels of disclosure a
little more intuitive.
The two sets of parameters considered here lead, for the case of perfect trade
disclosure, to these two extreme solutions. The prior intuition that these two scenarios lead to
very different solutions as
—£ turns out to be useful in understanding the behavior of the
solution at intermediate values of
rs
. These intermediate solutions are of interest, of course,
since they represent the more plausible representation of the inherently noisy nature of
disclosure. As we just saw, under perfect disclosure, in case I informed trader #1 trades
exclusively in the first round, while in case II, informed trader #1 trades exclusively in the
second round only.16
The equilibrium parameters for a these two cases, at a variety of disclosure levels, are
shown in Figures 2 through 10. The parameters for Case 1 are represented by the solid line
and the left hand side y-axis, while the parameters for Case 2 are represented by the dashed
line and the right hand side y-axis. In Figure 2, we examine the impact of changes in the
precision of disclosure on the level of O (the parameter utilized to characterize the information
taken from the first to the second round of trade, see Proposition 2). In the absence of any
adjustment in the trading strategy of trader #1 in the first round (i.e. a change in a), <I> is
inversely related to c n. For Case 1, this relationship persists even when the endogenous
adjustment in a to the changes in disclosure levels are taken into account. Since trader #1 view
trading in the first round as being more lucrative in this case, he is unwilling to adjust his
strategy in the first round too much. In Case 2, however, the relationship between Gn and O is
non-monotonic. Trader #1 is very interested in trading in the second round. At high levels of
(i.e. very noisy disclosure) he is not very concerned about the implications of his first round
trading on the equilibrium in the second round. However as Gn becomes smaller (i.e trade
disclosure becomes more precise), trader #1 consciously attempts to control the information
16
These exam ples have been replicated for a variety of different param eter values that have the same
characteristics, i.e. that in the lim it as disclosure becomes perfect, trader #1 prefers to trade exclusively in the
first round in Case I and exclusively in the second round in Case II. The results are sim ilar to those depicted in
Figures 2-10. Space constraints preclude the inclusion of the details, but details are available upon request.
20
revealed in the second round.by scaling back his trading activity. In order to better
understand the differences across the equilibria in the two cases, we discuss each one
separately next.
C ase 1
The implication of changes in the precision disclosure on the strategies of trader #1 in
the first round can be seen more clearly in Figure 3. Starting at high levels of
G n,
corresponding to very noisy disclosure, a reduction in the noisiness (an) initially causes trader
#1 to scale down the aggressiveness with which he trades in the first round (a).
Correspondingly, as seen in Figure 4, the market maker reduces the sensitivity of price to order
flow
(X ),
leading to an improvement in liquidity. Not surprisingly, the expected profits to trader
#1 from the first round decrease as well (Figure 8). However, as on is reduced beyond a certain
point, trader #1 now begins to trade more aggressively as the noisiness in disclosure is
reduced. Both the market-maker's pricing and the first round profits increase accordingly.
Intuitively, the rationale for this change in behavior can be explained based on the
consequences of reduced noisiness of trade reporting on the equilibrium in the second round.
The first round trading strategy of trader #1 is clearly dependent on the profits that he expects
to make in the second round. As disclosure becomes more precise, trading in the first round
effectively precludes any trading in the second round, since such disclosure results in the
trader no longer possessing an informational advantage relative to the market. In the situation
considered in Case I, trader #1 effectively "gives up" trading in the second round in order to
profit in the first round. This can be seen in Figure 9, where the expected profits to trader #1
from the second round of trade approach zero as disclosure becomes more precise (i.e.
Gn
approaches zero). The effect of increased transparency causes trader #1 to trade more
aggressively in the second round conditional on possessing superior information. However,
the reduced presence of trader #1 in the second round causes the market maker to lower the
sensitivity of price to order flow (T) leading to an improvement in liquidity.
strategy of trader #2 is non-monotonic in
G n, reflecting
The trading
the trade-offs discussed in Section 3.
21
C a se 2
In Case II, an reduction in the noisiness of trade disclosure causes trader #1 to reduce
trading in the first round to concentrate on the second round. As can be seen from Figures
3,4 and 7, trader #1, reduces a leading to lower profits from the first round, and the market maker
reduces
X,
leading to improved liquidity in the first round as well. In this case, the interesting
effects are in the second round of trade. When a n is high, lowering it initially causes trader #1
and trader #2 to trade more aggressively. Intuitively, at this stage, the increased disclosure
helps trader #2 by corroborating his information. Trader #2 consequently trades more
aggressively, and trader #1 responds. At this stage, the increased competition between the
two traders allows the market maker to actually reduce T, as can be seen in Figure 7. This is
also seen in Figures 9 and 10, where the expected profits to both traders decrease. However, as
cn
is decreased further, trader #1 scales down his trading in the first round. Since no
information has been revealed to the market, both trader #2 and the market maker know much
less. Consequently, trader #2 trades less aggressively (Figure 6) and the market maker
increases T (Figure 7). Trader #1 is able to reduce (3 (Figure 5) and still generate higher
expected profits in the second round. It is interesting to note that trader #2 prefers this
solution as well, since his profits also increase.
Both these cases together suggest that the impact of increased transparency on liquidity is
quite complex. In Case I, increasing transparency beyond a certain point actually worsens
liquidity in the first round, since trader #1 decides to cluster all his trades there. By contrast,
in Case 2, an increase in transparency leads to trader #1 deciding to trade primarily in the second
round, leading to improvement in liquidity in the first round, but at the expense of reduced
liquidity in the second round. This result is quite similar to that obtained by Admati and
Pfleiderer (1988) in the context of discretionary noise traders. These traders will naturally
gravitate towards periods of greatest liquidity, in order to minimize their losses. As shown in
this example, informed traders face similar incentives, especially in the presence of noisy trade
reporting. These traders naturally gravitate towards periods with the highest expected profits,
leading to a reduction in liquidity in that period, while liquidity improves in the other time
periods.
22
5. Implications
This paper has developed a multiperiod model of trading to examine the consequences
of noisy trade reporting on the equilibrium trading strategies of informed traders, with a
special focus on the implications of this disclosure for liquidity (in the different rounds of
trading). The primary finding is that when informed traders can adjust their trading strategies
to reflect the disclosure environment in which they trade, an increase in the precision of trade
reporting can have ambiguous affects on liquidity. This is because the increased precision of
disclosure creates incentives for traders who obtain information early to concentrate their
trades in a single round. This effect is similar to that identified by Admati and Pfleiderer
(1988) in the context of clustering of trades by liquidity traders. The situation where the
early informed trader clusters in the later round can also be interpreted as herding.
In addition, we find that increasing the precision of trade disclosure has mixed
implications for traders who receive private information later that others. This is because
such disclosure has two distinct effects. First, it serves a useful role in corroborating the
private information available to the trader. Second, it provides additional information to the
market maker, who utilizes this information in setting prices, thereby reducing the value of
the private information available to the trader. Consequently, the trader who receives private
information late could find his profits either increasing or decreasing as a function of better
(less noisy) disclosure.
The implication of these results for the structure of markets and regulation clearly
depends on the assumed objectives (of the markets or regulators), an area of considerable
ambiguity. What is clear is that the often stated objective of perfect (noiseless) disclosure is
not necessarily the appropriate goal to adopt. Such disclosure, or regulations that increase the
precision of existing trade disclosure policies, have complex implications for the intertemporal
liquidity, volatility, volume, and the acceleration in information revelation in a security
market.
The analysis in this paper exogenously specifies the noisiness in the trade reporting
process in order to concentrate on the implications for market equilibrium. A useful
extension of this analysis would be to model the noisiness in the reporting process as a
function of the structure of the market itself. This could also provide insights on the role of
23
such trade reporting in a multi-market setting. Finally, it would also be useful to examine
other dimensions of the equilibrium, like volatility, the intertemporal correlation of prices, and
the incentives to collect information under different disclosure regimes. All of these represent
interesting directions for future research.
Figure 1
Non-monotonic Ity in ^
(An example)
Information conveyed by previous round (O)
Figure 2
Im p licatio n s of D isc lo su re on 4>
0.5
0.06
0.05
0.4
-
0.04
0.3
u
8
- 0 .0 3
U
e
0.2
0.02
\
:
0.1
0.01
1
0.0
0
1
1
1
1
2
1
1
3
>
1
1
4
Noisiness of Trade Disclosure^)
1
5
0.00
CN
8
a
Figure 3
Trade Intensity
(T ra d e r 1. R o u n d 1)
Noisiness of Trade Dsclosure
(on)
Figure 4
Price Setting
( R o u n d 1)
1.47
1.46
- 1.4
n
.
1
1 .2
1.45
/
1.44
~
1.43
o
1.39
/
/
-
■ 0 .6
/
0.4
/
0 .2
/
1.38
-
J
1.37
0 .0
i
0
•
i
1
^
n
/
i
1.40
1 .0
0 .8
/
:
1.41
"
jr
;
a 142 -
^
--------
:
-
*
i
2
i
i
3
i
i
i
4
Noisiness of Trade Disclosure (c^)
i
5
-
~
Figure 5
Trade Intensity
(Trader 1. Rouxl2)
5
0.62
4
0.60
0.58
3
0.56
<r
2
0.54
0.52
1
0.50
0
0.48
0
1
2
3
4
5
Noisiness of Trade Disclosure (o^)
Figure 6
Trade Intensity
( T r a d e r 2, R o t n d 2 )
0.50
“ 0.9
0.49
- 0.8
0.48
- 0.7
0.47
8
0.46
- 0.6
CC<
Ls
0.45
“ 0.5
0.44
“ 0.4
0.43
“ 0.3
“ 0.2
0.42
J______i------1______ i______I______i______I______i______I______i______I____ :
0
1
2
3
4
Noisiness of Trade Disclosure (o^)
5
8
U
cdl
Figure 7
Price Setting
( R o u n d 2)
1.1
1.0
0.9
0.8
-
u
u
0.7
0.6
u
0.5
0.4
0.3
0.2
0.1
0
1
2
3
4
5
Noisiness of Trade Disclosure (c^)
Figure 8
Expected Profits
[Round 1. Trader 1 )
1.48
:
\
1.4
1.47
.**
1.2
1.46
1.45
-
~
1.44
-
§
U
1.43
" 1
1 4 2
C
1.41
1.0
-------------- “
c n
0.8
/
/
1.39
/
0.4
1
0.2
J
-
1.37
0.0
i
0
1
* =
C
/
-
1.38
0.6
/
1.40
«
U
1
1
1
1
2
!
1
3
!
1
,
4
Noisiness of Trade Disclosure (c^)
1
5
-
Figure 9
Expected Profits
( T r a d e r 1, R o u n d 2)
Figure 10
Expected Profits
(T ra d e r 2, R o t n d 2 )
-
0 .5
___________________________________ -
0 .4
1.2 5
1.2 0
.
-
1 .1 5
0 .3
6
CM
a
o
cT
%
6
K
1 1 0
.2
" a
c T
1.0 5
0 .1
1.0 0
1
0 .0
i
i
>
i
.
i
.
i
i
i
0 .9 5
0
1
N o is in e s s
2
of T ra d e
3
4
D is c lo s u r e
5
(o ^ )
Figure 11
Expected Profits
( T r a d e r 1)
2.9
2.9
2.7
2.7
2.5
2.5
t5
>
V
C3
2.3
2.3
y
2.1
2 .1
1.9
1.9
.
c?
8
a
oT
cf
;
1.7 ;
/
;
/
;
1.7
1.5
1.5
i
i
0
i
1
i
i
2
3
i
.
i
4
5
Noisiness of Trade Disclosure (o^)
Figure 12
Total Profits
3.5 3.o
-
3.0
a
u
N>
cn
1
g
U
2.5
/
/
C
2.0
'
1.5
total
~
3.5
i
0
<
C
2.0
i
1
i
i
2
i
i
3
i
i
<
4
Noisiness of Trade Disclosure (an)
i
5
1.5
30
Appendix
Proof of Proposition 1 The unconditional distribution of the random variables [v, y„ y2, P,,
S] is multivariate normal, with mean zero and a covariance matrix,
Z=
^11
2,2
Z 2, Z 22
whose components are
<*v
Ov
av
2+of
2,.= <*v
av
o2
<*v
a 2+ o
a ,X ,a v
«l<*v
a,(°v+°0
2,2 =
a,X,a2
2 22 ”
(A.l)
a,Ov
a ^ a ^ + G ^ + ^a',
X,af(a^+oJ)
^,a?(G^+Gj)
a 2(a2+ a 2)
(A.2)
(A.3)
and Z21=E12t . Standard results from multivariate normality (Anderson (1984) imply that the
conditional mean is given by
= i 12.l 2!-'.[p,
s
]
(A.4)
while the conditional covariance matrix is given by
a=2,|-212. v 1-212r
Appropriate simplifications leads to the relevant terms.
(A.5)
31
Proof of Proposition 2: The first informed trader (#1) faces the following optimization
problem:
max£2jx2[v - |Xy - r(x2 + z2 + «2)}y,}
(A .6)
where E2 denotes the fact that expectations are being taken with respect to the posterior
distribution characterized in Proposition 1. The associated first order condition can be
rewritten to yield the following trading strategy.
x, =IT
tOv,-rp2(D,2
[eu-fc]
co,
(A.7)
Similarly, the second trader's strategy leads to
3,2 =
1 C0v2- r a 2(0,2
[®2 P2]
CO 22
2r
(A .8)
Finally, in the second round, the market maker sets prices to break even, i.e.
(A .9)
P2 = E2[v \x 2 + y 2 + u 2]
which leads to
P=Pv +
a 2(ovl + p2COv2
a 2con + P2a»22 + 2 a2P2a>l2 +a u2
[x2 + y 2 + u 2]
(A. 10)
A comparison of terms between the conjectured equilibrium, and the individually rational
strategies, leads to the following parameter restrictions being imposed in the second round of
trading.
«2 = 2r
covl - r p 2co,2
^v2 ^"^2^12
,P2=—
CO22
co,
2r
(A.l 1)
and
r =
a 2COvl + P2^v2
a 2c°i1+ P2C022 + 2a2p2co12+ o u2
(A. 12)
32
Proof of Proposition 3: We present only the proof for the profits of trader #1, since the
condition for trader #2 is similar. Conditional on P, and S and y,, the expected profits of
trader #1 are given as
n ,2 =£2[jc2[v -p v-r(j:2+ ^+M2)j0i]
(A.13)
which upon simplification leads to
n i2 = C*2 = Pa 2[®l —M-i]
(A.14)
where the last part of (A. 14) follows from the fact that x2=(3 [0, - |i,].
■
Proof of Proposition 4: Recall, from proposition 3, that the expected profits to trader #1 from
the second round of trading is
7i21= ra 2 [e ,-P i]2
(A-15)
At t=0, after having observed 0,, the trader's expected profit (from the second round) from
trading a quantity x, is
^ [ra = (e ,-v , ^ - v lsx)2|e,]
<A-16>
which can be rewritten as
^o£roc2(9i —\|/1p(A.(a:1+«,))—M
, is(-*’+ mii)) |®iJ
(A.17)
Taking expectations of the total profits equation (14) and differentiating with respect to x,
leads to
2
2 2"0i —2Xart + r<x2 - 2(Xv|f lp +Vis)®i "*■2(A,\|f lP +\|/15) |
°v+ai
1
1
which, when rearranged, yields (14) in the text.
(A.18)
33
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@FedFRASER