WPS6995
Policy Research Working Paper 6995
Strategic Information Revelation
and Capital Allocation
Alvaro Pedraza Morales
Development Research Group
Finance and Private Sector Development Team
July 2014
Policy Research Working Paper 6995
Abstract
It is commonly believed that stock prices help firms’ manag- prices are not as useful in guiding capital toward its most
ers make more efficient real investment decisions, because productive use, leading to inefficient investment decisions.
they aggregate information about fundamentals that is not Using a sample of U.S. publicly traded companies between
otherwise known to managers. This paper identifies a limita- 1990 and 2010, the paper documents a positive correla-
tion to this view. It shows that if informed traders internalize tion between the quality of managerial information and
that firms use prices as a signal, stock price informativeness stock price informativeness. Contrary to the conventional
depends on the quality of managers’ prior information. In view that less informed managers should rely more on stock
particular, managers with low quality information would prices when making investment decisions, the author finds
like to learn about their own fundamentals by relying on no differences in the sensitivity of investment to stock prices
the information aggregated in the stock price. However, for different levels of managerial information. The evidence
in this case, the profitability of trading falls for informed suggests that while firms do learn from prices, the learn-
speculators, who therefore reduce their trading volume, ing channel and its effects on real investment are limited.
reducing the informativeness of prices. As a result, stock
This paper is a product of the Finance and Private Sector Development Team, Development Research Group. It is part of
a larger effort by the World Bank to provide open access to its research and make a contribution to development policy
discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org.
The author may be contacted at apedrazamorales@worldbank.org.
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Strategic Information Revelation and Capital Allocation
Alvaro Pedraza Morales∗
JEL Classiﬁcation: G12, G23, C7, E22
Keywords: Asymmetric information, feedback eﬀects, strategic interactions.
∗
Email address: apedrazamorales@worldbank.org. I am indebted to Pete Kyle and John Shea for their invaluable guidance.
This study has beneﬁted from the comments of Sushant Acharya, Jeronimo Carballo, Pablo Cuba, Laurent Fresard, Anton
Korinek, Federico Manderlman, Luminita Stevens, Shu Lin Wee and Russ Wermers. I also thank seminar participants at
the Western Economic Association International, Federal Reserve Bank of Atlanta, Federal Reserve Board of Governors,
University of Maryland, World Bank, Bank of International Settlements and Bowdoin College for comments and suggestions.
1. Introduction
Do stock prices improve eﬃciency by directing capital towards more productive uses? A widely held
view, dating back at least to Hayek (1945), is that stock prices are useful signals since they aggregate
information about fundamentals that is not otherwise known to ﬁrms’ managers.1 In this sense, the stock
price of a company might be informative to the manager when making a real investment decision.2 In a
very practical way, informative prices enable superior decision-making (Fama and Miller (1972)).
In this paper I identify a limitation to this view. More precisely, in a model where informed traders
in secondary markets internalize that stock prices are signals to ﬁrms’ managers, I show that trading
volume and price informativeness depend on the quality of managers’ prior information. In other words,
the amount of private information that is aggregated into the stock price through the trading process is a
function of managers’ initial information. The learning channel is limited because prices are less informa-
tive for ﬁrms with low quality of managerial information, precisely the case in which managers would like
to learn more from the stock market. This happens despite the fact that some market participants are
endowed with perfect information. As a result, stock prices are not as useful in guiding capital towards
its most productive use.
Aside from identifying a fundamental limitation to the allocational role of stock prices theoretically, I
make two novel empirical contributions. Using a sample of U.S. publicly traded companies, I document a
positive correlation between the quality of managers’ information and stock price informativeness. I ﬁnd
a stronger correlation for ﬁrms with higher institutional ownership, which further suggests the presence of
strategic behavior. Also, contrary to the conventional view that less informed managers should rely more
on stock prices when making investment decisions, I ﬁnd no diﬀerences in the sensitivity of investment
to stock prices for diﬀerent levels of managerial information. The evidence suggests that while ﬁrms do
learn from prices, the learning channel and its eﬀects on real investment are limited.
I model learning from prices as follows. There is a continuum of publicly traded ﬁrms facing a real
investment opportunity with uncertain net present value. Firms use stock prices to update their prior
about their own fundamentals. Informed and noise traders submit demands for the ﬁrm’s shares in a
secondary market. Informed traders are strategic in that they internalize the eﬀect of their trades on
both prices and on the ﬁrms’ inference problem.
1
Subrahmanyam and Titman (1999) argue that prices are useful to managers because they aggregate investors’ signals
about future product demand.
2
This mechanism has received empirical support in the recent work of Durnev et al. (2004), Chen et al. (2007) and Bakke
and Whited (2010).
2
The main result of the model can be summarized as follows: When ﬁrms’ managers are less informed
a priori, informed traders realize that their expected trading proﬁts are lower because the ﬁrm is less
likely to undertake the project in the ﬁrst place. When trading is costly, an informed speculator does not
want to buy the stock of a ﬁrm with a potentially good investment project if the investment is likely to
be cancelled. Similarly, a trader will not short sell a public ﬁrm with a negative NPV project if the ﬁrm
is not likely to invest. Under these circumstances, traders with private information reduce their trading
volume, and in turn prices are less informative about the fundamental. Investment sensitivity to price is
lower in this case, as ﬁrms recognize that prices contain less information and are less inclined to rely on
the stock price to update their prior. Overall, investment eﬃciency falls as the market signal is not as
useful in helping managers distinguish between good and bad projects.
Stylized facts about speculative markets suggest that the best-informed traders are large. In the stock
market, arbitrageurs with private information about merger prospects buy and sell signiﬁcant percentages
of the outstanding equity of publicly held companies. It is well recognized in existing literature that large
traders take into account their eﬀect on prices in choosing the quantities they trade (Grinblatt and Ross
(1985) and Kyle (1985)). If an investor has superior information, attempts to use it will “publicize” some
of the information and instantly reduce its value. Price-taking behavior would be irrational in this case.
This paper extends this logic to argue that large traders internalize that prices are also signals to ﬁrms’
managers when making real investment decisions.
This paper is related to a growing literature that studies feedback eﬀects on equilibrium asset prices.3
The basic idea of this literature is that if ﬁrms use market prices when deciding on their actions, traders
should adjust their strategies to reﬂect this response. On the theoretical side, this paper is closely related
to Bond et al. (2010). The authors suggest that when agents (e.g. directors, regulators, or managers)
learn from stock prices, there is a complementarity between the agent’s direct sources of information and
his use of market data. However, in their model, trading in the secondary market is not modeled explicitly.
Instead, the stock price is set through a rational expectations condition, which is then used by the agent
who is taking a corrective action. The main drawback in their model is that a rational expectations
equilibria does not always exist, which limits the model’s predictions.4 In my model, trading by informed
speculators and ﬁrms’ real investment decisions are the outcome of a strategic game. By assuming that
informed speculators internalize the ﬁrms’ inference problem, I am able to show existence of equilibria
for any level of managerial information, contrary to the non-existence result in Bond et al. (2010). In
3
See Bond et al. (2012) for an excellent survey of this literature.
4
The authors interpret non-existence as indicating a loss of information transmitted by prices.
3
this sense, my model has a clear empirical prediction, namely that stock prices are less informative for
managers with low quality of information.5 Furthermore, my model allows me to study informed trading
behavior when there is learning from stock prices, a feature that is omitted in Bond et al. (2010).
Other papers studying feedback eﬀects include Leland (1992), Dow et al. (2011), Goldstein and Guem-
bel (2008), Edmans et al. (2011) and Goldstein et al. (2012). Leland (1992) shows that when insider
trading is permitted, prices better reﬂect information and expected real investment rises. Dow et al.
(2011) ﬁnd that information production in secondary markets is sensitive to the ex-ante likelihood of the
ﬁrm undertaking the project and Edmans et al. (2011) study asymmetric trading behavior between good
and bad information. Most of these papers assume a discrete space for ﬁrms’ fundamentals,6 typically
specifying two possible valuations for the investment project (i.g. high and low). My model assumes a
continuous space of ﬁrms’ fundamentals, which allows me to study the interaction between the quality
of managerial information, trading behavior, stock price informativeness and investment eﬃciency, issues
that to my knowledge have been overlooked by the existing literature.
This paper provides a novel explanation for why markets are limited in their ability to aggregate
information and guide real decisions. Shleifer and Vishny (1997) provide an alternative explanation based
on limits to arbitrage, in which the slow convergence of prices to fundamentals may deter speculators from
trading on their information. Other explanations rely on market frictions such as short selling constraints.
For example, Diamond and Verrechia (1987) show that short selling constraints aﬀect the speed of price
adjustment to private information. In my model, the ability of prices to fully reﬂect fundamentals and to
coordinate investment is crucially related to the precision of ﬁrms’ prior beliefs about their fundamental
value.
Empirical evidence that price informativeness is high in well-developed ﬁnancial systems and low in
emerging markets is presented by Morck et al. (2000). The authors argue that in countries with well-
developed ﬁnancial markets, traders are more motivated to gather information on individual ﬁrms. My
model oﬀers an alternative interpretation of their results. I argue that low price informativeness may
result from the failure of stock prices to aggregate information when feedback eﬀects are present and
ex-ante fundamental uncertainty is high, which is potentially the case for emerging markets.
Finally, this paper is related to the empirical literature that studies learning from prices. Durnev et
al. (2004), Chen et al. (2007) and Bakke and Whited (2010) show that investment sensitivity to prices
is higher for ﬁrms for which the stock price is more informative about fundamentals. My empirical work
5
This result holds independently of whether the investment decision is value-increasing or value-decreasing for the ﬁrm.
6
Another example is Goldstein and Guembel (2008), who study price manipulation when traders are uninformed.
4
addresses the determinants of stock price informativeness. The evidence suggests that stock prices are less
informative for ﬁrms with low quality information ex-ante. I also estimate a standard investment equation
as in Chen et al. (2007), and show that, contrary to the conventional view, less informed managers do
not rely more on stock prices to make investment decisions. Collectively, the evidence suggest that while
secondary market are a useful source of information, they are limited in their ability to guide real decisions.
The rest of this document is organized as follows. Section 2 introduces the model economy. Equilibrium
results are derived in section 3. This section includes a model extension where ﬁrms can incur a cost to
acquire information about the fundamental before observing the stock price. In section 4 I present the
empirical exercise and I conclude in section 5.
2. Model
The model consists of three periods, t ∈ {0, 1, 2}, with three types of continuum agents: ﬁrms, informed
speculators (one for each ﬁrm) and noise traders. Stocks for each ﬁrm are traded in a secondary mar-
ket. Each ﬁrm’s manager needs to decide whether to continue or abandon an investment project. The
investment decision is taken to maximize ﬁrm value (there is no shareholder/manager agency problem).
2.1. Firms
The economy is populated by a continuum of ﬁrms. At period t = 0 each ﬁrm is uncertain about its own
fundamental value θ, which determines its ﬁnal proﬁts (for instance, the ﬁrm may be uncertain about the
2 ).
viability of a project or future demand). θ is unobservable and ﬁrms have a common prior θ ∼ N (µθ , σθ
At t = 1 ﬁrms observe their own stock price q and decide whether to invest (d = i) or not (d = n). If
a ﬁrm decides to invest it pays a ﬁxed investment cost c > 0. In period t = 2 payoﬀs are realized for each
ﬁrm according to
θ−c
d=i
Πd = (1)
0
d=n
2.2. Financial Markets
For each stock there is one risk neutral informed speculator. He learns the ﬁrm’s fundamental value θ at
period t = 0. In this setting I am modeling the extreme case where the speculator is perfectly informed
and the ﬁrm is not. This simplifying assumption allows tractability. I conjecture that similar results
5
would hold if the speculator has some private information about the ﬁrm fundamental that is orthogonal
to the ﬁrm’s information. This would generate some learning from prices. At date 1, conditional on their
information, informed speculators submit a market order XI (θ) to a Walrasian auctioneer.7 I assume
that speculators do not observe the price when they trade, and hence submit a market order, as in Kyle
(1985). This captures the idea that speculators, when they trade, do not have the market information
that the ﬁrm will have when making the investment decision (recall that the ﬁrm bases its investment
decision on the price of the security). I impose no any additional constraints on the demands by the
informed speculators, such as short selling constraints. That is, the informed speculators either have deep
pockets or have access to ﬁnancing to buy or sell as many shares as they ﬁnd proﬁt maximizing.
The Walrasian auctioneer also observes a noisy supply curve from uninformed traders and sets a price
z , q ), a continuous
to clear the market. The noisy supply for each stock is exogenously given by XN (˜
˜ and a price q . The supply curve XN (˜
function of an exogenous supply shock z z , q ) is strictly decreasing
in z ˜ ∈ R is independent
˜ and increasing in q , so that supply is upward sloping in price. The supply shock z
2 ).
˜ ∼ N (0, σz
of other shocks in the economy, and z
The usual interpretation of noisy supply is that there are agents who trade for exogenous reasons,
such as liquidity or hedging needs. They are usually referred to as “noise traders”. In this setting, the
presence of noise traders guarantees that prices will not be fully revealing, as there can be diﬀerent prices
for the same fundamental value.
z , q ) takes the following functional form:
To solve the model in closed form, I assume that XN (˜
z, q) = q − z
XN (˜ ˜. The parameter captures the elasticity of supply with respect to the price. It
can be interpreted as the liquidity of the market: when is high, supply is very elastic with respect to the
price, and large shifts in informed demand are easily absorbed in the price without having much of a price
impact. This notion of liquidity is similar to that in Kyle (1985), where liquidity is considered high when
the informed trader has a low price impact. These basic features, i.e., that supply is increasing in price
and has a noisy component, are standard in the literature. It is also common in the literature to assume
particular functional forms to obtain tractability. The speciﬁc functional form assumed here is close to
Goldstein et al. (2012). The equilibrium price is given by the market clearing condition q − z
˜ = XI (θ).
In the last period t = 2, the informed speculators and noise traders earn a share of ﬁrms’ proﬁts
proportional to their stock ownership. The model timeline for each ﬁrm is depicted in Figure 1.
7
This order is not observed by the ﬁrm.
6
2.3. Equilibrium
I now turn to the deﬁnition of equilibrium in this economy.
Deﬁnition 1. Perfect Bayesian Nash Equilibrium An equilibrium with imperfect competition among
informed speculators and learning from prices is deﬁned as follows: (i) Each informed speculator chooses
a trading strategy XI (θ) that maximizes expected proﬁts subject to the market clearing condition XI (θ) =
z , q ) and the investment strategy by the ﬁrm. (ii) Each ﬁrm chooses an investment rule to maximize
XN (˜
expected payoﬀs given the observed stock price q , prior beliefs about their own fundamental value and
beliefs about the informed speculator trading strategy. (iii) Each player’s belief about the other players’
strategies is correct in equilibrium.
In other words, an equilibrium is a ﬁxed point in strategies where each ﬁrm sets a best response
(investment rule) to market prices, given prior beliefs and the informed speculator trading strategy, and
speculators set their optimal demands recognizing the price impact of their trades and the ﬁrms’ reaction.
3. Solving the Model
In this section, I explain the main steps that are required to solve the model. Using the market clearing
condition, I start by solving the optimal investment rule by the ﬁrm for a given stock price. I then
characterize the optimization problem of the informed speculator for the given investment rule. Finally,
given the investment rule by the ﬁrm and the trading rule by the informed speculator, I calculate the
ﬁxed point.
3.1. Firms
After observing the stock price, the ﬁrm’s posterior distribution on its fundamental value is
ϕ( q − XI (θ))ξ (θ)
ξ (θ | q ) = ∞ (2)
−∞ ϕ( q − XI (θ))ξ (θ)dθ
2 and ξ () is
where ϕ() is the density function of the normal distribution with mean 0 and variance σz
2.
the density function of a normal distribution with mean µθ and variance σθ
Proﬁt maximization implies that a ﬁrm with stock price q will invest if the expected proﬁt under the
∞
posterior is nonnegative, −∞ θξ (θ | q )dθ ≥ c. In this setting, the ﬁrm’s decision is a cutoﬀ rule, such that
∞
for any q ≥ q the ﬁrm will invest (d = i), where −∞ θξ (θ | q )dθ = c, and will not invest (d = n) if q < q .
7
Lemma 1. If ﬁrm managers conjecture a linear demand function by the informed speculators of the form
XI (θ) = a + bθ, then the cutoﬀ price function q is given by:
1 1 σ2
ql = a + cb − (µθ − c) z
2 (3)
b σθ
Proof in Appendix C.
Lemma 1 refers to the functional form of the cutoﬀ price when managers believe that informed spec-
ulators’ trades are linear in the fundamental.8 More precisely, q l in Lemma 1 is the ﬁrms’ best response
to linear demands by the informed speculators. The cutoﬀ price is set as an optimal weighting between
2 /σ 2 represents the ratio of
the prior information of the manager and the price signal. The fraction σz θ
the precision of the stock price to the precision of the manager’s prior information. When the managers’
2 /σ 2 → ∞, the cutoﬀ price is q → −∞
precision is large relative to the precision of the price signal, σz θ
(always invest) if µθ > c and q → ∞ (never invest) if µθ < c. In this case, the ﬁrm’s investment decision
is independent of the stock price, as the manager’s decision is based exclusively on whether the ex-ante
expected proﬁts of the project are positive or negative. When the ratio of precisions between the signal
and the prior is ﬁnite, managers set a ﬁnite q l , in which case the investment decision depends on the
observed stock price.
The cutoﬀ rule also depends on the conjectured trading strategy of the informed speculators, i.e.
the parameters a and b. For example, if ﬁrms believe that informed speculators set their demands
independently of the fundamental, e.g. b = 0, managers understand that the price signal contains no
idiosyncratic information that would be useful to infer the fundamental, and rely only on their prior to
make the investment decision.
3.2. Informed Speculators
The risk neutral informed speculators maximize the expected proﬁts of their trading strategies,
maxXI (θ) E [XI (θ)(Πd − q ) | θ], subject to the market clearing condition XI (θ) = XN (˜
z , q ) and the ﬁrm’s
investment rule described above. Since each informed speculator internalizes his market power, the
optimization problem is transformed to
XI (θ)2
max XI (θ)E [Πd | θ] − (4)
XI (θ)
8
Lemma 1 is intended primarily to help build up intuition for the model mechanism. In section 3.2 I solve the model
numerically, in which case speculators demands are not linear and Lemma 1 does not hold.
8
The ﬁrst term in (4) is expected total earnings given the investment proﬁts. The trader is perfectly
informed about the value of the ﬁrm’s project, so his expectation is taken with respect to whether the
ﬁrm will invest or not. The second term in (4) is the cost of the trading strategy.
For a ﬁrm with fundamental value θ, the probability that the stock price q is above the threshold q
∞ 1
is P r(q ≥ q | θ) = q ϕ( q − XI (θ))dq = Φ σz (XI (θ) − q ) , where Φ is the cumulative distribution of
the standard normal.
Deﬁnition 2. Let ψ (q, XI (θ)) be deﬁned as the probability that the stock price q of a ﬁrm with fundamental
θ is above the ﬁrm’s cutoﬀ rule: ψ (q, XI (θ)) ≡ P r(q ≥ q | θ).
The informed speculator’s optimization problem becomes
XI (θ)2
max XI (θ) [ψ (q, XI (θ))(θ − c)] − (5)
XI (θ)
To summarize, expected trading proﬁts depend on the probability that the ﬁrm undertakes the project.
When the informed speculator trades, he always incurs the trading cost, which is quadratic the number of
shares he demands, while the expected revenue is proportional to the likelihood that the ﬁrm undertakes
the project. This maximization captures the idea that an informed trader does not want to buy shares in
a ﬁrm with a good investment project if the investment is likely to be cancelled. Similarly, is not optimal
for a trader to short sell a ﬁrm with a negative NPV project if the ﬁrm is not likely to invest.
To learn more about the impact of strategic trading and ﬁrm learning, below I also consider the
following alternatives to the benchmark model:
Alternative #1 - Perfect information: Firms learn their true fundamental before making the
investment decision. In this case, only ﬁrms with proﬁtable projects (good fundamentals) will invest, i.e.
θ ≥ c. Since ﬁrms with bad projects θ < c don’t invest, speculators don’t trade these companies, while
1 (θ ) = (θ − c) shares for ﬁrms with good fundamentals. At time zero, the expected price is
buying XI 2
zero for bad ﬁrms and E0 q 1 = 1
2 (θ − c) for ﬁrms with positive NPV. Here the expectation E0 is taken
over the noise shock. In this case the ﬁrm’s stock price is half its expected proﬁt. The results follow from
market power, as the informed speculator recognizes that every unit he demands of the stock will increase
the price by a factor of 2 .
Alternative #2 - No learning from prices: Firms don’t use stock prices q to update their beliefs
about fundamentals.9 If the ex-ante expectation of the investment return is greater than the investment
9
For instance, this would be the outcome if the ﬁrm is required to make its investment decision at the same time that the
9
2 (θ ) = (θ − c). Here,
cost, i.e. µθ > c, all ﬁrms invest. In this case, the informed speculator demand is XI 2
informed speculators take a long position on ﬁrms with positive net present value and short positions on
ﬁrms with negative present value.
Alternative #3 - Speculators do not internalize ﬁrms’ updating: In this alternative setting,
informed speculators make their trading decision assuming that ﬁrms invest without updating their prior.
3 (θ ) = (θ − c). From Lemma 1, ﬁrms set their cutoﬀ
If µθ > c, informed speculators’ demands are XI 2
price as
2
2σz
q 3 = −(µθ − c) 2σ2
(6)
θ
and the probability of investment is
ˆ3 (θ) = Φ (θ − c) 2(µθ − c)σz
ψ + 2 (7)
2σz σθ
which is monotonically increasing in θ, indicating that ﬁrms with better fundamentals are more likely
to invest than ﬁrms with bad fundamentals. In this alternative model, ﬁrms use the stock price as a
signal and therefore make a better and more informed investment decisions. The result arises almost
by construction, because some market participant (the informed speculator) is endowed with perfect
information which makes the price a good signal to improve the ﬁrm’s decision. However, in what follows,
I show that this learning channel is limited when the informed speculator internalizes the ﬁrm learning
process.
I now turn to the solution of the benchmark model, when ﬁrms learn from stock prices and informed
speculators are strategic in that they internalize both the price eﬀect and the ﬁrm’s updating process.
Proposition 1 presents the equilibrium.
fundamental value is revealed to the informed speculator. In this case, the ﬁrm cannot use prices to update beliefs at the
time of investment.
10
Proposition 1. There exists an equilibrium with strategic informed traders and ﬁrms. The equilibrium
in strategies can be approximated around θ = c as:
(i) If the expected NPV of the investment project is non-negative under the ﬁrm’s prior, µθ ≥ c, then the
equilibrium demands by speculators and ﬁrms’ cutoﬀ rule are:
∗ (θ ) = (θ−c)
• Informed speculators’ demand: XI 2 Φ(γ )
2
• Firms’ cutoﬀ rule: q ∗ = −(µθ − c) 2σz 1
2 σ 2 Φ(γ )
θ
2(µθ −c)σz
where Φ() is the cumulative distribution of the standard normal and γ = 2
σθ
.
(ii) If the expected NPV of the investment project is negative under the ﬁrm’s prior, µθ < c, then informed
∗ (θ ) = 0, and ﬁrms never invest, q ∗ → ∞.
speculators do not trade in equilibrium, XI
Proof in Appendix C.
Proposition 1 refers to the informed speculators demands’ and the ﬁrm strategy approximated around
θ = c. While a closed form solution for the entire space of θ is not available, the approximated solu-
tion provides the relevant economic intuition because it refers to the investment decision and the trading
behavior for the marginal ﬁrm. More precisely, the approximated solution allows me to compare equi-
librium strategies between ﬁrms with fundamentals slightly above and below the investment cost. While
the comparative statics below are carried out with the linear approximation, exact numerical solutions
are presented later to establish the general validity of the results.10
Case (i) in Proposition 1 refers to the equilibrium when the ex-ante expectation of the ﬁrm’s proﬁt
is non-negative. Here, speculators’ demands and the cutoﬀ price are scaled by a factor of Φ(γ ) and
1
Φ(γ ) respectively, relative to Alternative #3 above, in which speculators do not internalize the ﬁrm
learning. From here onwards I will refer to γ as the precision of managers’ information, which is inversely
2 (managers’ uncertainty about fundamentals).
proportional to σθ
dq 3
Under Alternative #3, the cutoﬀ price is strictly decreasing in the managers’ precision, dγ = − σz < 0.
As discussed earlier, in this standard signal extraction problem, managers rely less on the stock price the
more conﬁdent they are on their prior. In the benchmark model, when traders internalize the ﬁrm’s
updating process, the cutoﬀ price varies with the managers’ precision as follows:
dq ∗ −Φ(γ ) + γ Φ (γ ) σz
= (8)
dγ Φ(γ )2
10
From here onwards, all the analytical results in the benchmark model make use of this approximation unless stated
otherwise.
11
∗
In this case the cutoﬀ price is also strictly decreasing in managers’ precision ( dq
dγ < 0, proof in
Appendix C). However the second term in the numerator in (8) has the opposite sign compared to a
standard signal extraction problem. In particular, if managers have a more precise prior, the are likely to
1
rely less on the stock price (given by the factor − Φ(γ ) ). However, this eﬀect is dampened by the factor
γ Φ (γ )
Φ(γ )2
, since the manager understands that in this scenario stock prices will have more private information,
as informed speculators trade more in absolute terms (XI (θ) ∼ Φ(γ )) when γ is higher. On the contrary,
when γ is low (less precise prior), the ﬁrm manager wants to rely more on the stock price to make the
investment decision by increasing q ∗ , but he understands that when γ is low, informed speculators reduce
their demands in absolute terms, which in turn makes the price signal less informative, dampening the
learning channel. I expand on this discussion in the next section.
Finally, case (ii) in Proposition 1 refers to the equilibrium when the ex-ante expectation of the ﬁrm’s
proﬁt is negative. When µθ < c, informed traders don’t trade and stock prices are determined solely
by noise traders. Since ﬁrm managers understand this, they ignore the stock price, making the invest-
ment decision based exclusively on their prior information, which results in the investment project being
canceled. While this result also holds when traders don’t internalize the ﬁrms’ updating process (Alter-
native #3), it is surprising that even when all agents understand the feedback from prices to investment,
stock prices cannot promote better ﬁrm decisions by overcoming the information gap between traders
and ﬁrms.11 Note that in this case, speculators cannot proﬁt on their information despite having perfect
knowledge of the fundamental. In this case, the informed trader would be better oﬀ taking over the ﬁrm
as a private equity investor whenever the true θ > c, since he could then use his information to make
eﬃcient decisions for the ﬁrm.
The results indicate that the extent of information revelation through prices is sensitive to the ex-ante
likelihood of the ﬁrm undertaking the project. Dow et al. (2011) has a similar result when studying
information production in ﬁnancial markets. In their model, a continuum of atomistic speculators pay a
cost of acquiring information as long as others are also paying this cost. This in turn depends on whether
the ﬁrm is likely to undertake the project in the ﬁrst place. The less likely a ﬁrm is to invest, the less
incentive traders have to produce information about the project. In my model, some speculators are
endowed with perfect information and incur trading costs (i.e. the price eﬀect of their own trades). While
the model in Dow et al. (2011) studies complementarities between traders and information acquisition, I
abstract from such concerns by assuming one informed speculator per ﬁrm. However, this simpliﬁcation
11
The key assumption for this result is that the informed trader has no direct communication with the ﬁrm.
12
allows me to expand to a continuous space of ﬁrms’ fundamentals instead of the discrete setting in Dow
et al. (2011) with only two possible valuations for the investment project (i.g. high and low). With this
extension, my model is suitable for analyzing the interactions between the quality of ex-ante managerial
information and trading behavior, stock price informativeness and investment eﬃciency. In what follows
I present a detailed analysis of these interactions.
3.3. Informed Trading
Above, I showed that when informed traders do not trade (XI (θ) = 0), the stock price depends on
noise trading alone, which makes the stock price uninformative about the fundamental. Larger absolute
informed speculator demands reﬂect the presence of informed trading in the stock, which leads to more
informative prices. In other words, price informativeness refers to the amount of information the speculator
reveals through the stock price, which in turns allows managers to learn about the fundamental value.
To be precise, price informativeness is proportional to informed speculators’ trading volume.
Deﬁnition 3. Let the informed trading volume VI (θ) in a stock with fundamental θ be deﬁned as the
absolute value of informed speculators’ demands: VI (θ) ≡ |XI (θ)|.
∂VI (θ)
Corollary 1. Informed trading volume decreases in managers’ uncertainty about the fundamental ∂σθ <
0, for ﬁrms with fundamental value close to the investment cost (θ in the neighborhood of c).
Corollary 1 indicates that informed trading volume is lower, and hence price informativeness is lower,
when ﬁrm managers are less informed ex-ante about the fundamental. This result is exclusive to the
benchmark strategic model, as in all three of the alternative models, informed trading volume is indepen-
dent of the precision of the ﬁrms’ prior. Figure 2 presents equilibrium demands by informed speculators
for diﬀerent levels of managerial uncertainty σθ , for the case µθ = 1.05, c = = σz = 1. The equilibrium
demands in Proposition 1 are a linear approximation around θ = c. Figure 2 displays exact numerical
solutions for the given model parameters. Consistent with Corollary 1, informed speculator demands
decrease in absolute value for larger values of managerial uncertainty.
There is one important distinction to be made. Price informativeness in the model is not the same
as prices being unbiased. Prices are unbiased if they reﬂect the correct expected value of the ﬁrm. Take
the case when µθ < c. According to Proposition 1, ﬁrms never invest and speculators do not trade in
equilibrium. The expected price at t = 0 is zero for any ﬁrm, independently of the fundamental. Hence,
expected prices are unbiased as they reﬂect the fact that the ﬁrm is not investing. However, in this
13
situation, prices are not informative, since they are not useful to the ﬁrm. In general, when the precision
of managers’ prior information falls, prices informativeness falls, even though expected prices correctly
reﬂect the real value of the ﬁrm (taking into account the investment decision).
According to Figure 2, equilibrium demands in the benchmark model are convex in θ for σθ > 0. This
result can be rationalized as follows: When speculators have positive information about a ﬁrm’s prospects,
every share they buy of that ﬁrm increases the price of the stock, signaling to the ﬁrm that it should
continue the project, which is the value-maximizing decision from the point of view of the speculators.
Thus, in this case the incentives of the speculators and the ﬁrm are perfectly aligned. Meanwhile, when
speculators have negative information about the ﬁrm’s fundamental (θ < c), their inclination would be to
short sell ﬁrm shares. However, speculators realize that every additional unit borrowed lowers the share
price, making the ﬁrm more likely to cancel the investment project, which in turn reduces the payoﬀ
of the short position. As a result, the informed speculators reduce their short position when they have
adverse information. This asymmetry in trading by informed speculators with positive or negative news
about a ﬁrm’s investment outlook was ﬁrst studied by Edmans et al. (2011) in a setting with a discrete
distribution of ﬁrms’ payoﬀ. While this is certainly an interesting result, asymmetric trading results from
higher order terms in the solution. The ﬁrst order eﬀect, namely the reason why informed speculators
optimally reduce their trading volume for ﬁrms with low quality information ex-ante, is due to the fact
that low quality information increases the likelihood that ﬁrms won’t invest and speculators lose money
whenever they trade and the ﬁrm does not invest.
3.4. Investment
I now consider the real side of the economy. Corollary 2 presents the ex-ante probability that a ﬁrm with
fundamental θ will invest.
Corollary 2. If µθ > c, for ﬁrms with fundamental value close to the investment cost (θ in the neighbor-
hood of c), the probability of undertaking the project in the benchmark model is
(θ − c) 2(µθ − c)σz 1
ψ ∗ (θ) = Φ Φ(γ ) + 2 (9)
2σz σθ Φ(γ )
2(µθ −c)σz
where Φ() is the cumulative distribution of the standard normal and γ = 2
σθ
. Proof in Appendix C.
The probability of investment is monotonically increasing in θ. Similar to Alternative #3, ﬁrms with
better projects are more likely to invest than ﬁrms with bad projects, suggesting that learning from prices
14
improves the ﬁrms’ decision. However, in the benchmark model, the slope of the investment decision
with respect to θ is scaled by a factor of Φ(γ ). This indicates that the amount of information revealed
through the stock price depends on γ . In particular, the slope around θ = c measures the eﬃciency of
the investment decision, or how well ﬁrms distinguish between good and bad investment projects. Figure
3 depicts the probability of investment for the three alternative models and the benchmark strategic
model. When the ﬁrm does not learn from prices (Alternative #2), the ﬁrm decides solely according to
its prior. When µθ > c, all ﬁrms invest and there is no distinction between diﬀerent types of projects.
The investment probability is one for all values of θ, and the slope around θ = c is zero. Under perfect
information (Alternative #1), the probability of investment is one for θ ≥ c and zero otherwise. In this
case, ﬁrms perfectly diﬀerentiate between positive and negative NPV projects. The slope is undeﬁned
around θ = c, but one can think of it as inﬁnity. For intermediate cases, a steeper slope around θ = c
indicates that managers are making better investment decisions. The main take away from this ﬁgure
is that the slope around θ = c rather than the level of the probability of investment (ψ (c)) measures
investment eﬃciency in the model.
Deﬁnition 4. Investment eﬃciency is deﬁned as the slope of the probability of investment with respect
∂ψ (θ)
to θ around the point θ = c: ∂θ |θ=c
Corollary 3. If µθ > c, the investment decision is less eﬃcient with strategic traders than in the non-
∂ψ 3 (θ) ∂ψ ∗ (θ)
strategic alternative, i.e. ∂θ |θ=c > ∂θ |θ=c . Proof in Appendix C.
Corollary 3 refers to the fact that the slope of ψ (θ) around θ = c is greater in Alternative #3 than in
the model with strategic behavior, as shown in Figure 3 for a particular set of parameters. More precisely,
learning from prices improves investment eﬃciency in both models, but this improvement is smaller in
the strategic model.
In a standard signal extraction model, an uninformed manager is more likely to rely on the outside
signal to learn about the fundamental. In such a case, one would expect that managers with less precise
prior information rely more on the stock price to make their investment decision. In other words, the
sensitivity of investment to the stock price should be higher for less informed managers. To study whether
that intuition still holds in the benchmark model, I calculate the correlation between the expected stock
price and the investment probability.
Deﬁnition 5. The correlation between the expected stock price and probability of investment is deﬁned
15
as follows:
1
Corr(q, ψ ) = XI (θ) − X I ψ (θ) − ψ dξ (θ)
1
where X I and ψ are averages of the expected price and investment probability respectively, taken with
respect to the space of fundamentals θ.
Corollary 4. The correlation between the expected stock price and the probability of investment is:
γ
1. In the benchmark model: Corr (q ∗ , ψ ∗ ) = 1
2 Φ(γ )φ Φ(γ )
2 + (µ − c)2
σθ θ
2. In the non-strategic model: Corr (q ∗ , ψ ∗ ) = 2
1 2 + (µ − c)2
φ (γ ) σθ θ
where Φ() and φ() are the cumulative and probability distribution functions of the standard normal re-
2(µθ −c)σz
spectively and γ = 2
σθ
. Proof in Appendix C.
Corollary 4 implies that, all else equal, the correlation between stock prices and investment is increasing
in σθ in both the benchmark model and the non-strategic alternative, as expected from the standard
intuition discussed earlier. That is, managers with lower quality of information a priori are more likely
to rely on their own stock price to make investment decisions. However, Corollary 4 implies that the
correlation between investment and stock prices increases less with respect to managerial uncertainty
∂Corr(q ∗ ,ψ ∗ ) ∂Corr(q 3 ,ψ 3 ) 12
(σθ ) in the strategic model than in the non-strategic alternative, i.e. ∂σθ < ∂σθ . Two
salient features of the equilibrium drive this result. First, informed trading volume is lower when managers
are less informed about the fundamental. Second, less informed ﬁrms increase the cutoﬀ price, but by
less in the strategic case than in the non-strategic alternative, because they internalize that prices are
less informative. In summary, in the strategic model managers rely less on the stock price to make the
investment decision than when informed speculators fail to internalize the learning channel. Figure 4
presents the correlation between the stock price and the investment probability for diﬀerent levels of
managerial uncertainty, for the benchmark model and for alternative 3.13 The correlation is increasing in
managerial uncertainty for both cases, but less so when traders internalize the ﬁrms’ learning from prices.
To summarize, the model has three main implications. (i) For lower quality of managers’ information
a priori, there is less trading volume by informed speculators and lower price informativeness (Corollary
1). (ii) The investment decision is less eﬃcient when traders internalize the fact that ﬁrms learn from
12
Another implication of Corollary 4 is that the correlation between the expected stock price and the probability of
investment is smaller in the benchmark model when traders are strategic than in the non-strategic model: Corr(q ∗ , ψ ∗ ) =
γ γ
Φ(γ )φ Φ(γ )
Corr(q 3 , ψ 3 ) and Φ(γ )φ Φ(γ )
≤1
13
The ﬁgure presents exact numerical solutions.
16
prices (Corollary 3). (iii) The correlation between expected stock price and investment is smaller when
informed speculators behave strategically (Corollary 4).
4. Empirical Evidence
In this section I present empirical evidence on the connection between ex-ante managerial uncertainty
about ﬁrms’ fundamentals and stock price informativeness. I also study how the correlation between
investment and stock prices varies for diﬀerent levels of managerial information. The empirical analysis
that follows is based on a sample of U.S. public ﬁrms from 1990 to 2010. For each ﬁrm I construct two
measures of managerial information and uncertainty, and one measure of stock price informativeness.
These measures are described below.
4.1. Managerial Information and Uncertainty
When making corporate decisions, managers gather information about the outlook of their ﬁrms and
the proﬁtability of new products and projects. In the model outlined above, uncertainty about a ﬁrm’s
fundamentals refers to the variance of the ﬁrm’s prior distribution of future proﬁts. To measure ﬁrm
uncertainty about fundamentals, one would need to know not only the ﬁrm’s point estimates of expected
proﬁts but the entire distribution. To my knowledge there are no surveys at the ﬁrm level with probability
distributions on future earnings. For this reason, I rely on two proxies to measure managerial uncertainty
about the outlook of their ﬁrm.
The ﬁrst measure is based upon analysts’ earnings forecasts. At the ﬁrm level, surveys of analysts’
forecasts typically report ﬁrst moments, e.g. expected earnings, proﬁts or sales. As a proxy for ﬁrm
uncertainty I instead use dispersion of analysts’ earnings forecasts from the Institutional Brokers Estimate
System (IBES). Empirical evidence suggests that a large fraction of the information used by analysts comes
from discussions with ﬁrm managers, which also suggests that analysts’ information is not news to the
ﬁrm (Bailey et al. (2003)). Analysts collect information from each ﬁrm and issue their own forecast. The
assumption is that managers with more precise information are more likely to convey such information
to analysts covering the ﬁrm, and thus, one would expect less disagreement in the analysts’ forecasts.
On the contrary, more uncertainty about fundamentals is likely to be reﬂected in more disagreement in
the forecasts issued by the analysts covering a ﬁrm.14 The caveat, of course, is that consensus among
14
For instance, analysts might be talking to diﬀerent managers within a ﬁrm, and dispersion among analysts’ forecasts
could thus reﬂect disagreement within the ﬁrm. Alternatively, disagreement among analysts could reﬂect precisely the fact
17
analysts need not imply a high degree of conﬁdence in their point estimates. However, there is a large
body of literature that has studied forecast dispersion, and on balance these studies conﬁrm that forecast
dispersion is a useful proxy for uncertainty.15
For each ﬁrm, I construct this proxy for uncertainty using all the forecasts issued by analysts within
a ﬁscal year. Following Gilchrist et al. (2005), dispersion is deﬁned as the logarithm of the ﬁscal year
average of the monthly standard deviation of analysts’ forecasts of earnings per share, times the number
of shares, scaled by the book value of total assets. That is,
12
j =1 Ntj SDtj /12
DISi,t = log (10)
ASSET Si,t
where t and j denote year and month respectively. Ntj is the number of shares outstanding, and SDtj
is the standard deviation of the per-share earnings forecasts for all analysts making forecasts for month j.
The second measure of managerial information quality is based on insider trading activity (Chen et
al. (2007) and Foucault and Fresard (2013)). Managers should be more likely to trade their own stock
and make a proﬁt on these trades if they are more conﬁdent in their information. Although managers
don’t always trade on information, the premise is that on average, managers with better information
will trade more. To build this proxy for managerial information, I obtain corporate insiders’ trades
from the Thomson Financial Insider Trading database. I measure the quality of managers’ information
with the intensity of a ﬁrm’s insider trading activity, IN SIDERit , calculated as the ratio of the ﬁrm’s
shares traded by insiders in a year to the total number of ﬁrm shares traded. As in other studies I only
include open market stock transactions initiated by the top ﬁve executives (CEO, CFO, COO, President
and Chairman of the Board).16 Since IN SIDERit is a measure of absolute insider trading activity, it
captures managerial information but not the direction of such information, that is, whether the ﬁrm
has a positive or a negative outlook. My second proxy for the ﬁrm’s uncertainty about fundamentals is
1 − IN SIDER. While insider trades may reveal managers’ ﬁrm-speciﬁc information not embodied in
share prices, a potential drawback to this proxy for uncertainty is that the lack of insider trading might
that analysts don’t put a lot of weight on what they are hearing from the ﬁrm when ﬁrms provide noisy information.
15
Earlier papers using forecast dispersion to proxy for uncertainty include Bomberger and Frazer (1981), Lambros and
Zarnowitz (1987) and Barron and Stuerke (1998). Using the Survey of Professional Forecasters (SPF), Lambros and Zarnowitz
(1987) show a positive correlation between forecast dispersion and uncertainty, where uncertainty is proxied by the spread
of the probability distribution of point forecasts. IBES distributes only point forecasts, but the SPF provides both point
forecasts and the histogram of forecasts for GDP, unemployment, inﬂation, and other major macroeconomic variables. In
recent papers, Avramov et al. (2009) and Guntay and Hackbarth (2010) study forecast dispersion as a measure of uncertainty
about ﬁrms’ future earnings. Guntay and Hackbarth (2010) ﬁnd that dispersion is positively associated with credit spreads,
and it appears to proxy largely for future cash ﬂow uncertainty.
16
Foucault and Fresard (2013) and Peress (2010).
18
simply indicate that market prices are close to insider’s beliefs about the fundamentals of a ﬁrm, rather
than indicating low precision of managerial information. Nonetheless, we should expect better informed
managers who are more conﬁdent about the quality of their information to trade more.
4.2. Price Informativeness
To measure the amount of ﬁrm speciﬁc information contained in stock prices I use price nonsynchronicity.
Speciﬁcally, I measure price informativeness for a ﬁrm as the share of its daily stock return variation that
2 , where R2 is the R2 from the regression in year t of ﬁrm i’s daily
is ﬁrm-speciﬁc, deﬁned as P Iit = 1 − Rit it
returns on market and industry returns.17 The idea, ﬁrst suggested by Roll (1988), is that trading on
ﬁrm-speciﬁc information makes stock returns less correlated in the cross-section and thereby increases the
fraction of total volatility due to idiosyncratic shocks.18 This measure is related to price informativeness
in the model, in that increased informed trading volume should increase the idiosyncratic volatility of a
ﬁrm’s stock price.
Firms are matched to their speciﬁc three digit SIC industry. I exclude ﬁrm-year observations with
less than $10 million book value of equity or with less than 30 days of trading activity in a year. I
used CRSP data to measure stock returns and Compustat to measure book values. I exclude ﬁrms in
ﬁnancial industries (SIC code 6000-6999) and utility industries (SIC code 4000-4999). The ﬁnal sample
consists of an unbalanced panel with 5,607 ﬁrms and 33,610 ﬁrm-year observations of uncertainty and
price informativeness between 1990 and 2010. I detail the construction of all the variables in Table 1 and
Table 2 presents summary statistics. To reduce the eﬀect of outliers all variables are winsorized at 1% in
each tail. Finally, I scale all variables by their standard deviation so that the estimated coeﬃcients are
directly informative about the economic signiﬁcance of the eﬀects.
4.3. Empirical Methodology
To estimate the relationship between stock price informativeness and fundamental uncertainty, I consider
the following baseline speciﬁcation:
P Iit = αi + δt + βU N CERit + γXit + it , (11)
17
The market index and industry indices are value-weighted averages excluding the ﬁrm in question. This exclusion
prevents spurious correlations between ﬁrm and industry returns in industries that contain few ﬁrms.
18
This measure has been used extensively in the literature studying feedback between prices and managerial decisions. See
for example Durnev et al. (2004), Chen et al. (2007) and Foucault and Fresard (2013).
19
where the subscripts i and t represent respectively ﬁrm i and year t. The dependent variable is price
nonsynchronicity, my proxy for price informativeness. The explanatory variable U N CERit measures
managers’ uncertainty about fundamentals as captured by one of the proxies discussed in subsection
4.1. The vector X includes control variables such as ﬁrm size, measured as the natural logarithm of the
book value of assets, level of cash ﬂows and the number of analysts issuing forecasts for each ﬁrm. In
addition, I account for time-invariant ﬁrm heterogeneity by including ﬁrm ﬁxed eﬀects (αi ) and time-
speciﬁc eﬀects by including year ﬁxed eﬀects (δt ). The coeﬃcient β measures how managerial uncertainty
about fundamentals is related to stock price informativeness over time and across ﬁrms.
Table 3 reveals that the coeﬃcients on the dispersion between analysts’ earnings forecasts and insider
trading are signiﬁcantly negative in all speciﬁcations. While this does not establish causality, it suggests
a robust negative correlation between uncertainty and price nonsynchronicity even after controlling for
year and ﬁrm ﬁxed eﬀects. Of course this interpretation depends on the assumption that analysts’
earnings forecast dispersion and insider trading capture the quality of managers’ information. That is,
to the extent that forecast dispersion and insider trading capture managers’ uncertainty about the ﬁrms’
fundamentals, prices seem to be less informative about ﬁrm-speciﬁc information when the precision of
managers’ information is low.
In the model, the negative relationship between uncertainty and price informativeness is derived
from two main assumptions. The ﬁrst is that there are feedback eﬀects from prices to ﬁrms’ decisions.
The second is that both traders and ﬁrms are strategic. This provides a potential test exploiting a priori
cross-sectional diﬀerences in the relationship between uncertainty and price informativeness. In particular,
institutional investors are typically better informed than individual investors,19 and their trades are more
likely to have price eﬀects. This suggests that stocks with larger institutional ownership should exhibit
a stronger negative correlation between uncertainty and price informativeness. I test this hypothesis by
adding to the baseling regression the interaction between the measure of uncertainty and the percentage
of shares held by institutional investors in any given stock (U N CERit × IN STit )
P Iit = αi + δt + β0 U N CERit × IN STit + β1 U N CERit + β2 IN STit + γXit + it . (12)
Table 4 shows that the magnitude of the correlation between uncertainty and price informativeness
is indeed greater for ﬁrms with a larger share of institutional ownership. The results are similar for both
19
Economies of scale imply that institutional investors can acquire information at a lower cost per share traded than
individual investors.
20
proxies of ﬁrm fundamental uncertainty. Overall, the results suggest uncertainty and price informativeness
are most strongly related for ﬁrms for which strategic behavior is most likely to be present. As shown in
the model, such strategic behavior has important implications for how well markets reveal information,
and for how ﬁrms use stock prices when making investment decisions.
4.4. Investment sensitivity to price
The model above suggested that there are limits to ﬁrms’ ability to use stock prices as a guide in making
real investment decisions. This is because stock price informativeness is not exogenous with respect to the
precision of the managers’ prior information. When a manager is less informed a priori, strategic informed
traders optimally reduce their trading volume, making stock prices less informative about the fundamental.
In turn, managers themselves end up relying less on the price signal relative to the alternative case when
informed traders are non-strategic.
To test whether the quality of managerial information aﬀects the sensitivity of investment to the stock
price, I estimate a variant of a standard empirical investment equation as follows:
Iit = αi + δt + β1 Qit−1 + β2 Qit−1 · U N CERit−1 + γXit + it (13)
The dependent variable, Iit , is the ratio of capital expenditures in that year to lagged ﬁxed assets.
Following other studies on the sensitivity of investment to stock price, I use Tobin’s average Q as a
proxy for a ﬁrm’s market value. Average Q is deﬁned as a ﬁrm’s stock price times the number of shares
outstanding plus the book value of assets minus the book value of equity, divided by the book value of
assets. The vector X includes control variables known to correlate with investment decisions such as cash
ﬂows and ﬁrm size. To test for the eﬀect of the quality of managerial information on the relationship
between investment and stock price, I interact lagged managerial uncertainty (U N CERit−1 ) with Tobin’s
Q. I also control for the direct eﬀect of the quality of managerial information on investment. Following
the speciﬁcation in the previous section, I include ﬁrm ﬁxed eﬀects and year ﬁxed eﬀects.
Results are presented in Table 5. While the coeﬃcient β2 is positive for both proxies of managerial
uncertainty, these coeﬃcients are not statistically signiﬁcant, suggesting that there are no diﬀerences in
the sensitivity of investment to stock prices for diﬀerent levels of managerial information. Recall that the
model did predict a higher correlation between stock prices and investment for less informed managers
(as stock prices do contain some private information not possessed by managers), but this interaction is
21
limited in the benchmark model compared to the case when traders fail to internalize that ﬁrms learn
from prices.
In columns (1) and (3) I control for stock price informativeness and its interaction with Q. As in
previous studies (Chen et al. (2007), Bakke and Whited (2010) and Foucault and Fresard (2013)) I ﬁnd
that a ﬁrm’s investment is more sensitive to Tobin’s Q when its stock price is more informative. These
results suggest that ﬁrm managers learn from the private information aggregated into the stock price when
making investment decisions, but market prices does not seem to provide more guidance to managers with
low quality of information.
5. Conclusions
In this paper I ﬁnd limits to the ability of secondary markets to inform and guide ﬁrms’ real investment
decisions. The economy is modeled as a strategic game between ﬁrms and informed speculators. Before
making an investment choice, ﬁrms use stock prices to update their priors about their own fundamentals.
Informed traders are strategic in that they internalize the ﬁrms’ inference problem. In this setting, I show
that informed trading volume depends on the quality of managers’ prior information. In other words, the
amount of private information that is aggregated into the stock price through the trading process is a
function of managers’ initial information. Learning from prices is limited because prices are endogenously
less informative for ﬁrms with low quality of managerial information, which are precisely those ﬁrms that
would like to learn more from the stock market in the ﬁrst place. In turn, real investment eﬃciency falls
as the market signal is not as useful in helping managers diﬀerentiate between good and bad projects.
Using a sample of U.S. publicly traded companies, I document a positive correlation between the
quality of managers’ information and stock price informativeness. I show that less informed managers do
not rely more on the stock price to make investment decisions. Collectively, the evidence is suggestive of
limits to the ability of ﬁrms to learn relevant information from stock prices.
The model presented here may have implications for ﬁrms’ decision about whether to be ﬁnanced
through public or private equity. More speciﬁcally, depending on the information gap between traders and
the ﬁrm’s managers, the ﬁrm might beneﬁt from an IPO or might be better oﬀ being held privately. In the
same way, the quality of managers’ information should determine the choice of outside speculators to either
become private or public equity investors. When managers have low quality information, speculators’
trading proﬁts are limited, despite knowing the true value of the fundamental. In this case, the speculator
22
might be better oﬀ being a private equity investor in the ﬁrm, which would allow it to participate directly
in the investment decision. On the contrary, when managers’ prior information is good, speculators’
trading proﬁts are potentially large, and they can fully beneﬁt from their information by trading on
secondary markets. A further analysis of the links between managerial information, outsider information
and the optimal form of equity ﬁnance is left for future work.
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25
A. Appendix: Tables
Table 1
Variable deﬁnitions
Variable Deﬁnition Source
1 − R2 Price informativeness, deﬁned as one minus R2 from CRSP
regressing a ﬁrm’s daily returns on market and industry
indices over year t
SIZE Logarithm of the book value of assets Compustat
IN SIDER Ratio of ﬁrm’s shares traded by insiders in a given year Thompson Financial
to the total number of shares traded (in percentage terms). Insider Trading
Insider traders refers to open market stock transactions by Database and CRSP
the top ﬁve executives (CEO, CFO, COO, President and
Chairman of the board)
IN ST Percentage of shares held by institutional investors Thompson Financial
AN ALY ST Number of analysts issuing forecasts for each stock IBES
DIS Analyst forecast dispersion, deﬁned as the natural logarithm IBES and Compustat
of the standard deviation of analysts’ forecasts of earnings
per share, times the number of shares, scaled by the book
value of total assets as in Gilchrist et al. (2005)
CF Cash ﬂow, deﬁned as net income before extraordinary items + Compustat
depreciation and amortization expenses + R&D expenses
scaled by assets
Q Average Q, deﬁned as [Book value of assets - book value Compustat
of equity + market value of equity]/book value of assets
I Investment rate, deﬁned as the ratio of capital expenditures Compustat
to lagged ﬁxed assets
Table 2
Summary statistics
This table reports the summary statistics of the main variables used in the analysis. For each variable I present its mean, standard
deviation, 5th , 25th , 50th , 75th and 95th percentile. All variables are deﬁned in Table 1. The sample period is from 1990 to 2010.
Number
Variable of observations Mean St.Dev. 5% 25% 50% 75% 95%
1 − R2 34819 0.82 0.18 0.43 0.72 0.89 0.97 1.00
SIZE 34819 7.67 2.61 3.70 5.80 7.30 9.37 12.04
IN SIDER 34819 6.89 12.69 0.01 0.08 0.43 3.16 34.04
IN ST 34819 0.52 0.26 0.09 0.31 0.53 0.73 0.93
DIS 34819 -3.05 1.12 -4.89 -3.78 -3.07 -2.34 -1.15
CF 34819 1627 2502 -5.37 17.02 141 5706 7805
Q 35051 2.81 2.16 0.93 1.14 1.63 6.31 8.31
I 24155 0.29 0.42 0.49 0.11 0.19 0.34 0.79
26
Table 3
Price informativeness and managerial information
Deﬁnitiond of all variables are listed in Table 1. The dependent variable is Price Informativeness. Managers’ uncertainty is proxied
by the dispersion of analysts’ earnings forecasts (DIS ) and insider trading activity 1 − IN SIDER. T-statistics are in parentheses.
***/**/* indicate that the coeﬃcient estimates are signiﬁcantly diﬀerent from zero at the 1%/5%/10% level.
Measure of Uncertainty
DIS 1 − IN SIDER
Dependent variable: P I (1) (2) (3) (4) (5) (6)
U N CERit -0.10*** -0.18*** -0.06* -0.18*** -0.08*** -0.03***
(17.90) (26.30) (1.70) (31.11) (13.24) (6.39)
SIZEit -0.41*** -0.45*** -0.05*** -0.42*** -0.54*** -0.06***
(55.13) (27.87) (4.11) (58.29) (33.52) (4.42)
CFit 0.25*** 0.34*** 0.15*** 0.22*** 0.39*** 0.15***
(32.97) (18.91) (10.95) (29.76) (21.35) (10.95)
Firm ﬁxed eﬀects No Yes Yes No Yes Yes
Year ﬁxed eﬀects No No Yes No No Yes
R2 0.14 0.52 0.73 0.16 0.51 0.73
Number of Observations 24753 24753 24753 24921 24921 24921
Table 4
Price informativeness and managerial information: Interaction with institutional
ownership
Deﬁnitions of all variables are listed in Table 1. The dependent variable is Price Informativeness. Managers’ uncertainty is proxied by
the dispersion of analysts’ earnings forecasts (DIS ) and insider trading activity 1 − IN SIDER. SIZE and CF coeﬃcients are omitted.
T-statistics are in parentheses. ***/**/* indicate that the coeﬃcient estimates are signiﬁcantly diﬀerent from zero at the 1%/5%/10%
level.
Measure of Uncertainty
DIS 1 − IN SIDER
Dependent variable P I (1) (2)
U N CERit -0.02*** -0.02***
(3.40) (5.09)
U N CERit × IN STit -0.05*** -0.02***
(11.95) (4.36)
IN STit -0.22*** -0.22***
(28.30) (27.38)
Controls Yes Yes
Firm ﬁxed eﬀects Yes Yes
Year ﬁxed eﬀects Yes Yes
R2 0.73 0.74
Number of Observations 24753 24921
27
Table 5
Managerial information and investment sensitivity to price
Deﬁnitions of all variables are listed in Table 1. The dependent variable is investment. Managers’ uncertainty is proxied by the dispersion
of analysts’ earnings forecasts (DIS ) and insider trading activity 1 − IN SIDER. T-statistics are in parentheses. ***/**/* indicate
that the coeﬃcient estimates are signiﬁcantly diﬀerent from zero at the 1%/5%/10% level.
DIS 1 − IN SIDER
Dependent variable: I (1) (2) (3) (4)
Q 0.215*** 0.225*** 0.212*** 0.224***
(8.11) (8.45) (10.04) (10.51)
Q × PI 0.038*** 0.041***
(2.88) (3.24)
Q × U N CER 0.012 0.009 0.011 0.006
(0.74) (0.58) (0.82) (0.45)
PI -0.002 -0.001
(0.24) (0.05)
U N CER -0.023* -0.025** 0.009 0.007
(1.84) (2.03) (1.11) (0.84)
Controls Yes Yes Yes Yes
Firm ﬁxed eﬀects Yes Yes Yes Yes
Year ﬁxed eﬀects Yes Yes Yes Yes
R2 0.39 0.38 0.38 0.37
Number of Observations 19009 19009 19009 19009
28
B. Appendix: Figures
Figure 1. Timeline for each ﬁrm. Informed speculator (IS), Noise traders (NS).
(i) IS and NT submit
demands
IS learns the firm’s (ii) Firm observes
stock price and Firm’s output is
fundamental
decides whether realized
to invest or not
t=0 t=1 t=2
Figure 2. Equilibrium demands by informed speculators The ﬁgure shows informed speculators’ demands in the
model with strategic behavior and learning from prices for diﬀerent levels of managerial uncertainty (i.e. diﬀerent σθ ). The
ﬁgure presents numerical solutions with model parameters as follows: c = 1, µθ = 1.05, σz = 1 and = 1.
Informed Speculators’ Optimal Demands: X*
I
(θ)
1.5
1
0.5
0
−0.5
σθ=0
σθ=0.5
−1
σθ=1
σθ=4
−1.5
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
θ−c
29
Figure 3. Probability of investment. Probability of the ﬁrm undertaking the project under perfect information (al-
ternative #1), no learning from prices (alternative #2), no strategic behavior (alternative #3) and strategic behavior. The
parameters are c = 1, µθ = 1.05, σθ = 1, σz = 1, = 1, c = 1 and = 1.
Probability of Undertaking the Project: ψ(θ)
1
0.8
0.6
0.4
0.2
Perfect Information
No Learning
0 Non−strategic
Strategic
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
θ−c
Figure 4. Correlation between the stock price and the probability of investment. Correlation between price and
investment for diﬀerent levels of managerial uncertainty (σθ ) for the strategic and non-strategic (alternative #3) models.
The parameters are c = 1, µθ = 1.05, σz = 1 and = 1.
Corr(q,ψ)
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
Non−strategic
Strategic
0
0 0.5 1 1.5 2 2.5 3 3.5 4
σ2
θ
30
C. Appendix: Proofs
Proof of Lemma 1: If the ﬁrm manager conjectures a linear demand by the informed speculator of the
form XI (θ) = a + bθ, the posterior on the fundamental is:
ϕ( q − a − bθ)ξ (θ)
ξ (θ | q ) = ∞ . (14)
−∞ ϕ( q − a − bθ)ξ (θ)dθ
2 and ξ () is the
where ϕ() is density function of the normal distribution with mean 0 and variance σz
2 . Under the posterior, the cutoﬀ
density function of a normal distribution with mean µθ and variance σθ
rule is given by
∞
−∞ θϕ( q l − a − bθ )ξ (θ )
∞ =c (15)
−∞ ϕ( q l − a − bθ )ξ (θ )dθ
2
σz
1
Using the properties of the normal distribution I solve for the cutoﬀ price, q l = a + cb − 1
b (µθ − c) σ 2 .
θ
Proof of Proposition 1: The method of this proof is to iterate among best responses to ﬁnd the
ﬁxed point in strategies.
0 (θ ) = (θ−c)
Step 1a. Assume a linear function for the informed speculators’ demand XI 2 . From Lemma
1, the ﬁrm’s best response (cutoﬀ price) is
2
2σz
q 1 = −(µθ − c) 2σ2
θ
Step 1b. Using q 1 , I ﬁnd the optimal decision rule of the informed speculator. This is the solution
to the ﬁrst order condition from proﬁt maximization (5):
XI (θ) − q 1 1 XI (θ) − q 1 2
(θ − c) Φ + XI (θ)Φ − XI (θ) = 0
σz σz σz
For the reasons described in the main text, in this paper I am interested in the solution around θ = c.
Linearizing the FOC around θ = c, I can guess and verify that the demand function of the informed
speculators is of the form XI (θ) = k (θ − c). The ﬁrst order approximation of the FOC is:
q1 2k (θ − c)
(θ − c)Φ − − =0
σz
Φ(γ ) 2(µθ −c)σz
Solving for k, k = 2 , where γ = 2
σθ
. Informed speculators’ demands are:
31
1 (θ − c)
XI (θ) = Φ(γ )
2
Step 2a. Using the linear demands above X 1 (θ), I solve for the ﬁrm’s cutoﬀ rule.
2σz2 1
q 2 = −(µθ − c) 2 σ 2 Φ(γ )
θ
Step 2b. The iteration continues by assuming q 2 to ﬁnd the optimal decision rule of the informed
speculator. Following the linearization in step 1b, the informed speculators’ demands are:
1 (θ − c) 1
XI (θ) = Φ γ
2 Φ(γ )
Continuing the iteration procedure, in the n − th iteration the ﬁrm’s cutoﬀ rule and the speculators’
demands are:
2σz 2 1
q n = −(µθ − c)2 σ 2 f n−1 (γ )
θ
n (θ − c) n
XI (θ) = f (γ )
2
where f n (γ ) is a continued fraction of cumulative normal distributions of the form:
γ
f n (γ ) = Φ (16)
Φ γ
Φ γ
..
.
Lemma 2. If γ ≥ 0, the inﬁnite continued fraction f (∞) (γ ) converges to Φ(γ ). That is, limn→∞ f n (γ ) =
Φ(γ ). If γ < 0, limn→∞ f n (γ ) = 0.
Using Lemma 2, the ﬁxed point in strategies has the following form:
• If the expected NPV of the investment project is non-negative under the ﬁrm’s prior, µθ ≥ c then
32
the equilibrium demands by speculators and ﬁrms’ cutoﬀ rule are:
∗ (θ − c)
XI (θ) = Φ(γ )
2
2σ 2 1
q ∗ = −(µθ − c) 2 z2
σθ Φ(γ )
• If the expected NPV of the investment project is negative under the ﬁrm’s prior, µθ < c then,
∗ (θ ) = 0 and ﬁrms never invest, q ∗ → ∞.
informed speculators don’t trade in equilibrium XI
dq ∗
Proof that dγ < 0: From equation (8) it is suﬃcient to prove that g (γ ) ≡ −Φ(γ ) + γ Φ (γ ) < 0 for
γ ≥ 0. First, note that g () is monotonically decreasing in γ , g (γ ) = Φ (γ ) < 0. Also g (0) = − 1
2 . Hence
g (γ ) < 0.
Proof of Corollary 2: Replace the equilibrium demands by the informed speculators and ﬁrms’
cutoﬀ rule in the probability of investment: P r(q ≥ q | θ) = Φ 1 ∗ (θ ) − q ∗ ) .
(XI
σz
Proof of Corollary 3: Investment eﬃciency when informed speculators internalize the ﬁrm’s updat-
ing process is:
∂ψ ∗ (θ) γ
= Φ(γ )Φ (17)
∂θ θ=c 2σz Φ(γ )
Investment eﬃciency when informed speculators don’ internalize the ﬁrm’s updating process is:
ˆ(θ)
∂ψ
= Φ (γ ) (18)
∂θ θ=c 2σz
The g (γ ) can be deﬁned as the ratio between the investment eﬃciency measure in the non-strategic
case and the strategic case:
∂ψ b3 (θ)
∂θ
θ=c
g (γ ) ≡ (19)
∂ψ ∗ (θ)
∂θ
θ=c
This ratio is strictly decreasing in γ if γ > 0, that is
γ Φ (γ )Φ(γ )3 + γ 2 Φ (γ )2
g (γ ) = − < 0, if γ > 0
γ
Φ(γ )4 Φ Φ(γ )
Also, limγ →∞ g (γ ) = 1. Combining these two results, for any ﬁnite level of precision of managerial
information (i.e. γ ﬁnite) and µθ > c, the investment decision is less eﬃcient with strategic traders than
in the non-strategic benchmark.
33
Proof of Corollary 4: The ﬁrst order approximation of the probability of investment for a ﬁrm with
fundamental θ around θ = c can be written as: ψ 3 (θ) = Φ (γ ) + φ(γ )(θ − c) for the alternative case #3
γ γ
and ψ ∗ (θ) = Φ Φ(γ ) +φ Φ(γ ) (θ − c) for the benchmark model.
In the non-strategic model (Alternative #3), the correlation between the probability of investment
and stock prices, following Deﬁnition 5, is:
∞
φ(γ )
Corr(q 3 , ψ 3 ) = (θ − c)2 dξ (θ)
−∞ 2
φ(γ )
Integrating above I obtain Corr(q 3 , ψ 3 ) = 2
2 + (µ − c)2 . Similarly, for the strategic model
σθ θ
I use the linear approximation of the investment probability and integrate to calculate the correlation
between prices and investment.
34