1. real estate
2. buying and selling homes

The Paradox of Financial Fire Sales
James Dow*and Jungsuk Han†‡
March, 2015
Abstract
Why do fire sales happen for financial assets, even when there are well-capitalized
investors somewhere in the economy? We propose a theory of financial fire sales
based on a noisy rational expectations equilibrium framework. When informed
market participants are liquidity-constrained due to market-wide shocks, prices
become less informative. This creates an adverse selection problem, increasing
the supply of low-quality assets. This “Lemons” problem makes well-capitalized
uninformed market participants unwilling to absorb the supply, thereby freezing
the market. Our results shed light on the paradoxical nature of fire sales in which
capital moves out of the market when it is needed most and it is likely to have
higher returns.
Keywords: fire sales, adverse selection, market freeze, illiquidity, flight-to-quality,
informed trading
JEL Classification: G14, G21, D82, D83, D84
*
Department of Finance, London Business School, Regent’s Park, London, United Kingdom, NW1
4SA, E-mail: jdow@london.edu.
†
Department of Finance, Stockholm School of Economics, Drottninggatan 98, Stockholm, Sweden,
SE 111-60, E-mail: jungsuk.han@hhs.se.
‡
We thank Julian Franks, Pete Kyle, Paolo Sodini, Per Str¨omberg, and J¨orgen Weibull for helpful
comments and suggestions. We also thank seminar participants at BI Norwegian Business School,
London Business School, Stockholm School of Economics, and University of Gothenburg.
1
1
Introduction
In a fire sale, sellers are forced to sell assets at deep discounts because no one is willing to
buy them at fair prices. Sellers could be forced to sell because of financial distress, credit
market frictions, regulation, margin calls, etc. Why do fire sales happen? What makes
investors avoid buying assets that are apparently cheap? In the previous literature,
Shleifer and Vishny (1992) argue that fire sales can happen when industry experts with
higher private valuations do not have enough liquidity. Therefore, those assets are bought
at a discount by non-experts who cannot use these assets efficiently. This argument
naturally applies to real assets rather than financial securities because private valuations
of real assets, unlike those of financial securities, can differ significantly among investors.
Since financial securities typically require the holder only to collect cash flows, and not to
take any actions, it cannot explain significant differences in private valuations. Shleifer
and Vishny (1997), Gromb and Vayanos (2002), and Brunnermeier and Pedersen (2009)
argue that fire sales may occur even for financial securities due to limits to arbitrage. If
all investors are constrained in their buying capacities for various reasons, then they are
not able to buy apparently undervalued assets. However, this cannot explain many firesale episodes (such as the LTCM crisis and the collapse of the mortgage-backed securities
(MBS) markets during the 2007-8 financial crisis) where fire sales occurred even though
there were some well-capitalized investors somewhere in the world economy (e.g., Warren
Buffett, or sovereign wealth funds). Why wouldn’t well-capitalized investors want to step
in and buy undervalued assets whenever a fire sale starts, thereby preventing significant
price drops?
It could be that the only available well-capitalized investors are not specialists in
valuing the assets that are being sold. Their only expertise may lie in valuing other
types of assets. Hence, by buying, they could be exposing themselves to adverse selection
perpetrated by industry insiders. This could happen if the forced sellers choose to
sell their lowest-quality assets which, however, cannot be distinguished by the outside
investors from high quality assets, causing a classic “lemons” problem (e.g., Akerlof
(1970)). But if so, fire-sale discounts should should be severe when sellers are hit by
moderate shocks and have plenty of discretion on what to sell, while discounts should be
minor when sellers are hit by large shocks and need to sell all their holdings (Malherbe
(2014)). How can lemons problems explain fire sales during financial crises?
We give an intuitive answer to this question by examining the role of informed trading
2
in preventing adverse selection. We argue that informed traders who can buy assets
help to make market prices informative. If one trader is forced to sell, other traders
in the same market would be willing to buy the asset if underpriced. They prevent
high quality assets from being underpriced and trading at the same prices as low-quality
assets. Hence, informed buyers remove the incentive for sellers to favor selling low-quality
assets. But in case of a large shock to the market, this mechanism breaks down. A large
shock affecting traders who specialize in an asset prevents them from using their private
information to bid up undervalued assets: there are no buyers to keep prices informative.
Thus, liquidity shocks to informed market participants can make prices uninformative,
leading to adverse selection in which sellers supply only overvalued assets to the market.
This in turn leads uninformed agents, who are potential buyers for those assets, to
withdraw from the market even though they are not wealth-constrained. This leads to a
market freeze for high-quality assets because no one is willing to buy or sell them, while
agents holding these assets but subject to a liquidity shortage are forced to sell at fire
sale prices.
To formalize this argument, we develop an information-based theory of fire sales
using a noisy rational expectations equilibrium (REE) framework. We aim to achieve
the following goals in our paper. First, we explain the role of informed trading in fire
sales and market freeze. Unlike the traditional literature on adverse selection that only
features informed sellers, we highlight the role of informed buyers who compete to exploit
mispricing, and thereby make the price informative. Second, we answer the question of
why fire sales occur even when, somewhere in the economy, there is enough capital
to correct prices. In particular, we show that common liquidity shocks to informed
participants can make uninformed participants stay away from trading, thereby allowing
asset prices to fall below fundamental values. We also shed light on the paradoxical
nature of fire sales in which capital moves out of the market when it is needed most
and would seemingly earn higher returns. Third, we explain why a market freeze (or
severe illiquidity) happens at the same time as fire sales. That is, our paper explains
the “double whammy” situation where fire sales and low trading volume occur together
(see, for example, Tirole (2011) for a discussion of the double whammy during the recent
financial crisis).
Consider a two-period model with informed, but financially constrained, intermediaries and unconstrained, but uninformed, investors.1 There exists a marketable asset
1
We call these agents “intermediaries” because in practice they are likely to be financial intermedi-
3
with risky payoffs whose value is only known to the informed intermediaries. There are
two types of informed intermediaries. We assume some of them are sellers, much as
in traditional lemons models, while others are arbitrageurs. The sellers may receive a
liquidity shock forcing them to sell all their holdings, or, even if not forced to sell, they
may choose to sell in preference to liquidating other non-marketable assets they hold.
It is this choice that leads to the lemons problem. The liquidity shock introduces an
element of noise into the supply of the asset. The arbitrageurs are informed traders who
decide their trades depending on the price, but they are subject to liquidity shocks that
may limit the size of their trades.
The arbitrageurs compete to exploit arbitrage opportunities, thereby driving price
close to the fundamental value. Therefore, in a normal situation where arbitrageurs have
enough liquidity, the price reveal the fundamental value of the asset because informed
trading volume overwhelms the impact of noise in the supply. This allows intermediaries
in liquidity shortage (the sellers) to fund themselves by selling assets on their balance
sheet at fair prices regardless of the quality of those assets. This in turn makes uninformed investors willing to absorb the supply of assets without worrying about adverse
selection. On the other hand, in a crisis situation where intermediaries are liquidityconstrained due to a market-wide liquidity shock, prices become less informative. This
makes sellers only willing to supply low-quality assets to the market unless they are
forced to sell. This adverse selection problem in turn makes uninformed investors unwilling to absorb the supply of assets unless there is a drop in price to reflect a lemons
discount. Consequently, a market-wide liquidity shock can create fire sales by financial
institutions already in liquidity needs because they are forced to sell high-quality assets
at deep discounts. It also creates a market freeze because the supply of high-quality
assets decreases. We also show that price falls further because risk averse investors require a further risk premium to compensate for the risk of buying an asset whose price
is uninformative.
We further show that welfare is maximized when prices are equal to the fundamental
value for any realizations of the random variables because mispricing leads to misallocation of resources. Mispricing causes informed participants to inefficiently liquidate
otherwise valuable assets. In the case of overpricing of traded assets, intermediaries may
liquidate otherwise good businesses to speculate on the marketable asset. In the case of
underpricing of traded assets, intermediaries raise funds to fill in liquidity shortages by
aries such as banks, hedge funds.
4
inefficiently liquidating non-marketable assets rather than selling traded assets.
Our results also contribute to the debate on flight-to-quality by suggesting an alternative mechanism for flight-to-quality. When prices are uninformative, uninformed
investors lower their portfolio weights on the marketable asset because its payoffs are
perceived to be riskier. At the same time, they increase their portfolio weights on the
risk-free asset. Notice that this flight-to-quality mechanism does not involve any change
in preferences such as risk-aversion. It is rather a consequence of the change in the
amount of information about the fundamental value that is endogenously determined
by constrained informed trading. Furthermore, illiquidity does happen together with
flight-to-quality, but they are both consequences of liquidity shocks, rather than illiquidity being the cause for flight-to-quality.
The organization of the paper is as follows. Section 2 relates our paper to the
literature. Section 3 describes the basic model. Section 4 solves for the equilibrium.
Section 5 concludes.
2
Literature review
There exists a large volume of literature on both the theory and the empirics of fire
sales.2 On the empirical side, there is extensive evidence of fire sales across various
classes of assets and securities: (i) real assets (e.g., Pulvino (1998), Schlingemann, Stulz,
and Walkling (2002)), (ii) equities (e.g., Coval and Stafford (2007), Jotikasthira, Lundblad, and Ramadorai (2012)), (iii) bonds (e.g., Ellul, Jotikasthira, and Lundblad (2011),
Jotikasthira, Lundblad, and Ramadorai (2012)), (iv) structured products (e.g., Merrill, Nadauld, Stulz, and Sherlund (2014)), and (v) repos (e.g., Duarte and Eisenbach
(2014)).
As we discussed in the introduction, the previous theoretical literature suggests that
fire sales occur because of common liquidity shocks to industry experts (e.g., Shleifer
and Vishny (1992)), and limits to arbitrage (e.g., Shleifer and Vishny (1997), Gromb
and Vayanos (2002), and Brunnermeier and Pedersen (2009)). Several papers also focus
on the amplification mechanism of fire sales as in Gromb and Vayanos (2002), Geanakoplos (2003), Brunnermeier and Pedersen (2009), Krishnamurthy (2010) and Greenwood,
Landier, and Thesmar (2012). For example, Krishnamurthy (2010) shows that fire
2
Shleifer and Vishny (2011) and Tirole (2011) provide excellent surveys on the literature.
5
sales can occur due to the feedback effects between asset prices and balance sheets.
Counterparty risk can also contribute to the amplification mechanism of fire sales (e.g.,
Krishnamurthy (2010) and Caballero and Simsek (2013)). For example, Caballero and
Simsek (2013) study a model where fire sales occur because of counterparty risk in a
complicated network.
In his seminal paper, Akerlof (1970) shows that a market collapse (or freeze) can
happen due to adverse selection. If the quality of assets is only known to sellers, buyers’
valuation that depends on the average quality cannot satisfy sellers who own assets with
the highest quality. This makes them withdraw from the market, and subsequently lowers buyers’ expectation about the quality, and this in turn makes more sellers withdraw
from the market if they own assets with the highest quality among the remaining sellers.
This process can continue until there is no seller left, thus the market collapses. This
intuition has been extended and applied in the finance literature. In particular, there
have been several papers that emphasize the role of fire sales where sellers are forced to
sell due to distress. In that situation, buyers may face reduced information asymmetry
even when sellers are better-informed. However, they cannot tell whether the supply is
coming from liquidity-driven sales or information-driven sales if other sellers try to sell
when they get bad signals. This type of adverse selection can be source of market freeze.
For example, Bolton, Santos, and Scheinkman (2009) show that distressed sellers may
choose to sell earlier at fire sale prices to avoid potential adverse selection problem in the
future. Dang, Gorton, and Holmstrom (2012) suggest that a market freeze can occur
in debt market by extending argument by Myers and Majluf (1984). Although debt
securities are information-insensitive relative to equity, they can become information
sensitive when approaching a default state, thereby triggering information acquisition.
Such information asymmetries lead to adverse selection and market freeze. Malherbe
(2014) focuses on the self-fulfilling nature of liquidity dry-up. If sellers are forced to
sell their assets due to liquidity needs, adverse selection problem would not arise as in
Akerlof (1970). Therefore, sellers will not need to hoard liquidity if asset sales are driven
by liquidity. On the other hand, sellers will need to hoard liquidity if asset sales are
driven by information. Therefore, this creates multiple equilibria. Other authors have
demonstrated that a market freeze can happen when adverse selection is combined with
other frictions. For example, Guerrieri and Shimer (2014) show that market becomes
illiquid for high-quality assets due to adverse selection in the presence of search frictions.
Our theory is in stark contrast with the existing theories of fire sales or market freeze.
6
Akerlof (1970) shows how market freeze can happen in the presence of information
asymmetries. But, there is no informed trader who can correct market prices in that
argument. On the other hand, Grossman and Stiglitz (1980) show that the existence
of informed traders can improve the informativeness of prices, but the supply of assets
is inelastic to market prices. In our paper, Akerlof (1970) meets Grossman and Stiglitz
(1980), and this gives a mechanism that creates fire sales. This idea is illustrated in
Figure 1.
Noisy rational expectations equilibrium
(e.g., Grossman and Stiglitz, 1980)
Informed
arbitrageurs
Noise traders
Trading
Informed
sellers
Uninformed
investors
Supply
Market for Lemons
(e.g., Akerlof, 1970)
Figure 1: An illustration of our model: Akerlof (1970) meets Grossman and Stiglitz
(1980)
Our paper combines limits to arbitrage and adverse selection to provide a more
plausible mechanism of fire sales. We argue that informed experts set asset prices to
be fair, thus uninformed deep-pocketed buyers can supply liquidity to potential sellers
without trouble. During distress episodes, experts are constrained, thereby making
price uninformative. Consequently, uninformed buyers cannot supply liquidity, and this
in turn leads to fire sales because no one is willing (or able) to buy assets in fire sales at
fair prices. Our theory is different from Shleifer and Vishny (1992) because it does not
require private valuations. Our theory differs from Shleifer and Vishny (1997), Gromb
7
and Vayanos (2002), and Brunnermeier and Pedersen (2009) because it does not require
everyone to be constrained, and also our model generates adverse selection. Our theory
differs from Malherbe (2014) because a market freeze (or fire sale) occurs when sellers (or
experts) are highly distressed, and also because a market freeze arises due to the decrease
in informed trading (or limits to arbitrage). Malherbe (2014) argues that fire sales occur
because of increased adverse selection. However, for this argument to hold, sellers should
be only moderately distressed (i.e., more funding liquidity for sellers) during fire sales so
that they can choose to sell lemons instead of good assets. Yet fire sales typically occur
when sellers are highly distressed.
Fire sales and market freezes create negative externalities to the economy because
they distort resource allocations, and destabilize the market. In the literature, it has
been also argued that those externalities will not disappear by themselves in the absence
of publicly coordinated efforts. For example, Diamond and Rajan (2011) show that
distressed financial institutions do not have enough incentives to prevent their own fire
sales at a socially optimal level because of risk-shifting incentives. Many papers in the
literature suggest various policy implications to mitigate their adverse effects on the
economy (e.g., Bolton, Santos, and Scheinkman (2009), Krishnamurthy (2010), Shleifer
and Vishny (2010), Guerrieri and Shimer (2014)). Our paper also contributes to the
discussion of policy implications by looking at the problem in a new perspective that
involves the paradoxical nature of fire sales and market freeze.
Recent papers have extended the limits-to-arbitrage argument, and suggest that institutional impediments can slow down the speed of arbitrage capital (e.g., Mitchell,
Pedersen, and Pulvino (2007), Duffie (2010)). “Slow-moving capital” could be a reasons
why fire sales still occur when there is enough capital in the economy. Institutional impediments that contribute to slowing down capital movement may originate from various
sources such as searching frictions, taxes, regulations, and market segmentation. Our
paper contributes to the discussion of slow-moving capital by suggesting that relatively
small amount of informed capital (or “smart capital”) can facilitate capital movement
when it is unconstrained. However, the market can quickly become illiquid if informed
capital providers are subject to liquidity shocks, and this can be the source of slower
movement of capital than is case in normal times. We further show that informed capital
will not be at an ex-ante sufficient level to always prevent fire sales as long as capital is
costly. This is because the negative externalities created by fire sales are not internalized by each financial institution. Therefore, the existence of smart capital in the market
8
can be a double-edged sword; it creates financial stability and provides well-facilitated
liquidity with a relatively small amount of capital at a low cost. On the other hand,
stability and liquidity can be unwound quickly when smart capital is under distress.
Therefore, our results suggest an alternative amplification mechanism for systemic risk
in financial system.
3
3.1
Model
Basic setup
Consider a two-period economy (𝑡 = 1, 2). There exists a risky asset that is tradable
by all the participants in the financial market (henceforth, “the marketable asset”), and
a risk-free asset in infinitely elastic supply with an exogenously-given return 𝑟𝑓 . The
marketable asset is illiquid in the sense that it cannot be liquidated at 𝑡 = 1, but pays
a random liquidation value 𝑣 at 𝑡 = 2 with
{︃
𝑣=
𝑣𝐻 with probability 𝜌;
𝑣𝐿 with probability 1 − 𝜌;
where 𝑣𝐻 > 𝑣𝐿 . We call the marketable asset a “high-quality” asset in the event 𝑣 = 𝑣𝐻 ,
and “low-quality” otherwise. Examples of the marketable asset include most tradable
financial securities such as equities, corporate bonds, MBS, or CDO. For example, in
the case of a fixed income security such as corporate bond, one can interpret 𝑣𝐻 as the
promised payoff (face value plus coupon) of the security, 𝑣𝐿 as its recovery value, and
1 − 𝜌^ as the default probability.
There exists a continuum of participants: (i) sellers, (ii) arbitrageurs, (iii) investors,
and (iv) noise traders. We assume that each class of agents is present in a unit measure.
We denote 𝒮, 𝒜 and ℐ to be the set of sellers, arbitrageurs and investors, respectively.
The sellers and arbitrageurs are risk-neutral, capital-constrained, and informed (the
risk-neutrality assumption for these agents is for simplicity). On the other hand, the
investors are risk-averse, unconstrained, and uninformed.
The sellers participate to sell some or all of their endowment of the marketable asset
to meet liquidity needs or to raise funds for other investments. Each seller 𝑠 ∈ 𝒮 is
endowed with one unit of the marketable asset, and has liquidity position 𝑤𝑆 (available
9
cash minus short-term debt) which is positive or negative. Each seller also has access to
an investment opportunity (henceforth, “non-marketable asset”). The non-marketable
assets cannot be traded or transferred to other participants. The non-marketable asset
can be considered as the profit-generating operations of the firm. For simplicity, we
assume that the non-marketable assets generate an identical return 𝑟𝐼 at 𝑡 = 2 with
probability density function 𝑓𝐼 (·) and cumulative distribution function 𝐹𝐼 (·) with support [𝑟, 𝑟¯] where 𝑟 > 𝑟𝑓 and 𝑟¯ may be finite or infinite. This investment opportunity
may be operated in reverse; investing a negative amount represents liquidating existing
assets (which the sellers might wish to do in order to meet their liquidity obligation).
The sellers can observe the realization of 𝑣 and 𝑟𝐼 . We assume the sellers are subject
to a liquidity constraint requiring net cash flow from investing activities plus available
liquidity to be non-negative (violating this constraint would mean the seller is unable to
meet its obligations) as well as a short-sale constraint (i.e. they cannot sell more than
their endowment).
The arbitrageurs participate to generate profits by exploiting any mispricing by trading the asset. They are informed about the value of the marketable asset. Each arbitrageur 𝑎 ∈ 𝒜 has liquidity position 𝑤𝐴 which is strictly positive. 𝑤𝐴 can be thought
of as the arbitrageurs’s cash position and available credit (e.g., cash from pledging their
inventory of other assets). As in the case of the sellers, the arbitrageurs observe the
realization of 𝑣. We assume that the arbitrageurs are subject to margin requirements.
In a similar fashion to Brunnermeier and Pedersen (2009), we assume that arbitrageur
𝑎’s total margin on its position 𝑥𝑎 cannot exceed its available capital 𝑤𝐴 :
|𝑚𝑥𝑎 | ≤ 𝑤𝐴 ,
(1)
where 𝑚 is the dollar margin on the position.
The investors participate to transfer their liquidity to the future, and also potentially
to make trading profits. They do not know the value of the marketable asset. Each
investor 𝑖 ∈ ℐ is endowed with identical initial wealth 𝑤𝐼 at 𝑡 = 1, and has an identical
𝑖
constant absolute risk aversion (CARA) utility 𝑈 (𝑤2𝑖 ) = − 𝛾1 𝑒−𝛾𝑤2 where 𝑤2𝑖 is his wealth
at 𝑡 = 2. We assume that the investors are unconstrained in their capacity to invest
in available investment opportunities, i.e., they can borrow at 𝑟𝑓 if their initial funding
𝑤𝐼 is not large enough to cover investment expenses, and they can sell short. The
investors are uninformed about the realization of 𝑣, but in equilibrium they will learn
10
about it from the price of the marketable asset. For notational convenience, we define
𝑞(𝑝) ≡ 𝑃 𝑟(𝑣 = 𝑣𝐻 |𝑝) to be their updated belief the marketable asset is high quality.
Intuitively, the sellers can be considered to be firms that invest in projects and create
financial securities. A bank that originates and distributes loans is an example. The
arbitrageurs are firms that specialize in securities trading and have particular expertise
in trading the marketable asset but have limited capital. In practice such investors are
typically firms rather than individuals. Examples include investment banks that specialize in market-making, hedge funds that specialize in investing in the marketable asset
and similar assets, etc. Such arbitrageurs can be subject to liquidity constraints in the
form of regulatory requirements, margin calls, client capital withdrawals, etc. Elsewhere
in the economy, capital is not limited but the investors deploying this capital have no
expertise in the tradable asset. Such investors could be pension funds, insurance companies, financial institutions in other countries, sovereign wealth funds, wealthy investors
like Warren Buffett, etc—in other words, any investors who do not have particular expertise in valuing and trading the marketable asset. A more conceptual interpretation
is that the investors are the “representative consumer” of the economy who, although
he may choose to hold assets via intermediaries, has the option of investing directly.
The noise traders participate in the market for exogenous reasons. We denote 𝑧 to
be the demand of the noise traders, and assume that 𝑧 follows a probability distribution
with probability density function 𝑓𝜖 (·) with support on the real line. We assume that 𝑧 is
independent of asset value 𝑣. Although we have modeled noise by introducing a separate
class of traders, it is possible to add noise in other ways via utility maximizing agents.
For example, this could be done by making the return on the non-marketable asset
correlated across sellers, or giving investors a random endowment of the marketable asset
(as in Diamond and Verrecchia (1981)). The exogenous noise trade we have specified is
simpler.
We assume that 𝑓𝜖 (·) satisfies the following conditions for all 𝑧:
𝑓𝜖′′ (𝜖)𝑓𝜖 (𝜖) ≤ 𝑓𝜖′ (𝜖)2 ,
(2)
Condition in Eq. (2) is sufficient for us to show existence and uniqueness of equilibrium. This condition is satisfied by a wide variety of distributions that is unimodal such
as normal distribution.
11
The market opens with each seller deciding how many units of marketable asset to
sell out of its holdings. If there is no supply of the marketable asset from sellers, the
market does not open. That is, the market opens only when there is a positive supply
from sellers. Once the market is open, arbitrageurs and investors then condition their
demands on the price. The market clears by equating the aggregate volume from the
sellers to the demand from the other traders (arbitrageurs, investors, and noise traders).
3.2
Optimization problems
Each seller 𝑠 ∈ 𝒮 maximizes future expected profit by deciding whether to sell its
endowment of marketable assets and invest the proceeds in the non-marketable asset.3
We denote 𝑥𝑠 to be the units of marketable asset sold by seller 𝑠. Then, seller 𝑠 solves
an optimal trading problem given 𝑣 and 𝑙:
max 𝑣(1 − 𝑥𝑠 ) + (1 + 𝑟𝐼 )(𝑤𝑆 + 𝐸[𝑝|𝑣]𝑥𝑠 ).
𝑥𝑠 ∈[0,1]
(3)
Each arbitrageur 𝑎 ∈ 𝒜 maximizes profit by choosing a portfolio of the marketable
asset and the risk-free asset under the liquidity constraint. Arbitrageur 𝑎 solves the
following constrained optimization problem given 𝑝 and 𝑣:
max
𝑣𝑥𝑎 + (1 + 𝑟𝑓 )(𝑤𝐴 − 𝑝𝑥𝑎 ),
𝑎
(4)
|𝑚𝑥𝑎 | ≤ 𝑤𝐴 ,
(5)
𝑥
subject to
where Eq. (5) is the margin constraint.
Each investor 𝑖 ∈ ℐ maximizes expected utility of future wealth by choosing a portfolio that consists of the marketable asset and the risk-free asset. Investor 𝑖 solves the
following optimization problem given the price 𝑝:
]︂
(︁ [︀
]︀)︁⃒⃒
1
𝑖
max
𝐸 − exp −𝛾 𝑤𝐼 (1 + 𝑟𝑓 ) + (𝑣 − 𝑝(1 + 𝑟𝑓 ))𝑥 ⃒𝑝 .
𝛾
𝑥𝑖
[︂
3
Because 𝑟𝐼 > 𝑟𝑓 , the risk-free asset is always dominated by the non-marketable assets.
12
4
Equilibrium
4.1
Definition of equilibrium
As in standard REE models, investors infer the fundamental value from the price because
the equilibrium price is a function of the fundamental value and noise. Unlike such
models using the CARA-normal framework, however, the equilibrium price function
is nonlinear not only because of the distributions of the fundamental value and noise
trading but, more importantly, because of the financial constraints of the sellers and
the arbitrageurs. The investors in our model first infer net supply of the asset from
the price, then use it to infer the fundamental value.4 The net supply is defined to be
the sum of an informed trading component, which is the supply from the sellers minus
demand from the arbitrageurs, and a noise component, which is the demand from the
noise traders.
We define equilibrium in the standard manner by augmenting the equilibrium concept
with learning from net supply.
Definition 1. A noisy rational expectation equilibrium is a price 𝑝 and an allocation
(︀ 𝑠
)︀
𝑥 )𝑠∈𝒮 , (𝑥𝑎 )𝑎∈𝒜 , (𝑥𝑖 )𝑖∈ℐ such that (a) 𝑥𝑠 solves each seller 𝑠’s problem, (b) 𝑥𝑎 solves
each arbitrageur 𝑎’s problem, (c) 𝑥𝑖 solves each investor 𝑖’s problem, (d) 𝑝 clears the
market, i.e., supply from the sellers equals demand from the other agents:
∫︁
∫︁
𝑠
∫︁
𝑖
(6)
𝑎∈𝒜
𝑖∈ℐ
𝑠∈𝒮
𝑥𝑎 𝑑𝑎 + 𝜖,
𝑥 𝑑𝑖 +
𝑥 𝑑𝑠 =
and (e) 𝑝 is a sufficient statistic for 𝜉𝑝 , i.e.,
𝑞(𝑝) ≡ 𝑃 𝑟(𝑣𝐻 |𝑝) = 𝑃 𝑟(𝑣𝐻 |𝑝, 𝜉𝑝 ),
(7)
where 𝜉𝑝 denote the net supply of the marketable asset to the investors such that
∫︁
𝜉𝑝 ≡
∫︁
𝑠
𝑥 𝑑𝑠 −
𝑠∈𝒮
𝑥𝑎 𝑑𝑎 − 𝜖.
(8)
𝑎∈𝒜
4
Investors’ belief update based on net supply is parallel with Kyle (1985) in which the uninformed
market makers use order flows to infer the fundamental value. Unlike the market makers in a Kyle
model, however, the investors in our model do not directly observe net supply, but have to infer net
supply from the price.
13
We first conjecture that Eq. (7) holds, then, later, we verify that it is indeed true in
equilibrium.
We define a market freeze to be a situation in which the market fails to open with
a positive probability because no seller wants to sell the marketable asset. As we will
see, sellers always sell the low-quality asset whereas they may not want to sell the highquality asset sometimes. This is so because the low-quality assets can get overvalued
but the high-quality can get undervalued once prices become uninformative. Therefore,
we can measure the degree of market freeze by the probability that the market is open
for the high-quality asset, which is denoted by
𝜇𝐻 ≡ 𝑃 𝑟
(︁∫︁
⃒ )︁
⃒
𝑥 𝑑𝑠 > 0⃒𝑣𝐻 .
𝑠
𝑠∈𝒮
(9)
Because the low-quality asset is circulated in the market with probability one, 𝜇𝐻 can
be also interpreted as the relative circulation rate of the high-quality asset to that of the
low-quality asset.
Definition 2. There is a market freeze if the high-quality asset fails to fully circulate in
the market, i.e., 𝜇𝐻 < 1.
We define a fire sale to be an event in which sellers are selling their holdings at the
price below which they would not sell for informational reasons unless they are forced
to sell for non-informational reasons (i.e., liquidity needs).
Definition 3. The sellers engage in a fire sale if they sell their holdings of the high∫︀
quality asset at a discount rate greater than their minimum cost of capital, i.e., 𝑠∈𝒮 𝑥𝑠 𝑑𝑠 >
𝑣
.
0 when 𝐸[𝑝|𝑣𝐻 ] < 1+𝑟
The intuition behind these definitions is as follows. Since sellers are informed, one
could conjecture that they will normally sell overvalued assets and do not sell undervalued assets. Hence, a seller who sells an undervalued asset is trying to raise cash,
not selling for informational reasons. We call this a fire sale. When prices are higher
𝑣𝐻
than 1+𝑟
, the sellers are always willing to sell all their holdings regardless of information
𝑣𝐻
about the fundamental value. However, when prices are lower than 1+𝑟
, the sellers would
not want to sell unless the realization of 𝑟𝐼 is high enough that they want to sell the
apparently-undervalued high-quality asset not to give up more profitable non-marketable
assets.
14
Turning to the volume of sales, one might consider it natural in this model for sellers
to sell all their holdings. They can earn returns on the non-marketable asset that
dominate the returns required by the investors (i.e., 𝑟𝑠 is always greater than 𝑟𝑓 .). So,
if sellers are keeping back some of their holdings from the market, this suggests that
there is some kind of freeze in the functioning of the market. As we will show, it is
impossible for the marketable asset to trade in equilibrium at less than the present value
of the low-quality asset. This implies that sellers will always sell the low-quality asset.
But if the price is not revealing, some sellers may hold back the high-quality asset for
classic “lemons” motives. This is what we describe as a market freeze.
We can also quantify the magnitude of market freeze and fire sales. The degree of
market freeze is captured by 𝜇𝐻 which is the circulation rate of the high-quality asset.
On the other hand, the degree of fire sales is captured by the fire sale discount which is
the difference between the expected price of the high-quality asset minus its fundamental
𝑣𝐻
− 𝐸[𝑝|𝑣𝐻 ].
value, i.e., 𝛿 ≡ 1+𝑟
𝑓
4.2
4.2.1
Solving equilibrium
Demand and supply
There are two groups of informed market participants whose sales may reveal their
information: sellers and arbitrageurs. At a given level of 𝑟𝐼 , we let 𝑋(𝑝, 𝑣) denote their
net supply of the marketable asset as a function of 𝑝 and 𝑣:
𝑋(𝑝, 𝑣) ≡ 𝑋𝑆 (𝑣) − 𝑋𝐴 (𝑝, 𝑣),
(10)
∫︀
where 𝑋𝑆 (𝑣) ≡ 𝑠∈𝒮 𝑑 𝑥𝑠 𝑑𝑠 denotes the aggregate supply of the sellers given 𝑣, and
∫︀
𝑋𝐴 (𝑝, 𝑣) ≡ 𝑎∈𝒜 𝑥𝑎 𝑑𝑎 denotes the aggregate demand of the arbitrageurs given 𝑝 and 𝑣.
We start by describing the aggregate supply of the asset from the sellers.
Lemma 1. Given 𝑣, the sellers’ aggregate supply is as follows:
⎧
if 𝐸[𝑝|𝑣] <
⎪
⎨ 0
𝑋𝑆 (𝑣) ∈
[0, 1] if 𝐸[𝑝|𝑣] =
⎪
⎩
1
if 𝐸[𝑝|𝑣] >
𝑣
;
1+𝑟𝐼
𝑣
;
1+𝑟𝐼
𝑣
.
1+𝑟𝐼
(11)
Furthermore, the probability of the market opening for the high- and the low- quality
15
asset is given, respectively, by the following:
𝜇𝐻 = 1 − 𝐹 𝐼
(︀
)︀
𝑣
−1 ,
𝐸[𝑝|𝑣]
𝜇𝐿 = 1.
(12)
(13)
Proof. See Appendix.
To see this intuitively, note that each seller sells its holdings of the marketable asset
if the expected price exceeds the value of the asset, discounted at the opportunity cost
of capital. The opportunity cost of capital is the seller’s return on the non-marketable
asset (since we have assumed this is higher than the risk-free rate).
In other words, with high return on the non-marketable asset sellers will sell all their
endowment of the marketable asset to meet their liquidity needs and then invest the
surplus into the non-marketable asset. with a low return on the non-marketable asset
sellers will not sell their endowment of the marketable asset, and will meet their liquidity
needs by divesting from the non-marketable asset.
Next, given the price, arbitrageurs buy the marketable asset if undervalued, and sell
if overvalued. Because the arbitrageurs are identical, 𝑋𝐴 (𝑝, 𝑣) is equivalent to each arbitrageur’s optimal demand. The following is immediate from the arbitrageur’s problem
in Eq. (4):
Lemma 2. Given 𝑝 and 𝑣, the arbitrageurs’ aggregate demand is as follows:
{︃
𝑋𝐴 (𝑝, 𝑣) ∈
𝑤𝐴
𝑚
− 𝑤𝑚𝐴
if 𝑣 = 𝑣𝐻 ;
if 𝑣 = 𝑣𝐿 .
(14)
The arbitrageurs have perfectly elastic demands when the asset is correctly priced
(i.e. it is priced at discount rate 𝑟𝑓 ), subject to a minimum and a maximum. The
maximum they can demand is the quantity of the asset that exhausts their wealth, 𝑤𝑚𝐴 ,
and this is also their demand if the asset is underpriced. Likewise, the minimum they
can demand is − 𝑤𝑚𝐴 , which is also their demand if the asset is overpriced.
Once the market opens the investors trade with the arbitrageurs and the sellers who
are privately informed about the true value of 𝑣. Each investor attempt to infer 𝑣 from
the market clearing price 𝑝, thus, the demand of the asset is based on their posterior
∫︀
belief 𝑞(𝑝). Let 𝑋𝐼 (𝑝) ≡ 𝑖∈ℐ 𝑥𝑖 𝑑𝑖 denote the aggregate demand of the investors given 𝑝.
16
Lemma 3. Given 𝑝, the investors’s aggregate demand is given by
[︁ (︁ 𝑞(𝑝) )︁
(︁ 𝑣 − (1 + 𝑟 )𝑝 )︁]︁
1
𝐻
𝑓
𝑋𝐼 (𝑝) =
log
+ log
.
𝛾(𝑣𝐻 − 𝑣𝐿 )
1 − 𝑞(𝑝)
(1 + 𝑟𝑓 )𝑝 − 𝑣𝐿
(15)
Proof. See Appendix.
The investors’ demand is a standard CARA demand function for the case where
there only two possible outcomes for the terminal value of the asset. Because they are
risk-averse, they require a larger risk premium when they are less confident about the
expected payoff of the asset.
4.2.2
Learning
Recall that the market does not open in case there is no supply of the asset from the
sellers, i.e., 𝑋𝑆 (𝑣) = 0. Lemma 1 implies that the market may or may not be open
sometimes for a high-quality asset where the market is always open for a low-quality
asset. Therefore, the fact that the investors are participating in the market delivers some
information about the quality of the traded asset. Conditional on the market being open,
the investors’ probability assessment that the asset is of high quality is given by5
𝜌^ ≡ 𝑃 𝑟(𝑣 = 𝑣𝐻 |𝑋𝑆 (𝑣) > 0) =
𝜌𝜇𝐻
,
𝜌𝜇𝐻 + (1 − 𝜌)
(17)
where 𝜇𝐻 is the circulation rate of the high-quality asset such that
𝜇𝐻 = 1 − 𝐹 𝐼
(︁
)︁
𝑣𝐻
−1 .
𝐸[𝑝|𝑣𝐻 ]
(18)
Notice that the investors assess that the quality of the traded asset is likely to be
poorer as the expected undervaluation of the high-quality asset is greater (i.e., 𝐸[𝑝|𝑣𝐻 ] is
lower). This is related to “Lemons problem” arising from the sellers’ incentive of selling
their holdings only in the case the asset is of low quality unless they are forced to sell due
5
Using the Bayes’ rule, we have
𝜌𝑃 𝑟(𝑋𝑆 (𝑣) > 0|𝑣 = 𝑣𝐻 )
.
(16)
𝜌𝑃 𝑟(𝑋𝑆 (𝑣) > 0|𝑣 = 𝑣𝐻 ) + (1 − 𝜌)𝑃 𝑟(𝑋𝑆 (𝑣) > 0|𝑣 = 𝑣𝐿 )
(︀ 𝑣
)︀
𝐻
Then, Eq. (17) is immediate from Eq. (A.19) because 𝑃 𝑟(𝑋𝑆 (𝑣) > 0|𝑣 = 𝑣𝐻 ) = 1 − 𝐹𝐼 𝐸[𝑝|𝑣
− 1 and
𝐻]
𝑃 𝑟(𝑋𝑆 (𝑣) > 0|𝑣 = 𝑣𝐿 ) = 1.
𝑃 𝑟(𝑣 = 𝑣𝐻 |𝑋𝑆 (𝑣) > 0) =
17
to liquidity reasons (i.e., to raise funds for more profitable investment). Such adverse
selection problem becomes more severe as prices become more dislocated. Therefore,
the investors’ prior belief of the quality of the asset being high is adjusted to be 𝜌^ rather
than 𝜌.
The price is partially revealing and investors update their beliefs about the asset
value from the price. Equivalently, given the conjecture that price is a sufficient statistic
for 𝜉𝑝 (the net supply trading of the asset to the investors as defined in (11)) they update
their belief conditional on 𝜉𝑝 .
Lemma 4. The investors’ posterior belief conditional on 𝑝 is given by
𝑞(𝑝) =
𝜌^𝑓𝜖 (𝑋(𝑝, 𝑣𝐻 ) − 𝜉𝑝 )
.
𝜌^𝑓𝜖 (𝑋(𝑝, 𝑣𝐻 ) − 𝜉𝑝 ) + (1 − 𝜌^)𝑓𝜖 (𝑋(𝑝, 𝑣𝐿 ) − 𝜉𝑝 )
(19)
Proof. See Appendix.
Finally, we can verify that 𝑝 is a sufficient statistic for 𝜉𝑝 in equilibrium by finding a mapping 𝜉 from prices to net supply that gives equivalent information to directly
observing 𝜉𝑝 . That is, given 𝑝, the investors’ posterior belief is identical whether it
is conditioned on 𝜉𝑝 or 𝜉(𝑝), confirming the initial conjecture in Eq. (7). The result
is not obvious: in a noisy REE framework, updating beliefs based on price for general distributions of asset value is potentially complex. The canonical formulation is
a CARA-Gaussian model where one can infer the updating rule based on the conjecture that price is linear in the state variables (e.g. noise and asset value). Grossman
and Stiglitz (1980), Diamond and Verrecchia (1981) and Hellwig (1980) are examples.
Linearity is a special property which does not hold in general. In our model, price is
non-linear because of the Bernoulli distribution of asset value, the liquidity and short
sale constraints, and the general distribution of noise trading. Updating beliefs based
on net supply as well as price is straightforward, by Bayes’ rule, but in principle price
might not reveal net supply: a single price might correspond to two or more possible
values for net supply that lead to different posterior beliefs.6 Our assumption in Eq. (2)
on the distribution of noise trade, however, rules this out. Furthermore Eq. (2) is quite
a weak condition that is satisfied by any unimodal distribution, a class that nests the
commonly-used distributions. The details are in appendix.
6
It is fine for a price to correspond to two or more values for net supply as long as those values lead
to a single posterior belief.
18
4.2.3
Price informativeness and informed supply variation
When the informed agents (the sellers and the arbitrageurs) have more liquidity, prices
reveal fundamental value better. Let Δ𝑋 be the maximum difference in the information
component of net supply for the high- and low- quality assets:
Δ𝑋 ≡ 𝑋𝐴 (𝑣𝐿 ) − 𝑋𝐴 (𝑣𝐻 ) =
2𝑤𝐴
.
𝑚
Because Δ𝑋 measures the variability in the net supply of informed traders, we call it
the “informed supply variation.” It reflects the classic lemons intuition that good assets
are in smaller supply than lemons. We will show that price informativeness increases in
informed supply variation.
We can give an alternative representation of the investors’ posterior belief 𝑞(𝑝) given
𝑣 and 𝑧 using the definition of informed supply variation as follows:
Lemma 5. Given 𝑣 and 𝜖,
𝑞(𝑝) =
where
𝜌^
𝜌^ + 𝑓^𝜖 (𝑣, 𝜖)(1 − 𝜌^)
{︃
𝑓^𝜖 (𝑣, 𝜖) =
𝑓𝜖 (𝜖+Δ𝑋)
;
𝑓𝜖 (𝜖)
𝑓𝜖 (𝜖)
;
𝑓𝜖 (𝜖−Δ𝑋)
,
if 𝑣 = 𝑣𝐻 ;
if 𝑣 = 𝑣𝐿 ;
(20)
(21)
Proof. See Appendix.
The posterior belief 𝑞(𝑝) is close to one if the investors believe that the marketable
asset is almost certainly a high-quality asset, and 𝑞(𝑝) is close to zero if the investors
believe that it is almost certainly a low-quality asset. Notice that 𝑞(𝑝) is close to one if
𝑓^𝜖 (𝑣, 𝜖) is close to zero, and 𝑞(𝑝) is close zero if 𝑓^𝜖 (𝑣, 𝜖) is close to infinity. From Eq. (21),
one can observe that higher informed supply variation in general increases the chance of
revealing the fundamental value of the marketable asset. As Δ𝑋 increases, 𝑓^𝜖 (𝑣, 𝜖) gets
larger on average if 𝑣 = 𝑣𝐻 , and gets smaller on average if 𝑣 = 𝑣𝐿 .
We have assumed that the noise in the supply has unbounded support. Therefore,
prices won’t be fully revealing. However, when Δ𝑋 is sufficiently large, arbitrage activities push prices arbitrarily close to the fundamental value of the asset, thereby making
19
prices almost fully-revealing. Therefore, large enough informed supply variation will
reveal the fundamental value of the asset.
Revealing high-quality
Revealing low-quality
Distribution of
X(vH) - 
Distribution of
X(vL) - 
X(vH) - 
X(vL) - 
(a) larger informed supply variation Δ𝑋 (prices are almost fully-revealing)
Revealing high-quality
Revealing low-quality
Distribution of
X(vH) - 
Distribution of
X(vL) - 
X(vH) - 
X(vL) - 
(b) smaller informed supply variation Δ𝑋 (prices are less revealing)
Figure 2: An illustration of price informativeness under various realizations of 𝜖 at the
given level of 𝑝 and Δ𝑋
Figure 2 illustrates this idea graphically. When informed supply variation, Δ𝑋, is
large enough as in Figure 2.(a), the realization of 𝜉𝑝 overlaps very little between the two
cases with 𝑣 = 𝑣𝐻 and 𝑣 = 𝑣𝐿 . This makes 𝑣 almost fully revealed, then the price should
𝑣𝐻
be arbitrarily close to either the fundamental value of the high-quality asset ( 1+𝑟
) or
𝑓
𝑣𝐿
that of the low-quality asset ( 1+𝑟𝑓 ). On the other hand, when informed supply variation
is small as in the subfigure labeled (ii), the realization of 𝜉𝑝 overlaps between the two
20
cases with 𝑣 = 𝑣𝐻 and 𝑣 = 𝑣𝐿 . This makes 𝑣 revealed less in those more overlapping
𝑣𝐻
𝑣𝐿
ranges. This results in a noisier price that lies between 1+𝑟
and 1+𝑟
.
𝑓
𝑓
4.2.4
Equilibrium
The model always has an equilibrium that can be characterized as follows:
Proposition 1. (Existence and uniqueness of equilibrium) There always exists an equilibrium for any realization
of
(︁
)︁ the random variables 𝑣, 𝑟𝐼 and 𝜖. The market opens with
𝑣𝐻
probability 1−𝐹𝐼 𝐸[𝑝|𝑣𝐻 ] −1 when 𝑣 = 𝑣𝐻 , and opens with probability one when 𝑣 = 𝑣𝐿 .
Conditional on the market being open, the equilibrium price 𝑝 given 𝑣 and 𝜖 is uniquely
given by
)︀
1 (︀
𝑝=
𝜔(𝑣, 𝜖)𝑣𝐻 + (1 − 𝜔(𝑣, 𝜖))𝑣𝐿 ,
(22)
1 + 𝑟𝑓
where the weight 𝜔(𝑣, 𝜖) is given by
𝜌^𝑓𝜖 (𝜖)
𝜌^𝑓𝜖 (𝜖) + (1 − 𝜌^)𝑓𝜖 (𝜖 + Δ𝑋) exp(𝛾(𝑣𝐻 − 𝑣𝐿 )(1 −
𝜌^𝑓𝜖 (𝜖 − Δ𝑋)
𝜔(𝑣𝐿 , 𝜖) =
𝜌^𝑓𝜖 (𝜖 − Δ𝑋) + (1 − 𝜌^)𝑓𝜖 (𝜖) exp(𝛾(𝑣𝐻 − 𝑣𝐿 )(1 +
𝜔(𝑣𝐻 , 𝜖) =
𝑤𝐴
𝑚
− 𝜖))
𝑤𝐴
𝑚
− 𝜖))
;
(23)
.
(24)
Proof. See Appendix.
Proposition 1 shows the existence of equilibrium by proving that their exists a fixed
point for the price function Eq. (22). The weights (𝜔(𝑣, 𝜖), 1 − 𝜔(𝑣, 𝜖)) in the equilibrium
price in Eq. (22) reflect the investors’ pricing kernels given their posterior beliefs and
risk aversion. As the investors become more risk tolerant (i.e., 𝛾 goes to zero), the
weight approaches the posterior probability 𝑞(𝑝) (i.e., the price approaches the risk𝑣𝐿
neutral value). Notice that the price equals the value of low-quality asset (i.e., 1+𝑟
)
𝑓
𝑣𝐻
when 𝜔(𝑣, 𝜖) equals zero, and the value of high-quality asset (i.e., 1+𝑟
) when 𝜔(𝑣, 𝜖)
𝑓
equals one.
4.3
Fire sales and market freezes
In this section, we show that fire sales and market freeze can occur due to shocks to
the arbitrageurs’ capital. When the arbitrageurs have enough capital, we find that
21
there is no price dislocation (i.e., prices are arbitrarily close to the fundamental value
of the marketable asset). Because prices are informative, uninformed investors provide
liquidity to the market by absorbing the supply of the asset. In that case, sellers always
sell their holdings of the marketable asset rather than giving up on their profitable nonmarketable assets. Therefore, the supply of the marketable asset is insensitive to the
quality of the asset, and is driven only by the liquidity needs of sellers. However, in case
arbitrageurs’ capital is scarce, prices are dislocated. Because prices are not informative,
uninformed investors cannot provide liquidity to the full extent. In that case, sellers’
decisions depend on the asset quality. If the marketable asset is of low quality, they sell
it. However, if the marketable asset is of high quality, they only sell in case of severe
lack of liquidity (i.e., they are hit by a liquidity shock, or forgoing investment in nonmarketable assets is very costly.) Therefore, average asset quality per trading volume
goes down. Therefore, reduction in arbitrageurs’ capital leads to uninformative prices,
which in turn cause adverse selection.
We summarize our main results about fire sales and market freezes in the following
proposition:
Proposition 2. There exists a constant 𝑤¯𝐴 such that (i) (Fire sales) fire sales occur
below 𝑤¯𝐴 , i.e.,
{︃ 𝑣
𝑣𝐻
𝐻
, 1+𝑟
) if 𝑤𝐴 ≥ 𝑤;
¯
[ 1+𝑟
𝑓
(25)
𝐸[𝑝|𝑣𝐻 ] ∈
𝑣𝐻
𝑣𝐿
)
𝑤
<
𝑤,
¯
,
( 1+𝑟
𝐴
𝑓 1+𝑟
(ii) (Lemons’ problem) the average quality of traded asset becomes poorer below 𝑤¯𝐴 , i.e.,
{︃
𝜌^ ∈
{𝜌} if 𝑤𝐴 ≥ 𝑤;
¯
[0, 𝜌) 𝑤𝐴 < 𝑤,
¯
(26)
Proof. See Appendix.
Proposition 2 shows that there is no fire sale nor market freeze when there is enough
capital for the arbitrageurs. On the other hand, insufficient capital of arbitrageurs (or
liquidity shocks to arbitrageurs) can create double whammy of fire sales and market
freeze. In case of the high-quality asset, a reduction in arbitrage capital create a large
price reduction because it affects both supply and demand. That is, the investors are
less willing to absorb the net supply at any given level of prices.
Why do demand shrink so much when there is shock to the arbitrageurs’ capital?
22
Answering this question is the main purpose of our analysis in this subsection. We argue
that a reduction in arbitrage capital causes a direct effect and an indirect effect that
reduce demand, thereby precipitating price falls.
The direct effect, which we call “price informativeness effect” (or “noisy rational expectations equilibrium effect”) of a liquidity shock, is related to dislocating prices due
to the lack of informed trading. This effect is captured by our noisy REE framework
that interconnects price informativeness and liquidity shocks to informed trading. The
reduction of arbitrage capital creates an initial price fall by lowering the investors’ demand. This is because the investors cannot infer the quality of the traded asset as much
as before.
The indirect effect, which we call “adverse selection effect” (or “lemons effect”) of
a liquidity shock, is related to feedback effects between price dislocations and lemons
problem. When prices are noisy, the sellers will not sell the high-quality asset unless they
are forced to sell. That is, the initial price dislocation due to uninformative prices (i.e.,
the direct effect) worsens adverse selection of the sellers, thereby making the average
quality of the traded asset poorer. Because the average quality gets poorer, prices fall
further, but this in turn lowers the quality further by worsening adverse selection problem
further, and so on. Therefore, price become more dislocated as the overall quality of
traded assets become poorer. Such feedback mechanism causes “double whammy” of a
large fire sale discount and market freeze for the high-quality asset.
In summary, there is an initial fall in prices due to the price informativeness effect,
then, prices to fall further because of the adverse selection effect that is a consequence
of the first effect. This idea is illustrated in Figure 3. First consider an equilibrium
prior to the reduction in arbitrage capital. The equilibrium is determined at the point
𝐴 where demand and supply are matched given the initial size of arbitrage capital.
Now, suppose that there is a reduction in arbitrage capital, and this shift the supply
curve outward. Because the shock to arbitrage capital is common knowledge among
all participants, the investors incorporate that information, thus, their demand shifts
outward to accommodate such change. However, the demand is not going to shift
enough to support the same price at 𝐴 because prices are less revealing now. Therefore,
the price would be lowered to the point 𝐵. This in turn results in further decrease
in prices through lemons problem. That is, the reduction in arbitrage capital lowers
price informativeness, thus, this increases adverse selection. Therefore, demand shrinks
23
as a result of increased Lemons’ problem. As a result of the decrease in demand, the
equilibrium is determined at the point 𝐶 rather than 𝐵. Therefore, prices fall further
by the second impact on the demand.
Price
NREE
effect
Price
A
A
NREE
effect
B
B
Lemons
effect
C
Investors’
demand
Net supply
Investors’
demand
Net supply
Quantity
Quantity
(i) Price informativeness effect
(ii) Adverse selection effect
Figure 3: Fire sales: price informativeness effect vs. adverse selection effect
To quantify the two effects, we represent the price as a function of 𝑤𝐴 and 𝜌^ (by
dropping arguments such as 𝑣 and 𝜖) as follows:
]︁
1 [︁
𝑣𝐿 + (𝑣𝐻 − 𝑣𝐿 )𝜔(𝑤𝐴 , 𝜌^) ,
𝑝=
1 + 𝑟𝑓
(27)
where
𝜔(𝑤𝐴 , 𝜌^) ≡
𝜌^𝑓𝜖 (𝜖) + (1 − 𝜌^)𝑓𝜖 (𝜖 +
𝜌^𝑓𝜖 (𝜖)
2𝑤𝐴
) exp(𝛾(𝑣𝐻
𝑚
− 𝑣𝐿 )(1 −
𝑤𝐴
𝑚
− 𝜖))
.
(28)
𝑜𝑙𝑑
𝑛𝑒𝑤
When arbitrage capital is reduced from 𝑤𝐴
to 𝑤𝐴
, the change in the fire sale discount
(or the change in prices) can be decomposed into two components using under the new
representation:
𝐸[𝑝𝑜𝑙𝑑 |𝑣𝐻 ] − 𝐸[𝑝𝑛𝑒𝑤 |𝑣𝐻 ] = 𝐸
⏟
⏞
⏟
total effect
)︁⃒ ]︁
[︁ (𝑣 − 𝑣 ) (︁
⃒
𝐻
𝐿
𝑜𝑙𝑑 𝑜𝑙𝑑
𝑛𝑒𝑤
𝜔(𝑤𝐴
, 𝜌^ ) − 𝜔(𝑤𝐴
, 𝜌^𝑜𝑙𝑑 ) ⃒𝑣𝐻
1 + 𝑟𝑓
⏞
price informativeness effect
[︁ (𝑣 − 𝑣 ) (︁
)︁⃒ ]︁
⃒
𝐻
𝐿
𝑛𝑒𝑤
𝑜𝑙𝑑
𝑛𝑒𝑤
𝑛𝑒𝑤
+𝐸
𝜔(𝑤𝐴 , 𝜌^ ) − 𝜔(𝑤𝐴 , 𝜌^ ) ⃒𝑣𝐻 ,
1 + 𝑟𝑓
⏟
⏞
adverse selection effect
24
(29)
We can show that both effects are driving fire sales in the presence of shocks to
arbitrageurs’ capital.
Corollary 1. (Double whammy) The fire sale discount increases due to both the price
𝑛𝑒𝑤
informativeness effect and the adverse selection effect when 𝑤𝐴 is reduced to 𝑤𝐴
< 𝑤¯
𝑜𝑙𝑑
from 𝑤𝐴 ≥ 𝑤.
¯
Proof. See Appendix.
Expected price of high-quality asset
1.1
with adverse selection
without adverse selection
1.05
1
0.95
price informativeness effect
0.9
0.85
0.8
0.75
adverse selection effect
0.7
0.65
0.6
0
0.1
0.2
0.3
0.4
0.5
0.6
Arbitrage capital (wA )
0.7
0.8
0.9
1
(i) The expected price of the high-quality asset (𝐸[𝑝|𝑣𝐻 ])
0.85
with adverse selection
without adverse selection
Quality of traded asset (ˆ
ρ)
0.8
0.75
0.7
0.65
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Arbitrage capital (wA )
0.8
0.9
1
(ii) The quality of traded asset (^
𝜌)
Figure 4: The impact of liquidity shocks to arbitrageurs on the expected price and the
supply in case of the high-quality asset (𝑣 = 𝑣𝐻 )
Figure 4 illustrates the impact of liquidity shocks on the supply and the expected
price in case of the high-quality asset. The parameter values are set to be 𝑣𝐻 = 1, 𝑣𝐿 =
0.4, 𝜌 = 0.8, 𝑟𝑓 = 0, 𝛾 = 2, 𝜖 follows a normal distribution with mean zero and variance
(0.25)2 , and 𝑟𝐼 follows a truncated normal distribution with mean 0.5 and variance one
25
on the interval [0.01, 1]. The figure confirms our findings in Proposition 2. When there
is enough liquidity, the marketable asset is in a full supply and the price is always equal
to the fundamental value. As liquidity shock increases, however, the supply decreases
and the expected price falls. That is, market freeze and fire sales occur together due
to adverse selection. It is worth noting that when liquidity shocks get too big, there
is a reversal in the supply and the expected price. This is because adverse selection
completely disappears when there is no liquidity left for the arbitrageurs.
It is worth mentioning about the inverse hump shape of prices with respect to arbitrage capital. As is mentioned in Proposition 2, as arbitrage capital 𝑤𝐴 fall below 𝑤¯𝐴 ,
prices fall and quality becomes poorer than those with 𝑤𝐴 higher than 𝑤
¯𝐴 . However,
prices and the average quality is not monotone in 𝑤𝐴 . The reason is that arbitrageurs
are playing two roles in the economy. First, their informed trading reveals the true value
of 𝑣 to the investors through prices. Second, their informed trading increases adverse
selection in the net supply to the investors. As 𝑤𝐴 decreases, the first effect lowers
prices but the second effect raises prices. Therefore, depending on which effects are
more dominating prices may fall or rise. The first effect usually dominates the second
effect, and this is why prices in general falls as 𝑤𝐴 becomes lower than 𝑤¯𝐴 . However, the
second effect is most pronounced with some drop in 𝑤𝐴 , and that is when prices become
lowest because arbitrageurs now add more harm of adverse selection than the benefit of
revealing the true value. This is a generic feature of a noisy REE model which is not
particular in our model.
Our results contribute to the existing theories of adverse selection in the literature
by embedding informed trading into the framework. A typical model of market for
lemons such as Akerlof (1970) considers a case where prices are set based on the average
quality of the traded assets. Therefore, buyers typically do not learn from prices as
in our noisy REE framework. In our paper, an interesting difference arises from the
traditional adverse selection papers; although both sellers and arbitrageurs are informed
suppliers in the uninformed investors’ point of view, the impact of two participants
may go in opposite directions. The informed supply of sellers create adverse selection
problem, but the informed supply of arbitrageurs can resolve adverse selection. That
is, informed trading resolves adverse selection problem when unconstrained. This is so
because unconstrained informed trading can almost eliminate any information asymmetries by revealing the fundamental value of the traded assets. On the other hand, more
constrained informed trading can increase information asymmetries by making prices less
26
informative. With more mispricing driven by less informative prices, adverse selection
problem can get worsened on average. On the other hand, informed sellers selectively
supply only overvalued assets because it is more profitable. Therefore, uninformed buyers expect that the quality of assets is low, thus market for high quality assets collapses.
The recent literature such as Malherbe (2014) extends such idea by incorporating liquidity constraints for informed sellers. When informed sellers are in distress, they have
no choice but to supply all the assets. Therefore, uninformed buyers expect that the
quality of assets is higher than the case in which sellers are unconstrained, thus market
does not collapse. In this type of models, the lack of liquidity among informed financial
arbitrageurs actually prevents adverse selection instead of creating one.
4.4
Welfare
We can measure the welfare of the economy using the expected total payoffs by all the
existing assets. This includes the marketable asset, the non-marketable assets, and the
risk-free asset. Because the marketable asset merely changes hands through trades, any
change in the welfare arises from the allocations between the non-marketable assets and
the risk-free asset. Recall that the non-marketable assets are ex-ante operating at an
optimal level in the absence of liquidity constraints. Therefore, welfare is maximized
when such inefficient liquidations of the non-marketable assets are minimized. In the
previous subsections, we have shown that the arbitrageurs will trade the marketable
asset according to their liquidity needs when the price is equal to the fundamental
value. Then, it is easy to see that more price informativeness improves welfare by
helping efficient allocations of resources. Conversely, common liquidity shocks will make
price uninformative, thus result in reduction of welfare.
Such reduction in welfare due to inefficient liquidation of the non-marketable assets
is negative externality generated by fire sales. Mispricing caused by fire sales creates
incentives to sacrifice the efficiency of investment in two ways. First, those in liquidity shortage are forced to liquidate their non-marketable assets. Second, even those
in liquidity surplus choose to liquidate their non-marketable assets to raise funds for
speculating on the marketable asset.
We measure the welfare of the economy by the expected production by the sellers as
27
follows:
∫︁
𝑟¯
(𝜌(1 + 𝑟𝐼 )𝑝1𝑋𝑆 (𝑣𝐻 )>0 + (1 − 𝜌)(1 + 𝑟𝐼 )𝑝)𝑓𝐼 (𝑟𝐼 )𝑑𝑟𝐼 .
Π=
(30)
𝑟
We summarize our discussion on welfare in the following corollary that is a direct
consequence of Proposition 2.
Corollary 2. The expected welfare of the economy is maximized if 𝑤𝐴 ≥ 𝑤.
¯
As Proposition 2 implies welfare decreases with more mispricing. That is, liquidity
shocks reduce welfare by misallocating resources. In particular, when the asset quality
is high, liquidity shock creates welfare-reducing fire sales among constrained sellers. We
show this with a numerical example in Figure 5.
1.15
1.1
Expected welfa re (Π)
1.05
1
0.95
0.9
0.85
0.8
0.75
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Arbitrag e capita l (wA )
0.8
0.9
1
Figure 5: Expected welfare (parameter values are identical to the pervious numerical
examples)
5
Conclusion
In our paper, we develop an informed-based theory of fire sales using a noisy REE
framework. Our paper combines limits to arbitrage and adverse selection to provide
a more plausible mechanism of fire sales. In a normal situation when informed market participants are less liquidity-constrained, prices are informative. Thus, this allow
uninformed investors to absorb the supply of assets without worrying about adverse
selection. Therefore, distressed arbitrageurs can sell or buy according to their liquidity
28
needs. On the other hand, when informed market participants are liquidity-constrained
due to market-wide shocks, prices become less informative. This creates an adverse
selection problem, increasing the supply of low-quality assets. This “Lemons” problem
makes well-capitalized uninformed market participants unwilling to absorb the supply,
thereby freezing the market. This explains the “double whammy” situation where fire
sales and adverse selection occur together. Furthermore, we also shed light on the paradoxical nature of fire sales in which capital moves out of the market when it is needed
most and it is likely to have higher returns. Finally, our results have further implications
on welfare, financial fragility, and flight-to-quality.
Appendix
Proof of Lemma 3
Proof. An investor 𝑖 ∈ ℐ maximizes his expected utility:
𝐸𝑈 (𝑤2𝑖 ) = −
]︁
1 [︁
𝑖
𝑖
𝑞(𝑝)𝑒−𝛾[𝑤(1+𝑟𝑓 )+(𝑣𝐻 −𝑝(1+𝑟𝑓 ))𝑥 ] + (1 − 𝑞(𝑝))𝑒−𝛾[𝑤(1+𝑟𝑓 )+(𝑣𝐿 −𝑝(1+𝑟𝑓 ))𝑥 ] ,
𝛾
(A.1)
where 𝑞(𝑝) is the posterior belief of the investor conditional on the price. Because 𝑝 is a
sufficient statistic for 𝜉𝑝 , we have
𝑞(𝑝) = 𝑃 𝑟(𝑣 = 𝑣𝐻 |𝑝, 𝜉𝑝 ).
(A.2)
Then, the first order condition is equal to
[︂
𝜕𝐸𝑈 (𝑤2𝑖 )
𝑖
−𝛾𝑤(1+𝑟𝑓 )
=𝑒
𝑞(𝑝)(𝑣𝐻 − 𝑝(1 + 𝑟𝑓 ))𝑒−𝛾(𝑣𝐻 −𝑝(1+𝑟𝑓 ))𝑥
𝑖
𝜕𝑥
]︂
−𝛾(𝑣𝐿 −𝑝(1+𝑟𝑓 ))𝑥𝑖
+ (1 − 𝑞(𝑝))(𝑣𝐿 − 𝑝(1 + 𝑟𝑓 ))𝑒
= 0.
29
(A.3)
Notice that the second order condition is always satisfied because
[︂
𝜕 2 𝐸𝑈 (𝑤2𝑖 )
𝑖
−𝛾𝑤(1+𝑟𝑓 )
𝑞(𝑝)(𝑣𝐻 − 𝑝(1 + 𝑟𝑓 ))2 𝑒−𝛾(𝑣𝐻 −𝑝(1+𝑟𝑓 ))𝑥
= − 𝛾𝑒
𝑖
2
(𝜕𝑥 )
]︂
2 −𝛾(𝑣𝐿 −𝑝(1+𝑟𝑓 ))𝑥𝑖
+ (1 − 𝑞(𝑝))(𝑣𝐿 − 𝑝(1 + 𝑟𝑓 )) 𝑒
< 0.
(A.4)
Solving Eq. (A.3) for 𝑥𝑖 gives the optimal portfolio given 𝑝 as follows:
[︁ (︁ 𝑞(𝑝) )︁
(︁ 𝑣 − (1 + 𝑟 )𝑝 )︁]︁
1
𝐻
𝑓
𝑥 (𝑝) =
log
+ log
.
𝛾(𝑣𝐻 − 𝑣𝐿 )
1 − 𝑞(𝑝)
(1 + 𝑟𝑓 )𝑝 − 𝑣𝐿
𝑖
(A.5)
By aggregating each individual demand 𝑥𝑖 across all investors in ℐ, we can obtain
the aggregate demand of investors.
Proof of Lemma 4
Proof. According to the initial conjecture that 𝑝 is a sufficient statistic for 𝜉𝑝 , the investors can infer 𝜉𝑝 from 𝑝 to update their beliefs about 𝑣. We define 𝑞 * (𝑝, 𝜉𝑝 ) ≡ 𝑃 𝑟(𝑣 =
𝑣𝐻 |𝑝, 𝜉𝑝 ) to be the investors’ posterior belief conditional on 𝑝 and 𝜉𝑝 . Using Bayes’ rule,
the investors’ posterior beliefs that the marketable asset is of high quality can be derived
as follows:
𝜌^𝑙(𝜉𝑝 ; 𝑣𝐻 , 𝑝)
𝜌^𝑙(𝜉𝑝 ; 𝑣𝐻 , 𝑝) + (1 − 𝜌^)𝑙(𝜉𝑝 ; 𝑣𝐿 , 𝑝)
𝜌^𝑓𝜖 (𝑋(𝑣𝐻 ) − 𝜉𝑝 )
=
,
𝜌^𝑓𝜖 (𝑋(𝑣𝐻 ) − 𝜉𝑝 ) + (1 − 𝜌^)𝑓𝜖 (𝑋(𝑣𝐿 ) − 𝜉𝑝 )
𝑞 * (𝑝, 𝜉𝑝 ) =
(A.6)
where 𝑙(·; 𝑣, 𝑝) is the likelihood function of 𝜉𝑝 given 𝑣 and 𝑝.7
Now, we turn to the second step in which we prove that 𝑝 is indeed a sufficient
statistic for 𝜉𝑝 in equilibrium. Market clearing (see Eq. (6)) together with the definition
7
The numerator of the second expression in Eq. (A.6) is the likelihood of the quantity 𝜉𝑝 and price 𝑝
if the asset is high quality, since 𝜌^ is the probability the asset is high quality and 𝑓𝜖 (𝑋(𝑣𝐻 ) − 𝜉𝑝 ) is the
likelihood of the demand of the noise traders taking exactly the value that offsets a net supply of the
asset to the investors of 𝜉𝑝 , when informed demand takes the value 𝑋(𝑣𝐻 ) that corresponds to informed
traders knowing the asset is of high quality. Similarly the denominator has two terms, one of which is
the same as the numerator and the other being the similar term for the case of low asset quality.
30
of 𝜉𝑝 imply that for any value of 𝜉𝑝 and 𝑝,
𝜉𝑝 = 𝑋𝐼 (𝑝).
(A.7)
Whenever there exists a unique solution for 𝜉𝑝 that solves Eq. (A.7), there exists an
injective mapping from 𝑝 to 𝜉𝑝 . Therefore, the investors can infer 𝜉𝑝 correctly.
Eqs. (15) and (A.7) imply that
𝜉𝑝 =
[︁ (︁ 𝑞 * (𝑝, 𝜉 ) )︁
(︁ 𝑣 − (1 + 𝑟 )𝑝 )︁]︁
1
𝑝
𝐻
𝑓
log
+
log
.
*
𝛾(𝑣𝐻 − 𝑣𝐿 )
1 − 𝑞 (𝑝, 𝜉𝑝 )
(1 + 𝑟𝑓 )𝑝 − 𝑣𝐿
(A.8)
To prove that there exists a unique solution that solves Eq. (A.7), we first claim that
* (𝑝,𝜉 )
𝑝
𝑞 (𝑝, 𝜉𝑝 ) is decreasing in 𝜉𝑝 for any given 𝑝. It is true because 𝜕𝑞 𝜕𝜉
< 0 for all 𝜉𝑝 .
𝑝
*
To see this, we first obtain the first-order derivative of 𝑞 (𝑝, 𝜉𝑝 ) with respect to 𝜉𝑝 from
Eq. (A.6):
*
𝜕𝑞 * (𝑝, 𝜉𝑝 )
𝜕𝜉𝑝
)︀ (A.9)
(︀
𝜌^(1 − 𝜌^) 𝑓𝜖′ (𝑋(𝑣𝐻 ) − 𝜉𝑝 )𝑓𝜖 (𝑋(𝑣𝐿 ) − 𝜉𝑝 ) − 𝑓𝜖 (𝑋(𝑣𝐻 ) − 𝜉𝑝 )𝑓𝜖′ (𝑋(𝑣𝐿 ) − 𝜉𝑝 )
=−
.
(︀
)︀2
𝜌^𝑓𝜖 (𝑋(𝑣𝐻 ) − 𝜉𝑝 ) + (1 − 𝜌^)𝑓𝜖 (𝑋(𝑣𝐿 ) − 𝜉𝑝 )
′
Because 𝑓𝜖′′ 𝑓𝜖 ≤ 𝑓𝜖′ 2 (see Eq. (2)), 𝑓𝑓𝜖𝜖 is non-increasing (this can be verified by differenti′
ating 𝑓𝑓𝜖𝜖 ). Because net supply is larger when the asset is of low quality, 𝑋(𝑣𝐿 ) > 𝑋(𝑣𝐻 ),
and we have
𝑓𝜖′ (𝑋(𝑣𝐿 ) − 𝜉𝑝 )
𝑓 ′ (𝑋(𝑣𝐻 ) − 𝜉𝑝 )
≤ 𝜖
.
(A.10)
𝑓𝜖 (𝑋(𝑣𝐿 ) − 𝜉𝑝 )
𝑓𝜖 (𝑋(𝑣𝐻 ) − 𝜉𝑝 )
Note that by Eq. (A.10), the numerator is non-negative. As the denominator is strictly
* (𝑝,𝜉 )
𝑝
< 0.
positive, 𝜕𝑞 𝜕𝜉
𝑝
Notice that the rhs of Eq. (A.8) is continuous and non-increasing in 𝜉𝑝 and the rhs is
equal to infinity when 𝜉𝑝 ≤ 𝑋(𝑣𝐻 ) − 𝜖¯, and is equal to negative infinity if 𝜉𝑝 ≥ 𝑋(𝑣𝐿 ) − 𝜖.
Because, trivially, the lhs of Eq. (A.8) is continuous and strictly increasing in 𝜉𝑝 , there
(︀
)︀
exists a unique solution on the open interval 𝑋(𝑣𝐻 )−¯𝜖, 𝑋(𝑣𝐿 )−𝜖 that solves Eq. (A.8)
for any given 𝑝.8 That is, there exists a unique injective function that maps 𝑝 to 𝜉𝑝 .
We denote 𝜉(𝑝) to be the unique mapping that maps 𝑝 to 𝜉𝑝 . Then, we have 𝑞(𝑝) =
8
It can be easily verified that the rhs of Eq. (A.8) is non-increasing in 𝜉𝑝 given 𝑝 because it is
decreasing in 𝑞 * , and 𝑞 * (·) is non-decreasing in 𝜉𝑝 at any given level of 𝑝.
31
𝑞 * (𝑝, 𝜉(𝑝)). Using Eq. (A.6), we can finally represent the investors’ posterior belief as a
function of 𝑝 as desired:
𝑞(𝑝) =
𝜌^𝑓𝜖 (𝑋(𝑣𝐻 ) − 𝜉𝑝 )
.
𝜌^𝑓𝜖 (𝑋(𝑣𝐻 ) − 𝜉𝑝 ) + (1 − 𝜌^)𝑓𝜖 (𝜉𝑝 − 𝑋(𝑣𝐿 ))
(A.11)
Proof of Lemma 5
Proof. From the definition of 𝜉𝑝 and 𝑋(𝑣), we have
𝜉𝑝 = 𝑋(𝑣) + 𝜖.
(A.12)
𝑣𝐿
, 𝑣𝐻 ), substituting Eq. (A.12) in to Eq. (??) yields
When 𝑝 ∈ ( 1+𝑟
𝑓 1+𝑟𝑓
𝜌^𝑓𝜖 (𝜖 + 𝑋(𝑣) − 𝑋(𝑣𝐻 ))
𝜌^𝑓𝜖 (𝜖 + 𝑋(𝑣) − 𝑋(𝑣𝐻 )) + (1 − 𝜌^)𝑓𝜖 (𝜖 + 𝑋(𝑣) − 𝑋(𝑣𝐿 ))
𝜌^
=
,
𝜌^ + (1 − 𝜌^)𝑓^𝑧 (𝑝, 𝑣, 𝜖)
𝑞(𝑝) =
where
𝑓𝜖 (𝑝, 𝑣, 𝜖) =
𝑓𝜖 (𝜖 + 𝑋(𝑣) − 𝑋(𝑣𝐿 ))
.
𝑓𝜖 (𝜖 + 𝑋(𝑣) − 𝑋(𝑣𝐻 ))
(A.13)
(A.14)
We can represent Eq. (A.14) equivalently as in Eq. (21) for each case of 𝑣 ∈ {𝑣𝐻 , 𝑣𝐿 }.
Proof of Proposition 1
Proof. We solve for equilibrium in two steps. First, we fix the quality of the traded asset
𝜌^ as given. Then, we solve for a unique price 𝑝 that clears the market at any given level
of 𝑣 and 𝜖. Second, we show that there exists a unique value of 𝜌^ given the expected
price 𝐸[𝑝|𝑣] using the price function derived in the first step.
We start with the first step by fixing 𝜌^. For any given 𝑣 and 𝜖, the investors’ demand
has to be equal to the net supply of the asset due to the market clearing condition in
Eq. (6). Then, we get the following equation that should be satisfied by the equilibrium
32
price 𝑝:
[︁ (︁ 𝑞(𝑝) )︁
(︁ 𝑣 − (1 + 𝑟 )𝑝 )︁]︁
1
𝐻
𝑓
log
+ log
= 1 − 𝑋(𝑣) − 𝜖.
𝛾(𝑣𝐻 − 𝑣𝐿 )
1 − 𝑞(𝑝)
(1 + 𝑟𝑓 )𝑝 − 𝑣𝐿
(A.15)
Substituting Eq. (20) into Eq. (A.15) and solving for 𝑝, we can derive the equilibrium
price as follows:
)︀
1 (︀
𝑝=
𝜔(𝑣, 𝜖)𝑣𝐻 + (1 − 𝜔(𝑣, 𝜖))𝑣𝐿 ,
(A.16)
1 + 𝑟𝑓
where the weight 𝜔(𝑝, 𝑣, 𝜖) is given by
𝜌^
;
^
𝜌^ + (1 − 𝜌^)𝑓𝑧 (𝑣, 𝜖) exp(𝛾(𝑣𝐻 − 𝑣𝐿 )(1 − 𝑋(𝑣) − 𝜖))
𝑓𝜖 (𝜖 − 𝑋(𝑣) + 𝑋(𝑣𝐿 ))
𝑓^𝑧 (𝑣, 𝜖) =
.
𝑓𝜖 (𝜖 − 𝑋(𝑣) + 𝑋(𝑣𝐻 ))
𝜔(𝑣, 𝜖) =
(A.17)
(A.18)
Now, we turn to the second step that proves existence of the equilibrium supply from
the sellers. In equilibrium, the following should be satisfied:
)︀)︀
(︀
(︀ 𝑣
𝐻
−1
𝜌 1 − 𝐹𝐼 𝐸[𝑝|𝑣
𝐻]
)︀)︀
(︀ 𝑣
𝜌^ = (︀
.
𝐻
𝜌 1 − 𝐹𝐼 𝐸[𝑝|𝑣
−
1
+ (1 − 𝜌)
𝐻]
(A.19)
Notice that the rhs is greater than zero when 𝜌^ = 0, and is smaller than 𝜌 when
𝜌^ = 0. Because the rhs is continuous in 𝜌^, there should exist a solution that solves
Eq. (A.19) on a compact set [0, 𝜌]. This finishes the proof.
References
Akerlof, G. A., 1970, “The Market for “Lemons”: Quality Uncertainty and the Market
Mechanism,” Quarterly Journal of Economics, 84(3), 488–500.
Bolton, P., T. Santos, and J. A. Scheinkman, 2009, “Market and Public Liquidity,”
American Economic Review: Papers & Proceedings, 99(2), 594–599.
Brunnermeier, M. K., and L. H. Pedersen, 2009, “Market Liquidity and Funding Liquidity,” Review of Financial Studies, 22(6), 2201–2199.
33
Caballero, R. J., and A. Simsek, 2013, “Fire Sales in a Model of Complexity,” Journal
of Finance, 68(6), 2549–2587.
Coval, J., and E. Stafford, 2007, “Asset fire sales (and purchases) in equity markets,”
Journal of Financial Economics, 86(2), 479–512.
Dang, T. V., G. B. Gorton, and B. Holmstrom, 2012, “Ignorance, Debt and Financial
Crises,” Working paper.
Diamond, D. W., and R. G. Rajan, 2011, “Fear of Fire Sales, Illiquidity Seeking, and
Credit Freezes,” Quarterly Journal of Economics, 126(2), 557–591.
Diamond, D. W., and R. E. Verrecchia, 1981, “Information Aggregation in a Noisy
Rational Expectations Economy,” Journal of Financial Economics, 9(3), 221–235.
Dow, J., and J. Han, Forthcoming, “Contractual Incompleteness, Limited Liability, and
Asset Bubbles,” Journal of Financial Economics.
Duarte, F., and T. M. Eisenbach, 2014, “Fire-Sale Spillovers and Systemic Risk,” Working paper.
Duffie, D., 2010, “Asset Price Dynamics with Slow-Moving Capital,” Journal of Finance,
65(4), 1237–1267.
Ellul, A., C. Jotikasthira, and C. Lundblad, 2011, “Fear of Fire Sales, Illiquidity Seeking,
and Credit Freezes,” Journal of Financial Economics, 101(3), 596–620.
Geanakoplos, J., 2003, “Liquidity, Default, and Crashes: Endogenous Contracts in General Equilibrium,” in Advances in Economics and Econometrics: Theory and Applications, Eighth World Congress, Volume 2, ed. by M. Dewatripont, L. P. Hansen, and
S. J. Turnovsky. Cambridge University Press, Cambridge.
Greenwood, R., A. Landier, and D. Thesmar, 2012, “Vulnerable Banks,” Working paper.
Gromb, D., and D. Vayanos, 2002, “Equilibrium and welfare in markets with financially
constrained arbitrageurs,” Journal of Financial Economics, 66(2-3), 361–407.
Grossman, S. J., and J. E. Stiglitz, 1980, “On the Impossiblity of Informationally Efficient Markets,” American Economic Review, 70(3), 393–408.
34
Guerrieri, V., and R. Shimer, 2014, “Dynamic Adverse Selection: A Theory of Illiquidity,
Fire Sales, and Flight to Quality,” American Economic Review, 104(7), 1875–1908.
Hellwig, M. F., 1980, “On the Aggregation of Information in Competitive Markets,”
Journal of Economic Theory, 22, 477–498.
Jotikasthira, C., C. Lundblad, and T. Ramadorai, 2012, “Asset Fire Sales and Purchases
and the International Transmission of Funding Shocks,” Journal of Finance, 67(6),
2015–2050.
Krishnamurthy, A., 2010, “Amplification Mechanisms in Liquidity Crises,” American
Economic Journal: Macroeconomics, 3(2), 1–30.
Kyle, A. S., 1985, “Continuous Auctions and Insider Trading,” Econometrica, 53(6),
1315–1335.
Malherbe, F., 2014, “Self-Fulfilling Liquidity Dry-Ups,” Journal of Finance, 69(2), 947–
970.
Merrill, C. B., T. Nadauld, R. M. Stulz, and S. M. Sherlund, 2014, “Were There Fire
Sales in the RMBS Market?,” Working paper.
Mitchell, M., L. H. Pedersen, and T. Pulvino, 2007, “Slow Moving Capital,” American
Economic Review, 97(2), 215–220.
Myers, S. C., and N. S. Majluf, 1984, “Corporate financing and investment decisions
when firms have information that investors do not have,” Journal of Financial Economics, 13(2), 187–221.
Pulvino, T. C., 1998, “Do Asset Fire Sales Exist? An Empirical Investigation of Commercial Aircraft Transactions,” Journal of Finance, 53(3), 939–978.
Schlingemann, F. P., R. M. Stulz, and R. A. Walkling, 2002, “Divestitures and the
liquidity of the market for corporate assets,” Journal of Financial Economics, 64(1),
117–144.
Shleifer, A., and R. W. Vishny, 1992, “Liquidation Values and Debt Capacity: A Market
Equilibrium Approach,” Journal of Finance, 47(4), 1343–1366.
, 1997, “The Limits of Arbitrage,” Journal of Finance, 52(1), 35–55.
35
, 2010, “Asset Fire Sales and Credit Easing,” American Economic Review: Papers & Proceedings, 100(2), 46–50.
, 2011, “Fire Sales in Finance and Macroeconomics,” Journal of Economic Perspective, 25(1), 29–48.
Tirole, J., 2011, “Illiquidity and All Its Friends,” Journal of Economic Literature, 49(2),
287–325.
36