· we study flight from liquidity in otc markets . by extending the search -based otc market model...
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Very Preliminary Draft. Please do not cite or circulate without author’s permission.
Search Frictions and Flight from Liquidity in OTC Markets
Jaewon Choi University of Illinois at Urbana-Champaign
Sean Seunghun Shin Aalto University
This Draft: March 2017
Very Preliminary Draft. Please do not cite or circulate without author’s permission.
Search Frictions and Flight from Liquidity in OTC Markets
Abstract
We study flight from liquidity in OTC markets. By extending the search-based OTC market model of Duffie et al. (2007), we show that a liquid security can be priced lower, not higher, than an illiquid security during times of distress. The model features agents simultaneously trading two securities with identical cash flows but different levels of liquidity. The more liquid security is more prone to fire sales because sellers can quickly find buyers of the liquid security. Fire sales can drive the price of liquid security lower than that of the illiquid security if delay in finding buyers is too costly. We also document empirical evidence of flight from liquidity by examining on-the-run and off-the-run corporate bonds of the same issuers and same credit ratings. Consistent with the model, the prices of liquid on-the-run bonds become lower than those of illiquid off-the-run bonds, following liquidity events including periods of severe mutual fund outflows, downgrades of an issuer’s credit rating, and the 2008 financial crisis.
Keywords: Liquidity, Fire Sale, OTC markets, Corporate Bonds
1
1. Introduction
It is commonly accepted that investors flock to liquid securities during times of distress,
commonly referred to as flight to liquidity.1 The preference of investors for liquid securities during
such times would drive the prices of these securities relatively higher than those of illiquid ones.2
Flight to liquidity is arguably most salient features of financial markets and is also widely
documented.
On the other hand, a recent study by Boudoukh, Brooks, Richardson, and Xu (2016) suggest
that investors might flight from liquidity rather than to liquidity. They document that during the
recent European sovereign debt crisis, liquid European sovereign bond prices fell more than
illiquid bonds.3 Presumably, traders who want to minimize liquidation costs are likely to sell
liquid securities more, because those are easier to liquidate during distress times.4 Likewise, liquid
securities can be more prone to fire sales. It is particularly interesting to provide a theoretical
framework for over-the-counter (OTC) markets that would characterize the price implications of
fire sales for investors given with securities with varying degrees of market liquidity.
In this paper, we provide a theoretical framework that characterizes flight from liquidity in
OTC markets. In particular, we focus on the following features of the OTC markets to study flight
from liquidity. First, trading delays caused by high search frictions in finding counterparties in the
1 See, e.g., Longstaff (2004), Vayanos (2004), Beber, Brandt, and Kavajecz (2009), Goyenko, Subrahmanyam, and Ukhov (2011), Acharya, Amihud, and Bharath (2013), among many others. 2 For example, liquid on-the-run U.S. Treasury bonds became dramatically expensive relative to illiquid off-the-run counterparts due to worldwide flights to liquidity following the 1998 Russian default, which eventually caused the debacle of Long-Term Capital Management. See, for example, Scholes (2000) for the detailed account of the event. 3 Boudoukh, Brooks, Richardson, and Xu (2016) document that on-the-run government bonds of some European countries have become cheaper than off-the-run bonds during the recent European sovereign debt crisis 4 See, e.g., Scholes (2000), Coval and Stafford (2007), Ellul, Jotikasthira, and Lundblad (2011), Jotikasthira, Lundblad, and Ramadorai (2012), Manconi, Massa, and Yasuda (2012), Choi and Shin (2016), among many others.
2
OTC markets can amplify the price impacts of unexpected liquidity shocks.5 Second, there is a
recent empirical evidence by Boudoukh et al. (2016) suggesting the flight from liquidity in
sovereign bond markets. We attempt to provide a theoretical explanation for pricing mechanisms
behind these results and complement empirical evidence on the flight from liquidity by examining
corporate bond markets.
We first develop a theoretical framework to describe underlying mechanisms behind the flight
from liquidity. Our model is unique in that there are two assets with identical cash flows but with
different levels of liquidity in an OTC market. Hence, our model is an extension of the search-
based model of Duffie, Gârleanu, and Pedersen (2007) (hereafter, the DGP model), which is the
standard model for valuation in the OTC markets.
The novel implication of the model is that prices can be lower for liquid securities than for
illiquid securities following market-wide liquidity shocks. The key driving force behind this result
is the search-based nature of OTC markets. To trade a security in the OTC markets, an investor
has to identify a counterparty. When there are severe needs for liquidity, delay in searching is very
costly. Thus, investors are willing to sell liquid securities at discount, even at the price level lower
than illiquid securities. Buyers can take advantage of the discount due to heavy selling but prices
do not converge quickly to the fundamental value. First, the relative number of buyers is small
during times of distress. Second, due to search frictions, transactions do not occur immediately
and will take longer. Thus, price convergence can take a considerable period of time following a
liquidity shock. In normal times, however, the liquid security is priced higher than the illiquid
securities because of the liquidity premium.
5 See, for example, Duffie (2010).
3
We then examine the empirical implications of the model using corporate bond data from the
Trade Reporting and Compliance Engine (TRACE). We find that yields of more liquid bonds
increase more than less liquid bonds with almost identical cash flows following fire sale events,
such as large redemption requests received by the mutual fund industry, downgrades of an issuer’s
credit rating, or the recent financial crisis in 2008. Specifically, we examine yield spreads of an
issuer’s corporate bond pairs with similar maturities and same credit rating but different ages (illiq-
liq spreads henceforth) that tend to have different levels of liquidity.6
Our results show that the illiq-liq spreads become significantly negative following a liquidity
event. For example, the average illiq-liq spread in October 2008 was -0.4% following the
announcement of Lehman Brothers’ bankruptcy, suggesting that prices of liquid bonds were
cheaper than those of illiquid bonds during the time of distress in October 2008. This is consistent
with the model’s implication.
We also find that illiq-liq spreads are strongly positively related to net investment cash flows
to corporate bond mutual funds that hold sample corporate bond pairs used in the analyses, based
on a vector autoregression (VAR) framework. Specifically, a one standard deviation shock to
monthly investor outflows significantly decreases the illiq-liq spread by more than 70% of the
illiq-liq spread’s standard deviation over seven months following the shock. This result also
supports the model’s implication in that net investment outflows can force institutional investors
to sell liquid corporate bonds to meet their increased redemption requests.
Finally, we find that illiq-liq spreads become significantly negative after downgrades of
6 Similar approaches are used in other studies. See, e.g., Crabbe and Turner (1995), Helwege and Turner (1999), Helwege, Huang, and Wang (2014), among others.
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issuers’ credit ratings. When an issuer of a bond pair experiences a credit deterioration event, there
is large selling pressure to the bonds issued by the issuer. Consistent with the model’s prediction,
illiq-liq spreads become negative after downgrades and recover to non-negative levels later, being
consistent with the model’s implication.
This paper contributes to the literature on price pressures in bond markets (Greenwood and
Vayanos 2010, Ellul, Jotikasthira, and Lundblad 2011, Manconi, Massa, and Yasuda 2012,
D’Amico and King 2013, Goldstein, Jiang, and Ng 2015, Boudoukh, Brooks, Richardson, and Xu
2016, Choi and Shin 2016, Helwege and Wang 2016). Especially, this paper provides the
mechanism and rationale behind recent findings of Brooks, Richardson, and Xu (2016) that liquid
government bonds become cheaper during times of distress.
This paper also contributes to the literature on the pricing of liquidity (Amihud and
Mendelson 1988, Acharya and Pedersen 2005, among many others), the liquidity premium of
corporate bonds (Chen, Lesmond, and Wei 2007, Lin, Wang, and Wu 2011, Jong and Driessen
2012, Acharya, Amihud, and Bharath 2013, among many others), and that of sovereign bonds
(Cornell and Shapiro 1989, Amihud and Mendelson 1991, Longstaff 2004, Pasquariello and Vega
2009, Favero, Pagano, and Von Thadden 2010, Goyenko, Subrahmanyam and Ukhov 2011, among
many others) by documenting the seemingly counter-intuitive situation in relative prices of liquid
securities during times of flight from liquidity.
In addition, this paper contributes to the literature on the search-based asset-pricing (Duffie,
Gârleanu, and Pedersen 2005, Duffie, Gârleanu, and Pedersen 2007, Vayanos and Wang 2008,
Vayanos and Weill 2008, Weill 2008, Lagos and Rocheteau 2009, Lagos, Rocheteau, and Weill
2009, Feldhütter 2012, He and Milbradt 2014, among many others). Our model provides the novel
result that a flight from liquidity can occur during times of distress. Several papers also model
5
search-based markets with multiple assets, but their focuses and implications differ greatly from
those considered in this paper.7 One obvious advantage of our model is to provide a framework
in which liquid and illiquid securities in an OTC market can be fairly compared with each other.
Finally, to the best of our knowledge, our paper is the first paper to document the empirical
evidence on the flight from liquidity in the corporate bond markets.
The remainder of this paper is structured as follows. We develop the model in Section 2. In
Section 3, we present the results of the model, and in Section 4, we provide numerical examples
of those results. In Section 5, we discuss the model implications. In Section 6, we present empirical
evidence. We then conclude the study in Section 7.
2. The Model
In this section, we develop a model that extends the framework of valuation in the OTC
markets adopted by Duffie et al. (2007). The model helps to describe the mechanisms that make
liquid bonds relatively cheaper during times of distress.
2.1 Two Bonds with Different Liquidity in the Search-based Secondary Market
7 Vayanos and Wang (2008) document the concentration of investors toward the liquidity where there are two assets traded respectively in two segmented markets. Vayanos and Weill (2008) build a model which explains the on-the-run premia by introducing a search repo market for short positions. Weill (2008) documents the positive relationship between the liquidity on the expected returns by suggesting free float as a measure of illiquidity. In the model of Weill (2008), there are multiple assets traded but agents have to choose only one asset to trade and hold. None of previous three studies focus on the distressed market. By contrast, our model allow agents hold and trade two different assets simultaneously under the presence of aggregate shock.
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There are two long-lived bonds in the search-based secondary market. In this OTC market,
all bonds are bilaterally traded by risk-neutral agents. Both bonds are consols that pay a unit of
consumption per a unit of time. The two bonds are identical except for their liquidity. The liquidity
is modeled based on a search friction. To sell (or buy) a bond, agents must find a counterparty, and
this search follows the Poisson process. The Poisson intensity of searching for one bond (hereafter
bond 1) is 𝜆𝜆1 and that of searching for another bond (hereafter bond 2) is 𝜆𝜆2. We assume bond 1
is more liquid than bond 2. In other words, bond 1 has a higher counterparty search intensity than
bond 2 (i.e., 𝜆𝜆1 > 𝜆𝜆2).
The search processes of any agents for any bonds are independent. When an agent
successfully finds a counterparty, they can negotiate and trade one bond at a time at most, and
whether it is bond 1 or 2 is predetermined by the search steps. For example, if the counterparty is
searched during the searching process of bond 1 (with intensity 𝜆𝜆1), the agent can trade only bond
1, not bond 2.
2.2 Agents with Intrinsic Preference
As in Duffie et al. (2007), agents have two intrinsic type (high and low), indicating their
preference for bond ownership. H- (high) type investors prefer owning bonds, whereas L- (low)
type investors do not. Thus, L-type owners want to sell and H-type non-owners want to buy. The
trade is only successful between the L-type owner agent and H-type non-owner agent. L-type
agents can be viewed as those who have a need for cash (low liquidity) or a relatively lower use
for the bonds. This low preference is modeled as a bondholding cost 𝛿𝛿 that only L-type agents
must pay for owning a bond.
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The agent type can be switched by independent and identically distributed (i.i.d.)
idiosyncratic shocks with intensity 𝜆𝜆𝑑𝑑 (switching from H to L) or 𝜆𝜆𝑢𝑢 (switching from L to H).
The idiosyncratic shock can be viewed as a liquidity shock, such as a sudden redemption request
that forces the agent to sell the bond or a preference shock that makes the agent want to sell the
bond.
For simplicity, we assume that each agent can hold either 0 or 1 unit of each bond. Thus, no
short sales are allowed. To denote an agent’s type and ownership status for bonds 1 and 2, we use
three-digit type codes. For example, we assign the status code “hon” to H agents who currently
own bond 1 and do not own bond 2. The full set of type codes is 𝑇𝑇 =
{ℎ𝑜𝑜𝑜𝑜,ℎ𝑜𝑜𝑜𝑜,ℎ𝑜𝑜𝑜𝑜,ℎ𝑜𝑜𝑜𝑜, 𝑙𝑙𝑜𝑜𝑜𝑜, 𝑙𝑙𝑜𝑜𝑜𝑜, 𝑙𝑙𝑜𝑜𝑜𝑜, 𝑙𝑙𝑜𝑜𝑜𝑜}. The first letter, “h” or “l”, denotes an agent’s intrinsic
type H or L, respectively. The second two letters, “o” and “n”, denote an owner or a non-owner of
bond 1, respectively. The final letters, “o” and “n”, denote an owner or a non-owner of bond 2,
respectively. This code is not permanent for an agent. As an agent trades bonds and experiences
idiosyncratic shocks, her code can be changed occasionally. From now on, the “type” of agent
refers to one of the eight type codes in T and the “intrinsic type” of agent refers to the agent’s
preference status (i.e., H or L).
2.3 Bilateral Trades through Bargaining
When an agent finds a counterparty for a bond, a transaction price is determined by bargaining
for the bond between two agents. This bargaining process happens immediately and agents go back
to the searching queue immediately. Following Duffie et al. (2007), we assume a Nash bargaining
game (e.g., Nash, 1950) between two agents under the exogenously given bargaining power. This
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implies that the outcome of bargaining (i.e., price) is determined by the utility changes of the seller
and buyer and the bargaining power. Specifically,
𝑃𝑃 = ∆𝑉𝑉𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆(1 − 𝑞𝑞) + ∆𝑉𝑉𝐵𝐵𝑢𝑢𝐵𝐵𝑆𝑆𝑆𝑆𝑞𝑞 (1)
where 𝑞𝑞 ∈ [0, 1] is the bargaining power of the seller, ∆𝑉𝑉𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 is the change in the seller’s utility
from the transaction, and ∆𝑉𝑉𝐵𝐵𝑢𝑢𝐵𝐵𝑆𝑆𝑆𝑆 is the change in the buyer’s utility from the transaction. For
example, if the seller has all of the bargaining power, the buyer must fully pay her utility gain.
Agent utility is specified in Section 3.
2.4 Dynamics in the Fraction of Agents
There is a continuum of agents of mass one,
𝑖𝑖. 𝑒𝑒. , ∑ 𝜇𝜇𝜎𝜎𝜎𝜎∈𝑇𝑇 (𝑡𝑡) = 1 for ∀ t (2)
where 𝜇𝜇𝜎𝜎(𝑡𝑡) denotes the fraction of type 𝜎𝜎 ∈ 𝑇𝑇 agents at time t.
The total fractions of 𝑠𝑠1 and 𝑠𝑠2 agents are initially endowed bonds 1 and 2, respectively.
𝑖𝑖. 𝑒𝑒. , ∑ 𝜇𝜇𝜎𝜎𝜎𝜎∈𝑇𝑇𝑜𝑜1 (𝑡𝑡) = 𝑠𝑠1 for ∀ t (3)
∑ 𝜇𝜇𝜎𝜎𝜎𝜎∈𝑇𝑇𝑜𝑜2 (𝑡𝑡) = 𝑠𝑠2 for ∀ t (4)
where the subsets of agents owning bond 1 is defined as 𝑇𝑇𝑜𝑜1 = {ℎ𝑜𝑜𝑜𝑜,ℎ𝑜𝑜𝑜𝑜, 𝑙𝑙𝑜𝑜𝑜𝑜, 𝑙𝑙𝑜𝑜𝑜𝑜} and that
owning bond 2 is defined as 𝑇𝑇𝑜𝑜2 = {ℎ𝑜𝑜𝑜𝑜, ℎ𝑜𝑜𝑜𝑜, 𝑙𝑙𝑜𝑜𝑜𝑜, 𝑙𝑙𝑜𝑜𝑜𝑜}.
The type of agent can be changed by either the idiosyncratic liquidity shock (with intensity
𝜆𝜆𝑢𝑢 or 𝜆𝜆𝑑𝑑) or the successful trade. First, type lxy and hxy (𝑥𝑥,𝑦𝑦 ∈ {𝑜𝑜,𝑜𝑜}) agents become type hxy
9
and lxy agents from the shock with intensity 𝜆𝜆𝑢𝑢 and 𝜆𝜆𝑑𝑑, respectively. Therefore, for example, loo
agents become hoo agents at a rate of 𝜆𝜆𝑢𝑢𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡). Second, L-type bond 1 (or bond 2) owners
become L-type non-owners after they sell bond 1 (or bond 2). For example, when a loo agent and
an hno agent meet for bond 1, the loo agent becomes an lno agent and the hno agent becomes an
hoo agent. We assume the counterparty is searched from the entire pool of agents and that the
probability of being searched is the same for all agents. Thus, the total rate of transactions between
loo and hno agents for bond 1 is 2𝜆𝜆1𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡)𝜇𝜇ℎ𝑛𝑛𝑜𝑜(𝑡𝑡). Without loss of generality, the rate of
transactions between any two types of agent is similarly calculated.
As a result, the fractions of each type of agent follow the dynamics described by the following
eight equations.
��𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡) = 2𝜆𝜆1𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡)𝜇𝜇ℎ𝑛𝑛𝑜𝑜(𝑡𝑡) + 2𝜆𝜆1𝜇𝜇𝑆𝑆𝑜𝑜𝑛𝑛(𝑡𝑡)𝜇𝜇ℎ𝑛𝑛𝑜𝑜(𝑡𝑡) + 2𝜆𝜆2𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡)𝜇𝜇ℎ𝑜𝑜𝑛𝑛(𝑡𝑡) + 2𝜆𝜆2𝜇𝜇𝑆𝑆𝑛𝑛𝑜𝑜(𝑡𝑡)𝜇𝜇ℎ𝑜𝑜𝑛𝑛(𝑡𝑡) − 𝜆𝜆𝑑𝑑𝜇𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡) + 𝜆𝜆𝑢𝑢𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡)
(5)
��𝜇ℎ𝑜𝑜𝑛𝑛(𝑡𝑡) = 2𝜆𝜆1𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡)𝜇𝜇ℎ𝑛𝑛𝑛𝑛(𝑡𝑡) + 2𝜆𝜆1𝜇𝜇𝑆𝑆𝑜𝑜𝑛𝑛(𝑡𝑡)𝜇𝜇ℎ𝑛𝑛𝑛𝑛(𝑡𝑡) − 2𝜆𝜆2𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡)𝜇𝜇ℎ𝑜𝑜𝑛𝑛(𝑡𝑡) − 2𝜆𝜆2𝜇𝜇𝑆𝑆𝑛𝑛𝑜𝑜(𝑡𝑡)𝜇𝜇ℎ𝑜𝑜𝑛𝑛(𝑡𝑡) − 𝜆𝜆𝑑𝑑𝜇𝜇ℎ𝑜𝑜𝑛𝑛(𝑡𝑡) + 𝜆𝜆𝑢𝑢𝜇𝜇𝑆𝑆𝑜𝑜𝑛𝑛(𝑡𝑡)
(6)
��𝜇ℎ𝑛𝑛𝑜𝑜(𝑡𝑡) = 2𝜆𝜆2𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡)𝜇𝜇ℎ𝑛𝑛𝑛𝑛(𝑡𝑡) + 2𝜆𝜆2𝜇𝜇𝑆𝑆𝑛𝑛𝑜𝑜(𝑡𝑡)𝜇𝜇ℎ𝑛𝑛𝑛𝑛(𝑡𝑡) − 2𝜆𝜆1𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡)𝜇𝜇ℎ𝑛𝑛𝑜𝑜(𝑡𝑡)− 2𝜆𝜆1𝜇𝜇𝑆𝑆𝑜𝑜𝑛𝑛(𝑡𝑡)𝜇𝜇ℎ𝑛𝑛𝑜𝑜(𝑡𝑡) − 𝜆𝜆𝑑𝑑𝜇𝜇ℎ𝑛𝑛𝑜𝑜(𝑡𝑡) + 𝜆𝜆𝑢𝑢𝜇𝜇𝑆𝑆𝑛𝑛𝑜𝑜(𝑡𝑡)
(7)
��𝜇ℎ𝑛𝑛𝑛𝑛(𝑡𝑡) = −2𝜆𝜆1𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡)𝜇𝜇ℎ𝑛𝑛𝑛𝑛(𝑡𝑡) − 2𝜆𝜆2𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡)𝜇𝜇ℎ𝑛𝑛𝑛𝑛(𝑡𝑡) − 2𝜆𝜆1𝜇𝜇𝑆𝑆𝑜𝑜𝑛𝑛(𝑡𝑡)𝜇𝜇ℎ𝑛𝑛𝑛𝑛(𝑡𝑡)− 2𝜆𝜆2𝜇𝜇𝑆𝑆𝑛𝑛𝑜𝑜(𝑡𝑡)𝜇𝜇ℎ𝑛𝑛𝑛𝑛(𝑡𝑡) − 𝜆𝜆𝑑𝑑𝜇𝜇ℎ𝑛𝑛𝑛𝑛(𝑡𝑡) + 𝜆𝜆𝑢𝑢𝜇𝜇𝑆𝑆𝑛𝑛𝑛𝑛(𝑡𝑡)
(8)
��𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡) = −2𝜆𝜆1𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡)𝜇𝜇ℎ𝑛𝑛𝑜𝑜(𝑡𝑡) − 2𝜆𝜆1𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡)𝜇𝜇ℎ𝑛𝑛𝑛𝑛(𝑡𝑡) − 2𝜆𝜆2𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡)𝜇𝜇ℎ𝑜𝑜𝑛𝑛(𝑡𝑡)− 2𝜆𝜆2𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡)𝜇𝜇ℎ𝑛𝑛𝑛𝑛(𝑡𝑡) + 𝜆𝜆𝑑𝑑𝜇𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡) − 𝜆𝜆𝑢𝑢𝜇𝜇𝑆𝑆𝑛𝑛𝑛𝑛(𝑡𝑡)
(9)
��𝜇𝑆𝑆𝑜𝑜𝑛𝑛(𝑡𝑡) = −2𝜆𝜆1𝜇𝜇𝑆𝑆𝑜𝑜𝑛𝑛(𝑡𝑡)𝜇𝜇ℎ𝑛𝑛𝑜𝑜(𝑡𝑡) − 2𝜆𝜆1𝜇𝜇𝑆𝑆𝑜𝑜𝑛𝑛(𝑡𝑡)𝜇𝜇ℎ𝑛𝑛𝑛𝑛(𝑡𝑡) + 2𝜆𝜆2𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡)𝜇𝜇ℎ𝑜𝑜𝑛𝑛(𝑡𝑡)+ 2𝜆𝜆2𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡)𝜇𝜇ℎ𝑛𝑛𝑛𝑛(𝑡𝑡) + 𝜆𝜆𝑑𝑑𝜇𝜇ℎ𝑜𝑜𝑛𝑛(𝑡𝑡) − 𝜆𝜆𝑢𝑢𝜇𝜇𝑆𝑆𝑜𝑜𝑛𝑛(𝑡𝑡)
(10)
��𝜇𝑆𝑆𝑛𝑛𝑜𝑜(𝑡𝑡) = 2𝜆𝜆1𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡)𝜇𝜇ℎ𝑛𝑛𝑜𝑜(𝑡𝑡) + 2𝜆𝜆1𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡)𝜇𝜇ℎ𝑛𝑛𝑛𝑛(𝑡𝑡) − 2𝜆𝜆2𝜇𝜇𝑆𝑆𝑛𝑛𝑜𝑜(𝑡𝑡)𝜇𝜇ℎ𝑜𝑜𝑛𝑛(𝑡𝑡)− 2𝜆𝜆2𝜇𝜇𝑆𝑆𝑛𝑛𝑜𝑜(𝑡𝑡)𝜇𝜇ℎ𝑛𝑛𝑛𝑛(𝑡𝑡) + 𝜆𝜆𝑑𝑑𝜇𝜇ℎ𝑛𝑛𝑜𝑜(𝑡𝑡) − 𝜆𝜆𝑢𝑢𝜇𝜇𝑆𝑆𝑛𝑛𝑜𝑜(𝑡𝑡)
(11)
��𝜇𝑆𝑆𝑛𝑛𝑛𝑛(𝑡𝑡) = 2𝜆𝜆1𝜇𝜇𝑆𝑆𝑜𝑜𝑛𝑛(𝑡𝑡)𝜇𝜇ℎ𝑛𝑛𝑜𝑜(𝑡𝑡) + 2𝜆𝜆1𝜇𝜇𝑆𝑆𝑜𝑜𝑛𝑛(𝑡𝑡)𝜇𝜇ℎ𝑛𝑛𝑛𝑛(𝑡𝑡) + 2𝜆𝜆2𝜇𝜇𝑆𝑆𝑛𝑛𝑜𝑜(𝑡𝑡)𝜇𝜇ℎ𝑜𝑜𝑛𝑛(𝑡𝑡)+ 2𝜆𝜆2𝜇𝜇𝑆𝑆𝑛𝑛𝑜𝑜(𝑡𝑡)𝜇𝜇ℎ𝑛𝑛𝑛𝑛(𝑡𝑡) + 𝜆𝜆𝑑𝑑𝜇𝜇ℎ𝑛𝑛𝑛𝑛(𝑡𝑡) − 𝜆𝜆𝑢𝑢𝜇𝜇𝑆𝑆𝑛𝑛𝑛𝑛(𝑡𝑡)
(12)
10
Each equation represents the changes in fraction of one type of agent. The first two terms represent
trades of bond 1. The next two terms represent trades of bond 2. The final two terms represent the
idiosyncratic shocks that change the agents’ intrinsic types.
The following proposition states that there is a unique, stable steady-state equilibrium for
the fractions of agents regardless of the initial values.
Proposition 1. Existence of Unique Stable Steady-state Solution
For ∀ μ(0)∈[0,1]8, satisfying assumptions in Equations (2)-(4), there exists a unique steady-
state solution μ∈[0,1]8 to the system of Equations (5)-(12) and μ(t) μ as t ∞ where μ(t) is
the 8 x 1 vector for the fractions of eight agent types.
The proof is in Appendix A.
2.5 Times of Distress: Aggregate Shocks
Although the price of bond 1 (the more liquid one) is expected to be higher in the normal
period, we are interested in how the prices of bonds 1 and 2 are affected during times of distress.
The time of distress is modeled as the aggregate shocks that make the majority of agents
simultaneously want to sell their bonds. The aggregate shocks can be viewed as the aggregate
liquidity shocks or aggregate preference shocks. The aggregate shock follows the Poisson process
with intensity 𝜁𝜁 . 𝜁𝜁 is exogenously given, and this process is independent from any other
11
processes in the model. The shocks affect agents’ intrinsic type. When the aggregate shock arrivals,
a fraction of H agents become L agents, regardless of their bond ownerships.8 This represents the
sudden reduction in liquidity of H agents or sudden changes in preference of H agents. The
preference of L agents is not directly affected by the aggregate shocks.
When the aggregate shock arrives, we assume that the distribution of agents becomes 𝝁𝝁∗ ∈
[0, 1],8 which is determined by the fraction of abnormally increased L agents and the steady-
state equilibrium (𝝁𝝁�) in Proposition 1. Specifically,
𝜇𝜇ℎ𝑥𝑥𝐵𝐵∗ = �1 − 𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ ���𝜇ℎ𝑥𝑥𝐵𝐵 (13)
𝜇𝜇𝑆𝑆𝑥𝑥𝐵𝐵∗ = ��𝜇𝑆𝑆𝑥𝑥𝐵𝐵 + 𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ ��𝜇ℎ𝑥𝑥𝐵𝐵 (14)
where 𝑥𝑥,𝑦𝑦 ∈ {𝑜𝑜,𝑜𝑜}. The 𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ in Equation (13) represents the extent of H agents in the steady-
state equilibrium affected by aggregate liquidity shocks. Thus, it represents the size of the
aggregate shocks. For simplicity, we assume that it is exogenously given. The post-shock
distribution is not time dependent. This means that regardless of the arrival timings of the shock,
the post-shock distribution is always 𝝁𝝁∗, which is determined by the equilibrium mass and size of
shocks as described in Equations (13) and (14). This simplifying assumption is needed for the
tractability of the model. We define 𝜋𝜋ℎ𝑥𝑥𝐵𝐵(𝑡𝑡) as the probability that the aggregate shock changes
H agents to L agents given that the shock arrived at t. Then, 𝜋𝜋ℎ𝑥𝑥𝐵𝐵(𝑡𝑡) is determined as follows:
𝜋𝜋ℎ𝑥𝑥𝐵𝐵(𝑡𝑡) = 1 −𝜇𝜇ℎ𝑥𝑥𝑥𝑥∗
𝜇𝜇ℎ𝑥𝑥𝑥𝑥(𝑡𝑡) (15)
8 The agent type can be also changed by the aggregate shock. However, this is not considered in deriving Equation (5)-(12), because when the aggregate shock arrives the process of agent distribution is initialized to 𝝁𝝁∗ by the assumption. In other words, new dynamics begin with 𝝁𝝁(0) = 𝝁𝝁∗ if the aggregate shock arrives.
12
where 𝑥𝑥,𝑦𝑦 ∈ {𝑜𝑜,𝑜𝑜}.
3. The Solutions
3.1 Lifetime Utility of Agents
Through this model, we are interested in deriving the prices of bonds 1 and 2. To do that, we
take the following two steps. First, we derive an agent distribution process, which is described in
Equations (5)-(12). Then, given the distribution of agents at time t, we derive the dynamics in the
lifetime utility of the agents. The derivation is detailed in Appendix B. As a result, the utility of
the agents follows the dynamics of the following eight equations. Each of the eight equations
represents the changes in utility of each type of agent at time t.
��𝑉ℎ𝑜𝑜𝑜𝑜 = 𝑟𝑟𝑉𝑉ℎ𝑜𝑜𝑜𝑜 − 𝜆𝜆𝑑𝑑(𝑉𝑉𝑆𝑆𝑜𝑜𝑜𝑜 − 𝑉𝑉ℎ𝑜𝑜𝑜𝑜) − 𝜁𝜁�(1 − 𝜋𝜋ℎ𝑜𝑜𝑜𝑜)𝑉𝑉ℎ𝑜𝑜𝑜𝑜(0) + 𝜋𝜋ℎ𝑜𝑜𝑜𝑜𝑉𝑉𝑆𝑆𝑜𝑜𝑜𝑜(0) − 𝑉𝑉ℎ𝑜𝑜𝑜𝑜� − 2 (16)
��𝑉ℎ𝑜𝑜𝑛𝑛 = 𝑟𝑟𝑉𝑉ℎ𝑜𝑜𝑛𝑛 − 𝜆𝜆𝑑𝑑(𝑉𝑉𝑆𝑆𝑜𝑜𝑛𝑛 − 𝑉𝑉ℎ𝑜𝑜𝑛𝑛) − 2𝜆𝜆2𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜�𝑉𝑉ℎ𝑜𝑜𝑜𝑜 − 𝑉𝑉ℎ𝑜𝑜𝑛𝑛 − 𝑃𝑃𝑆𝑆𝑜𝑜𝑜𝑜ℎ𝑜𝑜𝑛𝑛,2� −2𝜆𝜆2𝜇𝜇𝑆𝑆𝑛𝑛𝑜𝑜�𝑉𝑉ℎ𝑜𝑜𝑜𝑜 − 𝑉𝑉ℎ𝑜𝑜𝑛𝑛 − 𝑃𝑃𝑆𝑆𝑛𝑛𝑜𝑜ℎ𝑜𝑜𝑛𝑛,2� − 𝜁𝜁�(1 − 𝜋𝜋ℎ𝑜𝑜𝑛𝑛)𝑉𝑉ℎ𝑜𝑜𝑛𝑛(0) + 𝜋𝜋ℎ𝑜𝑜𝑛𝑛𝑉𝑉𝑆𝑆𝑜𝑜𝑛𝑛(0) −𝑉𝑉ℎ𝑜𝑜𝑛𝑛� − 1
(17)
��𝑉ℎ𝑛𝑛𝑜𝑜 = 𝑟𝑟𝑉𝑉ℎ𝑛𝑛𝑜𝑜 − 𝜆𝜆𝑑𝑑(𝑉𝑉𝑆𝑆𝑛𝑛𝑜𝑜 − 𝑉𝑉ℎ𝑛𝑛𝑜𝑜) − 2𝜆𝜆1𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜�𝑉𝑉ℎ𝑜𝑜𝑜𝑜 − 𝑉𝑉ℎ𝑛𝑛𝑜𝑜 − 𝑃𝑃𝑆𝑆𝑜𝑜𝑜𝑜ℎ𝑛𝑛𝑜𝑜,1� −2𝜆𝜆1𝜇𝜇𝑆𝑆𝑜𝑜𝑛𝑛�𝑉𝑉ℎ𝑜𝑜𝑜𝑜 − 𝑉𝑉ℎ𝑛𝑛𝑜𝑜 − 𝑃𝑃𝑆𝑆𝑛𝑛𝑜𝑜ℎ𝑛𝑛𝑜𝑜,1� − 𝜁𝜁�(1 − 𝜋𝜋ℎ𝑛𝑛𝑜𝑜)𝑉𝑉ℎ𝑛𝑛𝑜𝑜(0) + 𝜋𝜋ℎ𝑛𝑛𝑜𝑜𝑉𝑉𝑆𝑆𝑛𝑛𝑜𝑜(0) −𝑉𝑉ℎ𝑛𝑛𝑜𝑜� − 1
(18)
��𝑉ℎ𝑛𝑛𝑛𝑛 = 𝑟𝑟𝑉𝑉ℎ𝑛𝑛𝑛𝑛 − 𝜆𝜆𝑑𝑑(𝑉𝑉𝑆𝑆𝑛𝑛𝑛𝑛 − 𝑉𝑉ℎ𝑛𝑛𝑛𝑛) − 2𝜆𝜆1𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜�𝑉𝑉ℎ𝑜𝑜𝑛𝑛 − 𝑉𝑉ℎ𝑛𝑛𝑛𝑛 − 𝑃𝑃𝑆𝑆𝑜𝑜𝑜𝑜ℎ𝑛𝑛𝑛𝑛,1� −2𝜆𝜆1𝜇𝜇𝑆𝑆𝑜𝑜𝑛𝑛�𝑉𝑉ℎ𝑜𝑜𝑛𝑛 − 𝑉𝑉ℎ𝑛𝑛𝑛𝑛 − 𝑃𝑃𝑆𝑆𝑜𝑜𝑛𝑛ℎ𝑛𝑛𝑛𝑛,1� − 2𝜆𝜆2𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜�𝑉𝑉ℎ𝑛𝑛𝑜𝑜 − 𝑉𝑉ℎ𝑛𝑛𝑛𝑛 − 𝑃𝑃𝑆𝑆𝑜𝑜𝑜𝑜ℎ𝑜𝑜𝑛𝑛,2� −2𝜆𝜆2𝜇𝜇𝑆𝑆𝑛𝑛𝑜𝑜�𝑉𝑉ℎ𝑛𝑛𝑜𝑜 − 𝑉𝑉ℎ𝑛𝑛𝑛𝑛 − 𝑃𝑃𝑆𝑆𝑛𝑛𝑜𝑜ℎ𝑛𝑛𝑛𝑛,2� − 𝜁𝜁�(1 − 𝜋𝜋ℎ𝑛𝑛𝑛𝑛)𝑉𝑉ℎ𝑛𝑛𝑛𝑛(0) + 𝜋𝜋ℎ𝑛𝑛𝑛𝑛𝑉𝑉𝑆𝑆𝑛𝑛𝑛𝑛(0) −𝑉𝑉ℎ𝑛𝑛𝑛𝑛�
(19)
��𝑉𝑆𝑆𝑜𝑜𝑜𝑜 = 𝑟𝑟𝑉𝑉𝑆𝑆𝑜𝑜𝑜𝑜 − 𝜆𝜆𝑢𝑢(𝑉𝑉ℎ𝑜𝑜𝑜𝑜 − 𝑉𝑉𝑆𝑆𝑜𝑜𝑜𝑜) − 2𝜆𝜆1𝜇𝜇ℎ𝑛𝑛𝑜𝑜�𝑉𝑉𝑆𝑆𝑛𝑛𝑜𝑜 − 𝑉𝑉𝑆𝑆𝑜𝑜𝑜𝑜 + 𝑃𝑃𝑆𝑆𝑜𝑜𝑜𝑜ℎ𝑛𝑛𝑜𝑜,1� −2𝜆𝜆1𝜇𝜇ℎ𝑛𝑛𝑛𝑛�𝑉𝑉𝑆𝑆𝑛𝑛𝑜𝑜 − 𝑉𝑉𝑆𝑆𝑜𝑜𝑜𝑜 + 𝑃𝑃𝑆𝑆𝑜𝑜𝑜𝑜ℎ𝑛𝑛𝑛𝑛,1� − 2𝜆𝜆2𝜇𝜇ℎ𝑜𝑜𝑛𝑛�𝑉𝑉𝑆𝑆𝑜𝑜𝑛𝑛 − 𝑉𝑉𝑆𝑆𝑜𝑜𝑜𝑜 + 𝑃𝑃𝑆𝑆𝑜𝑜𝑜𝑜ℎ𝑜𝑜𝑛𝑛,2� −2𝜆𝜆2𝜇𝜇ℎ𝑛𝑛𝑛𝑛�𝑉𝑉𝑆𝑆𝑜𝑜𝑛𝑛 − 𝑉𝑉𝑆𝑆𝑜𝑜𝑜𝑜 + 𝑃𝑃𝑆𝑆𝑜𝑜𝑜𝑜ℎ𝑛𝑛𝑛𝑛,2� − 𝜁𝜁(𝑉𝑉𝑆𝑆𝑜𝑜𝑜𝑜(0) − 𝑉𝑉𝑆𝑆𝑜𝑜𝑜𝑜) − 2(1 − 𝛿𝛿)
(20)
13
��𝑉𝑆𝑆𝑜𝑜𝑛𝑛 = 𝑟𝑟𝑉𝑉𝑆𝑆𝑜𝑜𝑛𝑛 − 𝜆𝜆𝑢𝑢(𝑉𝑉ℎ𝑜𝑜𝑛𝑛 − 𝑉𝑉𝑆𝑆𝑜𝑜𝑛𝑛) − 2𝜆𝜆1𝜇𝜇ℎ𝑛𝑛𝑜𝑜�𝑉𝑉𝑆𝑆𝑛𝑛𝑛𝑛 − 𝑉𝑉𝑆𝑆𝑜𝑜𝑛𝑛 + 𝑃𝑃𝑆𝑆𝑜𝑜𝑛𝑛ℎ𝑛𝑛𝑜𝑜,1� −2𝜆𝜆1𝜇𝜇ℎ𝑛𝑛𝑛𝑛�𝑉𝑉𝑆𝑆𝑛𝑛𝑛𝑛 − 𝑉𝑉𝑆𝑆𝑜𝑜𝑛𝑛 + 𝑃𝑃𝑆𝑆𝑜𝑜𝑛𝑛ℎ𝑛𝑛𝑛𝑛,1� − 𝜁𝜁(𝑉𝑉𝑆𝑆𝑜𝑜𝑛𝑛(0) − 𝑉𝑉𝑆𝑆𝑜𝑜𝑛𝑛) − (1 − 𝛿𝛿)
(21)
��𝑉𝑆𝑆𝑛𝑛𝑜𝑜 = 𝑟𝑟𝑉𝑉𝑆𝑆𝑛𝑛𝑜𝑜 − 𝜆𝜆𝑢𝑢(𝑉𝑉ℎ𝑛𝑛𝑜𝑜 − 𝑉𝑉𝑆𝑆𝑛𝑛𝑜𝑜) − 2𝜆𝜆2𝜇𝜇ℎ𝑜𝑜𝑛𝑛�𝑉𝑉𝑆𝑆𝑛𝑛𝑛𝑛 − 𝑉𝑉𝑆𝑆𝑛𝑛𝑜𝑜 + 𝑃𝑃𝑆𝑆𝑛𝑛𝑜𝑜ℎ𝑜𝑜𝑛𝑛,2� −2𝜆𝜆2𝜇𝜇ℎ𝑛𝑛𝑛𝑛�𝑉𝑉𝑆𝑆𝑛𝑛𝑛𝑛 − 𝑉𝑉𝑆𝑆𝑛𝑛𝑜𝑜 + 𝑃𝑃𝑆𝑆𝑛𝑛𝑜𝑜ℎ𝑛𝑛𝑛𝑛,2� − 𝜁𝜁(𝑉𝑉𝑆𝑆𝑛𝑛𝑜𝑜(0) − 𝑉𝑉𝑆𝑆𝑛𝑛𝑜𝑜) − (1 − 𝛿𝛿)
(22)
��𝑉𝑆𝑆𝑛𝑛𝑛𝑛 = 𝑟𝑟𝑉𝑉𝑆𝑆𝑛𝑛𝑛𝑛 − 𝜆𝜆𝑢𝑢(𝑉𝑉ℎ𝑛𝑛𝑛𝑛 − 𝑉𝑉𝑆𝑆𝑛𝑛𝑛𝑛) − 𝜁𝜁(𝑉𝑉𝑆𝑆𝑛𝑛𝑛𝑛(0) − 𝑉𝑉𝑆𝑆𝑛𝑛𝑛𝑛) (23)
For the simplicity of notation, the time subscriptions for ��𝑉s, 𝑉𝑉s, 𝜇𝜇s, 𝑃𝑃s, and 𝜋𝜋s are omitted.
Here, time t represents the time after the previous aggregate shock.
Neither idiosyncratic shock nor aggregate shock is insurable in this model. Thus, the utility
of the agents is type dependent and the market is incomplete. The prices of bonds are determined
by the utility of the agents who trade the bond, as shown in Equation (1). Thus, the prices (𝑃𝑃) in
Equations (16)-(23) depend on the types of agents participating in the transaction, and the
subscriptions for 𝑃𝑃 denote that information. For example, when loo and hno agents trade bond 1,
the price of bond 1 is defined as 𝑃𝑃𝑆𝑆𝑜𝑜𝑜𝑜ℎ𝑛𝑛𝑜𝑜,1 , where 𝑃𝑃𝑆𝑆𝑜𝑜𝑜𝑜ℎ𝑛𝑛𝑜𝑜,1 = (𝑉𝑉𝑆𝑆𝑜𝑜𝑜𝑜 − 𝑉𝑉𝑆𝑆𝑛𝑛𝑜𝑜)(1 − 𝑞𝑞) +
(𝑉𝑉ℎ𝑜𝑜𝑜𝑜 − 𝑉𝑉ℎ𝑛𝑛𝑜𝑜)𝑞𝑞. As there are four combinations of agent types in successful transactions for each
bond, four prices per bond are denoted in the model. We do not directly assume that the four prices
of the bond must be the same. After solving the model, however, we confirm that each bond is
indeed traded at one price at a given time. In other words, the equilibrium prices satisfy 𝑃𝑃𝑆𝑆𝑜𝑜𝑜𝑜ℎ𝑛𝑛𝑜𝑜,𝑖𝑖 =
𝑃𝑃𝑆𝑆𝑜𝑜𝑜𝑜ℎ𝑛𝑛𝑛𝑛,𝑖𝑖 = 𝑃𝑃𝑆𝑆𝑜𝑜𝑛𝑛ℎ𝑛𝑛𝑜𝑜,𝑖𝑖 = 𝑃𝑃𝑆𝑆𝑜𝑜𝑛𝑛ℎ𝑛𝑛𝑛𝑛,𝑖𝑖 (where i = 1 or 2) at time t. In the next section, we document the
closed forms of prices from the baseline model, where there is no aggregate shock.
3.2 Baseline Solution with No Aggregate Shocks
Before examining the effect of aggregate shocks, we solve the steady-state equilibrium of the
14
baseline model, in which there is no aggregate shock. In other words, we let 𝜁𝜁 = 0 in Equations
(16)-(23).
By Proposition 1, there is a unique steady-state equilibrium fraction of agents that can be
calculated from Equations (5)-(12) by imposing a steady-state condition of ��𝜇(𝑡𝑡) = 0. Then, under
the equilibrium mass of agents and steady-state condition of ��𝑉 = 0, Equations (16)-(23) become
a simple system of linear equations with full rank. Thus, all eight 𝑉𝑉s are solved in closed forms,
and the prices of the bonds are determined as follows.
The equilibrium price of bond 1 is equal to
𝑃𝑃𝑆𝑆𝑜𝑜𝑜𝑜ℎ𝑛𝑛𝑜𝑜,1 = 𝑃𝑃𝑆𝑆𝑜𝑜𝑜𝑜ℎ𝑛𝑛𝑛𝑛,1 = 𝑃𝑃𝑆𝑆𝑜𝑜𝑛𝑛ℎ𝑛𝑛𝑜𝑜,1 = 𝑃𝑃𝑆𝑆𝑜𝑜𝑛𝑛ℎ𝑛𝑛𝑛𝑛,1 = 1𝑆𝑆− 𝛿𝛿
𝑆𝑆�𝜆𝜆𝑑𝑑+𝑆𝑆(1−𝑞𝑞)+2𝜆𝜆1(𝜇𝜇𝑙𝑙𝑜𝑜𝑜𝑜+𝜇𝜇𝑙𝑙𝑜𝑜𝑙𝑙)(1−𝑞𝑞)�
𝑆𝑆+𝜆𝜆𝑑𝑑+𝜆𝜆𝑢𝑢+2𝜆𝜆1(𝜇𝜇𝑙𝑙𝑜𝑜𝑜𝑜+𝜇𝜇𝑙𝑙𝑜𝑜𝑙𝑙)(1−𝑞𝑞)+2𝜆𝜆1(𝜇𝜇ℎ𝑙𝑙𝑙𝑙+𝜇𝜇ℎ𝑙𝑙𝑜𝑜)𝑞𝑞 (24)
Also, the equilibrium price of bond 2 is equal to
𝑃𝑃𝑆𝑆𝑜𝑜𝑜𝑜ℎ𝑜𝑜𝑛𝑛,2 = 𝑃𝑃𝑆𝑆𝑜𝑜𝑜𝑜ℎ𝑛𝑛𝑛𝑛,2 = 𝑃𝑃𝑆𝑆𝑛𝑛𝑜𝑜ℎ𝑜𝑜𝑛𝑛,2 = 𝑃𝑃𝑆𝑆𝑛𝑛𝑜𝑜ℎ𝑛𝑛𝑛𝑛,2 = 1𝑆𝑆− 𝛿𝛿
𝑆𝑆�𝜆𝜆𝑑𝑑+𝑆𝑆(1−𝑞𝑞)+2𝜆𝜆2(𝜇𝜇𝑙𝑙𝑜𝑜𝑜𝑜+𝜇𝜇𝑙𝑙𝑙𝑙𝑜𝑜)(1−𝑞𝑞)�
𝑆𝑆+𝜆𝜆𝑑𝑑+𝜆𝜆𝑢𝑢+2𝜆𝜆2(𝜇𝜇𝑙𝑙𝑜𝑜𝑜𝑜+𝜇𝜇𝑙𝑙𝑙𝑙𝑜𝑜)(1−𝑞𝑞)+2𝜆𝜆2(𝜇𝜇ℎ𝑙𝑙𝑙𝑙+𝜇𝜇ℎ𝑜𝑜𝑙𝑙)𝑞𝑞 (25)
These prices are basically the same as the results in Duffie et al. (2007). In that sense, this
baseline solution is not particularly interesting. However, it is worth noting that the four prices of
a bond turn out to be same in equilibrium, although it is not directly assumed in the model. The
price increases in 𝜆𝜆𝑥𝑥 for all 𝜆𝜆𝑥𝑥 greater than a constant lower bound determined by the other
parameters of the model if 𝑠𝑠𝑥𝑥 < 𝜆𝜆𝑢𝑢/(𝜆𝜆𝑢𝑢 + 𝜆𝜆𝑑𝑑) for any 𝑥𝑥 ∈ {0,1}. The condition 𝑠𝑠𝑥𝑥 < 𝜆𝜆𝑢𝑢/(𝜆𝜆𝑢𝑢 +
𝜆𝜆𝑑𝑑) means there is less than one bond 𝑥𝑥 per one high type agent in equilibrium. The proof is
almost the same as that in Duffie et al. (2007), and hence it is omitted here.
3.3 The Closed Form Solution
15
To solve the system of differential equations (Equations (16)-(23)), we convert it into a vector
form as follows.
��𝑽(𝑡𝑡) = 𝐴𝐴1(𝝁𝝁(𝑡𝑡))𝑽𝑽(𝑡𝑡) − 𝐴𝐴2 − 𝐴𝐴3(𝝁𝝁(𝑡𝑡))𝑽𝑽(0) (26)
where A1 and A3 are 8 x 8 matrices that depend on 𝝁𝝁(𝑡𝑡), the distribution of agents, and 𝐴𝐴2 is
a 8 x 1 constant matrix that depends on other constant parameters in the model.
Equation (26) is a simple ordinary differential equation with a closed form solution, except it
includes 𝑽𝑽(0) (i.e., the initial value of 𝑽𝑽(𝑡𝑡)). To solve this, we first obtain the closed form
solution by treating 𝑽𝑽(0) as a constant. We then calculate 𝑽𝑽(0) using the solution and substitute
back into the closed form solution. Thus, the following two equations represent the closed form
solution.
𝑽𝑽(𝑡𝑡) = � exp (−� 𝐴𝐴1�𝝁𝝁(𝑢𝑢)�𝑑𝑑𝑢𝑢𝑠𝑠
𝑡𝑡) ∙ (𝐴𝐴2 + 𝐴𝐴3�𝝁𝝁(𝑠𝑠)�𝑽𝑽(0)) 𝑑𝑑𝑠𝑠
∞
𝑡𝑡 (27)
𝑽𝑽(0) = �I8 − � 𝑒𝑒𝑥𝑥𝑒𝑒 �−� 𝐴𝐴1�𝝁𝝁(𝑢𝑢)�𝑑𝑑𝑢𝑢𝑠𝑠
0� ∙ 𝐴𝐴3�𝝁𝝁(𝑠𝑠)� 𝑑𝑑𝑠𝑠
∞
0�−1
∙ (� 𝑒𝑒𝑥𝑥𝑒𝑒 �−� 𝐴𝐴1�𝝁𝝁(𝑢𝑢)�𝑑𝑑𝑢𝑢𝑠𝑠
0� ∙ 𝐴𝐴2 𝑑𝑑𝑠𝑠
∞
0)
(28)
It is difficult to interpret the solution with symbolic terms; the form of the solution is too
complicated, as shown in Equations (27) and (28). Thus, we examine the numerical examples in
the next section.
16
4. Numerical Illustrations
This section provides numerical examples to see how the difference in liquidity affects bond
prices during times of distress.
4.1 Model Parameters
We suppose that 𝜆𝜆1 = 2,500 and 𝜆𝜆2 = 125. This choice of searching intensities implies
that the average times needed to sell are 0.31 and 6.15 days for bonds 1 and 2, respectively,
conditional on the equilibrium mass of agents. λd and λu, the intensities of idiosyncratic shocks,
are assumed to be 0.2 and 2, respectively. This means that H and L agents maintain their intrinsic
types for 5 and 0.5 years on average, respectively. The aggregate shock arrives with an intensity
(𝜁𝜁) of 0.05. To focus on the difference in liquidity, we assume other parameters like the amounts
outstanding of bonds 1 and 2 (𝑠𝑠1 and 𝑠𝑠2, respectively), holding costs (𝛿𝛿), bargaining powers (𝑞𝑞),
and extent of aggregate shock (𝜋𝜋ℎ𝑜𝑜𝑜𝑜∗ , 𝜋𝜋ℎ𝑜𝑜𝑛𝑛∗ , 𝜋𝜋ℎ𝑛𝑛𝑜𝑜∗ , and 𝜋𝜋ℎ𝑛𝑛𝑛𝑛∗ ) are neither bond specific nor
ownership status specific. Specifically, We suppose that 𝑠𝑠1 = 𝑠𝑠2 = 0.75, 𝛿𝛿 = 2.5, 𝑞𝑞 = 0.5, and
𝜋𝜋ℎ𝑜𝑜𝑜𝑜∗ = 𝜋𝜋ℎ𝑜𝑜𝑛𝑛∗ = 𝜋𝜋ℎ𝑛𝑛𝑜𝑜∗ = 𝜋𝜋ℎ𝑛𝑛𝑛𝑛∗ = 0.7. The discount rate (𝑟𝑟 ) is 0.1. These choices of parameters
similarly follow Duffie et al. (2007), except that we assume a larger extent of aggregate shock to
emphasize the distress from market-level shock. We summarize all of the parameters in the
following table.
[Table 1]
17
4.2 Calibrations
The steady-state fractions of agents (𝝁𝝁�) are determined as in Table 2 Panel A, under the
parameters in Table 1. The agent distribution remains in 𝝁𝝁� if there is no additional aggregate
shock. After the aggregate shock, the agent masses become the post-shock distribution (𝝁𝝁∗) in
Table 2 Panel B. In equilibrium, a majority of the agents (67%) are high type agents who own both
bonds 1 and 2 (i.e., hoo type). While only 0.03% (=0.0002/0.75) of bond 1 is misallocated to L
agents, 0.47% (=0.0035/0.75) of bond 2 is misallocated to L agents, as the search friction of bond
2 is higher than that of bond 1. After the aggregate shock, the loo type has the highest fraction
(47.19%). The fractions of potential sellers for bonds 1 and 2 are 52.50% and 52.60%, respectively.
[Table 2]
Figure 1 shows the prices of bonds 1 and 2 around the aggregate shock. The y-axis represents
the price (in terms of unit consumption) and the x-axis represents the calendar time in years. During
the “normal times” (i.e., a long time after the last aggregate shock, when the effect of the last shock
has mostly disappeared), the price of bond 1 is 7.30 and the price of bond 2 is 6.90. The prices of
both bonds are less than the ideal price 10 (=1/r), where the market has infinite liquidity and no
aggregate shock. The presence of searching costs and aggregate shock are two sources of the
discount. Bond 2 is priced much lower (by about 5%)9 because the search cost is higher than bond
1. Both bonds 1 and 2 are priced lower than the baseline model with no aggregate shock because
9 (7.30-6.90)/7.30.
18
the risk of aggregate shock lowers the prices.10 Compared with the baseline model, the prices of
bonds 1 and 2 drop by about 27% and 25%, respectively. The price of bond 1 drops more, reflecting
that the aggregate shock decreases it to a greater extent.
[Figure 1]
We assume that the aggregate shock arrives at t = 0.5. Right after the aggregate shock, the
price of bond 1 drops from 7.30 to 1.99 (72.73%) and the price of bond 2 drops from 6.90 to 2.26
(67.24%). The more liquid bond (i.e., bond 1) experiences a bigger drop in price, and as a result
the liquid bond is priced lower than the relatively illiquid bond (i.e., bond 2). The prices are
recovered in the following few years. The recovery of the liquid bond is faster than that of the
illiquid bond. The price of bond 1 is recovered above the price of bond 2 about 0.28 year after the
shock. The price of bond 1 is almost recovered to the “normal” level 0.7 year after the shock. The
price of bond 2 is almost recovered after about 1.5 years. Furthermore, the presence of aggregate
shocks decreases the bond prices.
The main driving force of the results is the difference in abnormal selling pressures caused
by the difference in liquidity. The selling pressures increase more for bond 1 than for bond 2 due
to the lower search friction, although the amounts of sellers and buyers participating in the market
similarly increase for bonds 1 and 2.11 The effect of search friction is highlighted for the agents
10 In the baseline model with the parameters in Table 1, the prices of bonds 1 and 2 are 9.96 and 9.25, respectively. 11 To clarify that the difference in amounts of agents is not the key driving force behind the preceding results, we note that the amounts of buyers and sellers are similar for bonds 1 and 2. In other words, the shock increases the quantity of sellers and decreases the quantity of buyers for both bonds 1 and 2 to similar extents. The post-shock fraction of L-type owners of bonds 1 and 2 are similar (52.50% and 52.60% respectively). Furthermore, H-type non-owners of
19
who hold both bonds. The fraction of loo agents is abnormally elevated from 0.002 to 0.4719 and
accounts for about 90%12 of the sellers of bonds 1 and 2 right after the shock. At the same time,
the post-shock fractions of buyers are only 0.0478 for bond 1 and 0.0487 for bond 2. The loo agents
are more likely to sell bond 1 first, as it is easier to search for the counterparty. Specifically,
conditional on loo agents successfully selling a bond immediately after the aggregate shock, there
is a 95.15%13 chance of selling bond 1 and only a 4.85% chance of selling bond 2. Therefore, they
exert greater selling pressure on bond 1 than on bond 2, which is importantly reflected in the post-
shock prices. As time passes after the shock, the distribution of agents is recovered to the normal
(steady-state equilibrium) level, and so the selling pressures are recovered to the normal level.
To ascertain how the abnormal selling pressure affects prices more specifically, we look at
how agents’ lifetime utility changes after the aggregate shock. As in Equation (1), the price of bond
1 is equal to (𝑉𝑉𝑆𝑆𝑜𝑜𝑜𝑜 − 𝑉𝑉𝑆𝑆𝑛𝑛𝑜𝑜)(1 − 𝑞𝑞) + (𝑉𝑉ℎ𝑜𝑜𝑜𝑜 − 𝑉𝑉ℎ𝑛𝑛𝑜𝑜)𝑞𝑞 and the price of bond 2 is equal to
(𝑉𝑉𝑆𝑆𝑜𝑜𝑜𝑜 − 𝑉𝑉𝑆𝑆𝑜𝑜𝑛𝑛)(1 − 𝑞𝑞) + (𝑉𝑉ℎ𝑜𝑜𝑜𝑜 − 𝑉𝑉ℎ𝑜𝑜𝑛𝑛)𝑞𝑞 . Hence, the difference between the two prices is
determined by (𝑉𝑉𝑆𝑆𝑜𝑜𝑛𝑛 − 𝑉𝑉𝑆𝑆𝑛𝑛𝑜𝑜 ) and (𝑉𝑉ℎ𝑜𝑜𝑛𝑛 − 𝑉𝑉ℎ𝑛𝑛𝑜𝑜 ). Thus, we discuss how 𝑉𝑉ℎ𝑜𝑜𝑛𝑛 and 𝑉𝑉ℎ𝑛𝑛𝑜𝑜 are
affected by the shock first, and then discuss the utility of other types of agents.
The hon and hno agents have owner positions for one bond and non-owner positions for the
other bond. In terms of owner position, the utility drops because the prospect of selling decreases
due to the abnormally elevated fraction of sellers. Bond 1 owners experience greater drops in utility
than bond 2 owners due to the greater abnormal selling pressure, which is amplified by the higher
bonds 1 and 2 have similar post-shock fractions (4.78% and 4.87%, respectively). 12 𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜∗ /(𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜∗ + 𝜇𝜇𝑆𝑆𝑜𝑜𝑛𝑛∗ ) = 89.89%, 𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜∗ /(𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜∗ + 𝜇𝜇𝑆𝑆𝑛𝑛𝑜𝑜∗ ) = 89.71%. Thus, loo agents are the major sellers in the post-shock distribution, as the majority of agents shocked (67.40%) are hoo agents. The fraction of loo agents is the most abnormally increased (from ��𝜇𝑆𝑆𝑜𝑜𝑜𝑜 = 0.02% to 𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜∗ = 47.19%). 13 2𝜆𝜆1𝜇𝜇𝑙𝑙𝑜𝑜𝑜𝑜
∗ �𝜇𝜇ℎ𝑙𝑙𝑜𝑜∗ +𝜇𝜇ℎ𝑙𝑙𝑙𝑙
∗ ��2𝜆𝜆1𝜇𝜇𝑙𝑙𝑜𝑜𝑜𝑜
∗ �𝜇𝜇ℎ𝑙𝑙𝑜𝑜∗ +𝜇𝜇ℎ𝑙𝑙𝑙𝑙
∗ �+2𝜆𝜆2𝜇𝜇𝑙𝑙𝑜𝑜𝑜𝑜∗ �𝜇𝜇ℎ𝑜𝑜𝑙𝑙
∗ +𝜇𝜇ℎ𝑙𝑙𝑙𝑙∗ ��
=95.15%.
20
search frequency. For H-type owners, this effect is conditional on the possibility of idiosyncratic
shocks, which may make them to want to sell their holdings.
For non-owner positions, the agents experience utility increases, as they can expropriate the
abnormal selling pressures. Bond 1 non-owners have more of a chance to expropriate than bond 2
non-owners because it is easier to sell and buy bond 1. For L-type non-owners, this effect is
conditional on the possibility of idiosyncratic shocks, which may turn them into H-type non-
owners.
In summary, due to the aggregate shock, hon and hno agents who are not directly affected by
the shock experience immediate decreases in utility from the owner position of one bond and
immediate increases from the non-owner position of the other bond. The net utility changes are
positive, but the hno agents experience more increases in lifetime utility than the hon agents.14 A
similar argument holds for the lon and lno agents. Thus, (𝑉𝑉𝑆𝑆𝑜𝑜𝑛𝑛 − 𝑉𝑉𝑆𝑆𝑛𝑛𝑜𝑜) and (𝑉𝑉ℎ𝑜𝑜𝑛𝑛 − 𝑉𝑉ℎ𝑛𝑛𝑜𝑜) become
negative, and bond 1 is priced lower than bond 2, conditional on whether the abnormal selling
pressures are large enough.
Meanwhile, agents who hold both assets experience decreases in utility from the aggregate
shocks. The loo agents experience a severe loss in lifetime utility from the aggregate shock,15 as
the possibility of selling the assets becomes significantly lower due to the abnormal increase in the
mass of loo agents. Furthermore, the hoo agents who are directly affected by the shock experience
large drops in utility as their type changes from hoo to loo.16 The utility of hoo agents who are
14 𝑉𝑉ℎ𝑛𝑛𝑜𝑜 jumps upward from 11.33 to 15.92 (increases by 40.51%) and 𝑉𝑉ℎ𝑜𝑜𝑛𝑛 jumps upward from 11.67 to 15.62 (increases by 33.84%). 15 Before the shock, Vloo=18.48. After the shock, Vloo=15.87. 16 If a hoo agent becomes a loo agent due to the shock, her utility drops from 𝑉𝑉ℎ𝑜𝑜𝑜𝑜=18.63 to 𝑉𝑉𝑆𝑆𝑜𝑜𝑜𝑜=15.87.
21
unaffected by the shock also drops, but only slightly17 as a result of the probability of becoming
loo agents due to the idiosyncratic shock.
Finally, the agents who do not hold bonds experience increases in their utility due to the shock,
as they expropriate the abnormal selling pressures. The hnn agents who are not directly affected
by the shock experience the most significant increases in utility.18 The extent of the increases is
greater than that of the hon and hno agents. The lnn agents also experience increases in their utility,
reflecting the probability of expropriating the selling pressures when they become hnn agents by
obtaining new capital from the idiosyncratic shock.19
The price recovery results from the evolution in agent fractions through the refinancing
channel. The fractions of H and L agents recover to the equilibrium level as they trade bonds and
experience the idiosyncratic shocks. The abnormally elevated fraction of L agents (the sellers)
decreases to equilibrium as the new capital arrives (with intensity λu). As time goes by, more L
agents become H agents who are willing to buy the bond. In this situation, bond 1 can be bought
faster than bond 2. Thus, the price of bond 1 recovers faster than that of bond 2 as more agents
obtain new capital.
During normal times when the agent fractions return to equilibrium level, bond 1 is priced
higher. Owning bond 1 is more valuable than owning bond 2, as the expected utility loss from the
holding costs is low due to the higher liquidity, under the normal fractions of agents (i.e., steady-
state equilibrium of agent fractions). This is the well-known benefit of liquidity. The expected
utility loss from the aggregate shock cannot dominate this effect during the normal times, as the
17 𝑉𝑉ℎ𝑜𝑜𝑜𝑜 drops from 18.63 to 17.90. 18 𝑉𝑉ℎ𝑛𝑛𝑛𝑛 jumps upward from 4.37 to 13.63. 19 𝑉𝑉𝑆𝑆𝑛𝑛𝑛𝑛 jumps upward from 4.34 to 11.65.
22
probability of aggregate shock is low enough.
The aggregate shocks have the price effect described previously, provided there is an
abnormally large fraction of sellers and a small fraction of buyers. In the model, the extent of the
aggregate shock is modeled by the parameter 𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ , which represents the portion of H agents
affected by the aggregate shock (e.g., see Section 2.5). If 𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ is small, only a small fraction of
sellers increases from the aggregate shocks, and the fraction of agents quickly returns to the normal
level. Thus, if 𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ is sufficiently small, the benefits of liquidity (e.g., faster recovery and lower
inventory risks due to high search intensity) dominate the costs of liquidity during times of distress.
Figure 2 replicates Figure 1 except that 𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ is now equal to 10%. In this case, the benefits of
liquidity dominate even when the aggregate shock arrives and the traditional liquidity premium is
observed during entire periods.
[Figure 2]
To ascertain the effects of the extent of aggregate shocks more specifically, we calibrate the
model using the parameters in Table 1, except we change 𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ from 0% to 95% by 1%.20 Figure
3 shows the post-shock distribution of agent fractions (i.e., 𝝁𝝁∗) across different sizes of aggregate
shock (𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ ). As the size of the shock increases, the fraction of L-type agents, especially loo agents,
increases.
20 We stop the calibration at 𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ =95% because when 𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ becomes larger than 95%, the market fails when the aggregate shock arrives, as the price of bond 1 becomes less than zero.
23
[Figure 3]
Figure 4 shows the relationships between the extent of aggregate shock and prices in the
normal (i.e., long time after the last shock) and distressed (i.e., immediately after the shock) times.
During both normal and distressed times and for both bonds 1 and 2, the price and extent of the
shock have a negative relationship. When the size of the shock increases, the extent of the selling
pressure becomes larger, the prices immediately following the shock become smaller, and the
prices in “normal” times become smaller, reflecting the possibility of larger price drops. In normal
times, the price of bond 1 is always larger than the price of bond 2 (e.g., see Figure 4 Panel A). In
other words, during normal times, the more liquid bond always gets priced higher, as expected.
However, during times of distress, the price of bond 1 becomes lower than the price of bond 2
when the extent of the aggregate shock is large enough (e.g., see Figure 4 Panel B). Specifically,
the value of liquidity (price of bond 1 minus price of bond 2) becomes negative if 𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ > 41%
and decreases monotonically when 𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ increases from 41% through 95%. Therefore, our result
is consistent with the claim that the more liquid bond is more fragile to the large aggregate shock
where most investors want to leave the market.
[Figure 4]
Figure 5 shows the prices of bonds 1 and 2 during normal times across various search
24
intensities. We solve the model by changing the search intensity of bond 1 (𝜆𝜆1) from 250 through
6,250, while fixing the search intensity of bond 2 (𝜆𝜆2) at 125. In Panels A and B, We assume the
extent of the aggregate shock is very small (𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ =10%) and large enough (𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ =70%),
respectively. Other parameters are the same as in Table 1. Regardless of the extent of the aggregate
shock, the price of bond 1 during normal times increases as its liquidity increases. When the search
intensity is very large and the aggregate shock is very small, the price is almost 10 (=1/r).
[Figure 5]
Figure 6 shows the prices of bonds 1 and 2 during distressed times across various search
intensities. Figure 6 is generated in a similar fashion to Figure 5, but shows the “post-shock prices”
(i.e., prices immediately following the shock). In Figure 6 Panel A, We assume the extent of the
aggregate shock is very small (𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ =10%). The relationship between liquidity and the post-shock
price is positive. That is, the more liquid bond gets priced “higher,” as in the normal times. The
benefits of liquidity (from the lower searching cost) dominate the harm of liquidity (from the
greater extent of selling pressures) during times of distress, as the extent of selling pressures and
the effects of aggregate shock on the price are small.
Under the presence of large aggregate shock, however, we obtain a totally different result.
Figure 6 Panel B shows that the relationship between liquidity and the post-shock price is negative,
not positive. That is, contrary to normal times, the more liquid bond is priced “lower” during times
of distress. The discount of bond 1 monotonically increases as the search intensity increases.
Furthermore, the marginal increase in the discount of bond 1 decreases as the search intensity
25
increases.
Combining Figures 5 and 6, our results indicate that the value of liquidity is conditional on
the market condition. The liquidity is beneficial to the bond price during normal times and weakly
stressed times, in which only a small portion of investors is forced to sell. However, the liquidity
can be harmful during times of distress, conditional on whether the extent of distress is enough to
force a large fraction of investors to sell their holdings.
[Figure 6]
5. Model Implications
The main implication of our model is that if trading delays exist due to the search frictions,
the large extent of fire sales relative to buying pressures can cause a flight from liquidity. This
implies that more liquid securities may be traded at lower prices than less liquid securities with
almost identical cash flows during market-wide liquidity events. For example, the U.S. corporate
bond markets following the Lehman Brothers bankruptcy during the 2008 financial crisis could be
a good testing ground for the flight from liquidity. Significant selling pressures from institutional
fire sales in the U.S. corporate bond markets arose following the Lehman Brothers collapse.21 Our
model implies that during the financial crisis a more liquid on-the-run bond was priced lower than
a less liquid off-the-run bond issued by the same firm with the same credit rating and time to
21 See, e.g., Ellul, Jotikasthira, and Lundblad (2011), Manconi, Massa, and Yasuda (2012), and Choi and Shin (2016).
26
maturity.
Furthermore, Figure 6 implies that a level of liquidity (i.e., extent of search friction)
negatively affects the price under the market-wide shock with severe institutional fire sales, but
positively affects the price if the buying pressure is high relative to the selling pressure from fire
sales. This implication is consistent with recent findings of Boudoukh et al. (2016), who document
that a more liquid government bond can be priced lower than its illiquid counterpart. They examine
13 developed countries’ sovereign bond markets22 and find flights from liquidity in most of their
sample countries except for Germany and the U.S.23 In Germany and the U.S., the flight from
liquidity is not observed because investors flock to the safe haven (i.e., flight to quality). In other
words, during the European sovereign debt crisis, the relative extent of selling pressure in the
Germany and U.S. government bond markets is not large enough to cause a flight from liquidity
due to the flight to quality. This situation is consistent with Figure 2.
In addition, our model suggests that similar results may be observed in other OTC markets,
such as the corporate bond markets. For example, during times of distress, within the safe-haven
group of bonds (with high credit quality), the level of liquidity may positively affect the price due
to increased demands from a flight to quality, while the liquidity may affect prices negatively
within the other groups of bonds such as junk bonds, which are exposed to larger selling pressure.
In other words, investors may flock to relatively liquid safe-haven bonds from relatively liquid
junk bonds during the recession periods. This implication is consistent with Acharya, Amihud, and
Bharath (2013), who document that in stressed regimes with shocks in the stock and bond markets,
22 Australia, Belgium, Canada, Denmark, France, Italy, Germany, Japan, Netherlands, Spain, Sweden, the United Kingdom, and the U.S. 23 Boudoukh et al. (2016) document that more liquid government bonds are priced higher in Germany and the U.S. during the European sovereign debt crisis.
27
rises in market illiquidity positively affect the prices of investment grade bonds on average, but
negatively affect the prices of junk bonds.
In sum, our model implies that the pricing of liquidity critically depends on economic
conditions such as the extent of delays in trading and the extent of fire sales relative to buying
pressures.
6. Empirical Analysis
Our interest is in providing empirical evidence to support the implication that liquid bonds
can be priced lower than illiquid bonds with identical credit risks due to the flight from liquidity
seen during the asset fire-sale episodes.
6.1 Empirical Methodology
We examine the yield spreads of corporate bond pairs issued by the same firm with similar
maturities but different ages. The age of a bond is widely documented and used as a proxy for its
liquidity.24 As a bond gets older, it becomes less liquid, as larger fractions of the amounts issued
are absorbed by investors’ buy-and-hold portfolios. In this way, we can separate out the effect of
liquidity.
24 See, e.g., Sarig and Warga (1989), Alexander, Edwards, and Ferri (2000), Hong and Warga (2000), Schultz (2001), Elton, Gruber, Agrawal, and Mann (2004), Houweling, Mentink, and Vorst (2005), and Ericsson and Renault (2006) among many others.
28
Specifically, we first pick the bond most recently issued by a firm with an age younger than
2 years.25 We then attempt to find a matching bond issued by the same firm with an age older than
3 years. To be successfully matched, the two bonds (“young” and “old” bonds) should have the
same credit ratings26 and option features.27 Among the remaining bonds, we attempt to pick one
bond with the closest time-to-maturity. In addition, we require the maximum difference in time-
to-maturity to be smaller than 1 year. Finally, if multiple bonds still remain through this matching
process, we pick the oldest bond. We obtain one matched old bond per one young bond at most.
The sample of matched pairs is updated daily.
Our main variable of interest is the yield spread of the pairs of bonds. For each pair of bonds,
we calculate the yields of the illiquid bond (i.e., the older bond in the pair) minus the yields of the
liquid bond (i.e., the younger bond in the pair) each day. Hereafter, we refer to the spreads as illiq-
liq spreads. We use daily yields based on the actual transaction prices of the day. Thus, by
construction, the illiq-liq spreads require transactions of both liquid and illiquid bonds to be well
defined. Using same-day trades has the following advantages. First, it ensures that the spreads are
not affected by time-fixed effects (including unobservable ones). Second, it mitigates concerns
where the spreads are driven by changes in yields of liquid bonds only and the illiquid bonds are
not traded at all. We use monthly averages (or medians) of daily yields in our main analyses to
reduce noise and increase statistical powers.
25 Houweling, Mentink, and Vorst (2005) document that any threshold values of age between 4 months and 2 years can be used to classify a bond as a “young” liquid bond or an “old” illiquid bond. We use the cut-off of 2 years to maximize our sample size. 26 We ensure that the two bonds have the same seniority. 27 Such as callable, putable, sinking fund, asset-backed, and enhancement features.
29
6.2 Data
Our data source for the corporate bond yields is the Trade Reporting and Compliance Engine
(TRACE) database. We use bond pricing data from March 2005, as the coverage of the TRACE
becomes fully comprehensive after February 2005.28 Our bond yields data end in the first quarter
of 2015. To filter the reporting errors in TRACE, We follow the filtering procedures described by
Dick-Nielsen (2009, 2014). After the filtering, we exclude retail-sized trades (i.e., trades with
volumes below $100,000) following Dick-Nielsen, Feldhutter, and Lando (2012).
We obtain bond-specific information from the Mergent Fixed Income Securities Database
(FISD), including ages, credit ratings,29 maturity, amounts outstanding, and other characteristics.
To be included in our sample, the bonds must be non-convertible U.S. fixed coupon bonds.30 In
addition, we use the S&P long-term-issuer-level ratings from Compustat for the issuer ratings.
We use the Center for Research in Security Prices (CRSP) survivorship-bias-free mutual fund
database to estimate the mutual fund flows. Specifically, the monthly net flow of funds is estimated
as:
𝑓𝑓𝑙𝑙𝑜𝑜𝑓𝑓𝑗𝑗,𝑡𝑡 = 𝑇𝑇𝑇𝑇𝐴𝐴𝑗𝑗,𝑡𝑡−𝑇𝑇𝑇𝑇𝐴𝐴𝑗𝑗,𝑡𝑡−1∗�1+𝑆𝑆𝑗𝑗,𝑡𝑡�𝑇𝑇𝑇𝑇𝐴𝐴𝑗𝑗,𝑡𝑡−1
(29)
where 𝑇𝑇𝑇𝑇𝐴𝐴𝑖𝑖,𝑡𝑡 is the total net assets for fund j at the end of month t and 𝑟𝑟𝑗𝑗,𝑡𝑡 is the monthly returns
28 We are following the FINRA news release in defining the date of full implementation of TRACE (see, e.g., http://www.finra.org/newsroom/2005/nasds-fully-implemented-trace-brings-unprecedented-transparency-corporate-bond-market). Recent papers such as Jostova, Nikolova, Philipov, and Stahel (2013) also use February 2005 as the full coverage date. 29 We use S&P credit ratings for bonds and assign 21 to the AAA rating. 30 To be classified as corporate bonds, we require a Mergent FISD bond type (CCOV, CDEB, CLOC, CMTN, CMTZ, CP, CPAS, CPIK, CS).
30
for fund j over month t. To be included in our sample of mutual funds, a fund must be a corporate
bond mutual fund.31 At the start of each month, we require them to hold both bonds of at least one
pair used to estimate the illiq-liq spreads during the month. For mutual fund holdings, we use the
Morningstar database, which consists of quarterly holdings between July 2002 and December 2014.
Each month, we use the holding data at the end of the previous quarter.
6.3 Descriptive Statistics
The matching described in Section 6.1 yields 1,018 unique pairs of bonds from 442 unique
issuers between March 2005 and March 2015.32 During our 121-month sample period, the average
number of unique pairs in a month with available illiq-liq spreads for at least 1 day is about 67.01
pairs. The numbers of sample pairs exhibit growth patterns, as both the numbers and dollar
amounts of corporate bonds outstanding increase,33 and once a new bond is issued and included
in the sample, the pair is likely to remain for the next 2 years. The minimum number of sample
pairs during a month is 31 (July 2006). The maximum number of sample pairs during a month is
129 (March 2015).
[Table 3]
31 The Lipper objective code is one of A, BBB, HY, SII, SID, and IID. The CRSP objective code starts with “IC.” We exclude index, ETF, and ETN funds. 32 We exclude pairs of bonds if no illiq-liq spread is observed for the pair during our sample period. 33 According to SIFMA, U.S. corporate bond issuances doubled in dollar value from 2005 to 2015.
31
Table 3 provides sample statistics for the bond characteristics. On average, younger bonds in
the sample of pairs have an average age of 0.27 year, while the older bonds have an average age
of 5.01 years. Meanwhile, the time-to-maturities of the liquid and illiquid bonds are 5.66 and 5.60
years on average, respectively. The ages and time-to-maturities are estimated at the date when the
pair first appeared in the sample. Consistent with our construction, the bonds in a pair have very
similar time-to-maturities but very different ages. Furthermore, about 80% of the bonds in our
sample are investment-grade bonds. Although 13 pairs of bonds with S&P ratings are not available,
we require their seniority (obtained from the Mergent FISD database) to be the same. Thus, we
include them to increase the sample size.34
6.4 Empirical Results
6.4.1 Time Series Evidence of Negative Illiq-liq Spreads During Times of Distress
The first implication of our model is that liquid bonds can be priced lower than illiquid bonds
during times of distress, such as the 2008 financial crisis or the Taper Tantrum in 2013.
[Figure 7]
Figure 7 Panel A shows the time series of monthly average illiq-liq spreads between March
34 However, our results remain intact regardless of the inclusion of these bonds.
32
2005 and March 2015. Before the 2008 financial crisis, the spreads are positively significant, as
normally expected for the liquidity premium. The spreads then experience large drops during the
2008 financial crisis, dropping rapidly to the -0.4% level after the announcement of the Lehman
Brothers bankruptcy in September 2008. The negative spreads are significant at the 95% level.
This means the yields of the liquid bonds are significantly higher, and the prices lower, than those
of the illiquid bonds on average. This is consistent with the model implication of a flight from
liquidity. The yields recover to near-zero levels during the first half of 2009, and then to slightly
negative levels during mid-2010 and early 2011, when the European debt crisis began. They then
recover to a significantly positive level of 0.2%, higher than the post-crisis level. Finally, the
spreads tighten again after the Taper Tantrum when the Fed signals its monetary policy of
increasing the interest rate. Figure 7 Panel B shows the time series of monthly medians on the illiq-
liq spreads. It presents similar results, meaning that the averages are not derived from extreme
values.
In sum, Figure 7 supports the model implication of times of distress, especially the Lehman
Brothers bankruptcy during the 2008 financial crisis.
The characteristics of our sample pairs, such as the differences in ages and maturities between
two bonds in a pair, are obviously time varying because the bond compositions in our sample are
time varying. This may affect the results in Figure 7. To mitigate this concern, we provide time
series of average differences in time-to-maturity and age between bonds in our sample pairs, as
shown in Figure 8.
[Figure 8]
33
In Figure 8 Panels A and B, We plot the monthly averages of the differences in time-to-
maturity and age, respectively, of the bonds in a pair. During the 2008 financial crisis, the monthly
average difference in time-to-maturity consistently decreases and the average age difference
consistently increases. These results cannot explain the decreases in the illiq-liq spreads during the
same periods. This mitigates the concern that the negative illiq-liq spreads seen during the 2008
crisis in Figure 7 are derived by changes in the sample composition.
6.4.2 Investor Capital Flows to Corporate Bond Markets and Illiq-liq Spreads
Stale pricing provides an alternative hypothesis for the negative illiq-liq spreads. During the
crisis, the prices of liquid bonds fall, but the prices of illiquid bonds do not move correspondently
due to their staleness. If the negative illiq-liq spreads are driven by the staleness of illiquid bond
prices, there should be a mean reversion in the illiq-liq spreads over relatively short horizons, a
point also mentioned in Boudoukh et al. (2016). However, Figure 7 shows that the negative spreads
last over several months during the 2008 crisis. Provided we use the days where both illiquid and
liquid bonds are traded, it is difficult to use staleness to explain the negative spreads in Figure 7.
In this section, we directly examine the relationship between investor capital flows to
corporate bond markets and the illiq-liq spreads. Thus, we provide evidence that more directly
supports our implication that the negative spreads are driven by price pressures and probably not
explained by staleness. We use investor flows to corporate bond mutual funds as a proxy for the
investor capital flows. Just like the hoo (or loo) type agents in the model, we require a mutual fund
to hold both liquid and illiquid bonds in at least one pair at the start of a month to be included in
34
our estimation of the flows during the month. Thus, our results are probably not driven by
differences in the extent of mutual fund flows to liquid and illiquid bonds in a pair, but probably
driven by the ease with which liquid bonds are sold (i.e., flight from liquidity). We then consider
our sample mutual funds as one big fund and use aggregate flows.
[Figure 9]
Figure 9 shows the monthly mutual fund capital flows as percentages of the total net assets at
the end of last month and the average illiq-liq spreads, which are also plotted in Figure 7. As
expected, there are large outflows during the 2008 financial crisis, especially after the
announcement of the Lehman Brothers bankruptcy, and also after the Taper Tantrum in 2013.
During October 2008, the month following the announcement of the Lehman Brothers bankruptcy,
the outflows are as large as 3.42%. The outflows immediately following the Taper Tantrum (i.e.,
June 2013) are also as large as 2.52%. Yet, the illiq-liq spreads do not rapidly drop as in the 2008
crisis. The fire sales of corporate bonds are not as severe as in the 2008 crisis, when the entire
market crashed.35
We then directly examine the effect of investor flows to illiq-liq spreads using the VAR model
with one lag.36 Two variables in the VAR model have the following orders: monthly flows to
35 Choi and Shin (2016) also show that the price effects of corporate bond mutual fund fire sales are much larger during the 2008 crisis than during the Taper Tantrum period. 36 We choose a 1-month lag based on the Akaike information criterion (AIC) and Bayesian information criterion (SBIC).
35
corporate mutual fund investors and monthly average illiq-liq spreads, respectively.
[Table 4]
Table 4 shows the VAR results. The only difference between Panels A and B is that the latter
uses the median, instead of average, illiq-liq spreads. Table 4 shows that the monthly flows and
illiq-liq spreads are significantly positively related. In other words, the illiq-liq spreads decrease
when there are negative flows (i.e., outflows), holding everything else constant.
We then examine the effect of shocks in flows on illiq-liq spreads through impulse response
analyses. By imposing a Cholesky ordering of variables (1. flows, 2. spreads), we can examine the
effect of flow shocks that are orthogonalized to contemporaneous illiq-liq spreads.
[Figure 10]
In Figure 10, we provide the orthogonalized impulse response function (oIRF) of illiq-liq
spreads to flow shocks. Figure 10 Panel A shows that following a one standard deviation shock to
the flows, the spreads exhibit quite persistent responses over several months. The responses of the
illiq-liq spreads start from about 5% (month 0) and grow to 16% of the standard deviation (month
3) and gradually decrease during the last 24 months. The responses are statistically significant at
the 95% level from months 3 to 7. This persistence is consistent with the selling pressure stories,
according to which mutual fund outflows tend to be very persistent. Even when only those
36
significant responses are cumulated, the cumulative responses comprise about 70% of the standard
deviation over 5 months. If we include the first three months, the percentage exceeds 100%. Thus,
the economic effects of the flows are large enough to change the sign of the illiq-liq spreads.
In sum, the VAR analysis results support the flight from liquidity.
6.4.3 Issuer Credit Rating Downgrades and Illiq-liq Spreads
The model implications are not limited to the market level, as discussed in Section 5. In this
section, we use the downgrades of an issuer’s credit rating as fire-sale episodes among bondholders
of the issuer. Specifically, we use the S&P long-term-issuer-level credit ratings obtained from
Compustat. To be included our sample, an issuer that experienced the downgrade should have the
sample bond pair (matched as described in Section 6.1) during the month of the downgrade. We
fix the pair (founded during the month of the downgrade) and track the illiq-liq spreads of the pair
for 3 months before and 9 months after the downgrade month. Finally, our sample consists of 76
downgrade events of issuers with available bond pairs issued by the issuer for illiq-liq spreads
between March 2005 and December 2014.
[Table 5]
Table 5 Panel A shows the number of downgrade events across years (Column 1) and across
three types of events: downgrades from investment grades to investment grades, from investment
grades to speculative grades, and from speculative grades to speculative grades. As shown in Table
37
5 Panel B, our sample covers only a small part of all of the downgrades in Compustat. Furthermore,
on average, the issuers in our sample have higher credit qualities. This limitation results from
requiring the yields observation of matched bond pairs to calculate the illiq-liq spreads.
[Figure 11]
Figure 11 shows how the illiq-liq spreads change on average around the downgrade events.
Starting from 3 months before the downgrade, the spreads slowly decrease, probably due to the
information leaks. During and following the downgrades, the decrease accelerates to the -0.4%
level. It remains relatively steady 2-6 months after the downgrade, and then recovers 9 months
after the downgrade. This result is consistent with the lower pricing of liquid bonds during times
of distress.
[Figure 12]
Figure 12 shows how the extent of changes in the illiq-liq spreads differs across the extent of
the shocks. We assume the price pressure episodes are more reflective of bonds downgraded to
junk levels rather than investment-grade levels. Thus, we expect the changes in illiq-liq spreads to
be relatively small for downgrades within investment-grade ratings than for others. The results
support this expectation. In Figure 12 Panel A, the illiq-liq spreads turn negative at a relatively
small magnitude (from 0.2% to -0.2%), yet remain statistically significant. However, in Figure 12
38
Panels B and C, the illiq-liq spreads fall more dramatically (0.8% to -0.8% in Panel B, -0.4% to -
1.6% in Panel C).
7. Conclusions
In this paper, we show that as opposed to the well-documented flight to liquidity, the flight
from liquidity, where a liquid bond is priced lower, not higher, than an illiquid bond, can be
observed during times of severe market-wide distress in the OTC markets. We develop a model
that extends the search-based OTC market model of Duffie et al. (2007) by allowing agents in the
markets to simultaneously hold and trade two bonds with different levels of search friction. The
agents move away more from liquid bonds than from illiquid bonds when the market deteriorates
due to the ease with which liquid bonds are sold. In normal times, or when the extent of the distress
is small, we obtain the standard result that the liquid bond is priced higher than the illiquid bond.
We then provide empirical evidence supporting the implications of the model. We find that younger
liquid bonds are cheaper than older illiquid bonds following liquidity events, such as large outflow
shocks to mutual fund investors, downgrades of issuers’ credit ratings, and the 2008 financial crisis.
39
Appendix A. Proof of Proposition 1.
(Proof)
To simplify the proof, we take advantages of similarity between our model and Duffie et al.
(2007)’s model. First, We make a new set of equations N1 ≡ {(5)+(6), (7)+(8), (9)+(10),
(11)+(12)}. Specifically, for example, (5) means Equation (5) in this paper, and (5) + (6) means a
new equation generated by adding left hand sides and right hand sides of Equations (5) and (6),
respectively. Then, (5) + (6) can be simplified as following.
(𝜇𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡) + 𝜇𝜇ℎ𝑜𝑜𝑛𝑛(𝑡𝑡)) = 2𝜆𝜆1(𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡) + 𝜇𝜇𝑆𝑆𝑜𝑜𝑛𝑛(𝑡𝑡)) ∙ (𝜇𝜇ℎ𝑛𝑛𝑜𝑜(𝑡𝑡) + 𝜇𝜇ℎ𝑛𝑛𝑛𝑛(𝑡𝑡)) −𝜆𝜆𝑑𝑑(𝜇𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡) + 𝜇𝜇ℎ𝑜𝑜𝑛𝑛(𝑡𝑡)) + 𝜆𝜆𝑢𝑢(𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡) + 𝜇𝜇𝑆𝑆𝑜𝑜𝑛𝑛(𝑡𝑡))
(A1)
The sum of fraction of hno and hnn agents, i.e. 𝜇𝜇ℎ𝑛𝑛𝑜𝑜(𝑡𝑡) + 𝜇𝜇ℎ𝑛𝑛𝑛𝑛(𝑡𝑡), represents the total
fraction of H agents who don’t have the bond 1 at time t. Likewise, other terms in Equation (A1)
can be interpreted in a similar vein. Then, We notice that all terms related to the bond 2 is canceled
out and N1 describes the evolution in fraction of agents for only one bond (the bond 1) as in Duffie
et al. (2007). Together with this observation and the results of Duffie et al. (2005, 2007), the
following statement holds; there is a unique stable steady-state solution for the set of equations N1,
which is a constant solution determined by following steady-state condition (A2). The proof
closely follows the proof of Proposition 1 in Duffie et al. (2005), hence it is omitted here.
�𝜇𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡) + 𝜇𝜇ℎ𝑜𝑜𝑛𝑛(𝑡𝑡)� = �𝜇𝜇ℎ𝑛𝑛𝑜𝑜(𝑡𝑡) + 𝜇𝜇ℎ𝑛𝑛𝑛𝑛(𝑡𝑡)� =�𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡) + 𝜇𝜇𝑆𝑆𝑜𝑜𝑛𝑛(𝑡𝑡)� =�𝜇𝜇𝑆𝑆𝑛𝑛𝑜𝑜(𝑡𝑡) + 𝜇𝜇𝑆𝑆𝑛𝑛𝑛𝑛(𝑡𝑡)� =0 (A2)
We define 𝑎𝑎 to be the steady-state equilibrium fraction of ‘H type bond 1 owners’. Then, 𝑎𝑎
is well-defined in the interval (0, 1) and unique regardless of any admissible initial conditions.
Next, We define N2 ≡ {(5)+(7), (6)+(8), (9)+(11), (10)+(12)}. By following the same
arguments in the above, N2 describes the evolution in fraction of agents for only one bond (now it
40
is the bond 2). There exists a unique stead-state equilibrium satisfying the following condition
(A3).
�𝜇𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡) + 𝜇𝜇ℎ𝑛𝑛𝑜𝑜(𝑡𝑡)� = �𝜇𝜇ℎ𝑜𝑜𝑛𝑛(𝑡𝑡) + 𝜇𝜇ℎ𝑛𝑛𝑛𝑛(𝑡𝑡)� =�𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡) + 𝜇𝜇𝑆𝑆𝑛𝑛𝑜𝑜(𝑡𝑡)� =�𝜇𝜇𝑆𝑆𝑜𝑜𝑛𝑛(𝑡𝑡) + 𝜇𝜇𝑆𝑆𝑛𝑛𝑛𝑛(𝑡𝑡)� =0 (A3)
We define 𝑏𝑏 to be the unique steady-state equilibrium fraction of ‘H type bond 2 owners’.
Then, 𝑏𝑏 is well-defined in the interval (0, 1).
As in Equations (5) – (12), agents’ intrinsic types (H or L) are changed by idiosyncratic shocks
but not changed by transactions. Thus in the equilibrium, the total fraction of L agents at time t is
𝜆𝜆𝑑𝑑/(𝜆𝜆𝑢𝑢 + 𝜆𝜆𝑑𝑑). We define 𝑦𝑦 as following.
𝑦𝑦 ≡ 𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡) + 𝜇𝜇𝑆𝑆𝑜𝑜𝑛𝑛(𝑡𝑡) + 𝜇𝜇𝑆𝑆𝑛𝑛𝑜𝑜(𝑡𝑡) + 𝜇𝜇𝑆𝑆𝑛𝑛𝑛𝑛(𝑡𝑡) = 𝜆𝜆𝑑𝑑/(𝜆𝜆𝑢𝑢 + 𝜆𝜆𝑑𝑑) (A4)
From now on, we assume t is large enough to achieve (A2), (A3), and (A4).
By (A2) and (A3), the following equations hold.
𝜇𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡) =−𝜇𝜇ℎ𝑜𝑜𝑛𝑛(𝑡𝑡) =−𝜇𝜇ℎ𝑛𝑛𝑜𝑜(𝑡𝑡) =𝜇𝜇ℎ𝑛𝑛𝑛𝑛(𝑡𝑡) (A5)
𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡) =−𝜇𝜇𝑆𝑆𝑜𝑜𝑛𝑛(𝑡𝑡) =−𝜇𝜇𝑆𝑆𝑛𝑛𝑜𝑜(𝑡𝑡) =𝜇𝜇𝑆𝑆𝑛𝑛𝑛𝑛(𝑡𝑡)
By the definitions of 𝑎𝑎 and 𝑏𝑏, Equation (A4), and Equations (1) – (3), the equilibrium
fractions of agents in system N1 and N2 can be expressed as following.
𝜇𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡) + 𝜇𝜇ℎ𝑜𝑜𝑛𝑛(𝑡𝑡) = 𝑎𝑎 𝜇𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡) + 𝜇𝜇ℎ𝑛𝑛𝑜𝑜(𝑡𝑡) = 𝑏𝑏
(A6)
𝜇𝜇ℎ𝑛𝑛𝑜𝑜(𝑡𝑡) + 𝜇𝜇ℎ𝑛𝑛𝑛𝑛(𝑡𝑡) = 1 − 𝑦𝑦 − 𝑎𝑎 𝜇𝜇ℎ𝑜𝑜𝑛𝑛(𝑡𝑡) + 𝜇𝜇ℎ𝑛𝑛𝑛𝑛(𝑡𝑡) = 1 − 𝑦𝑦 − 𝑏𝑏
𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡) + 𝜇𝜇𝑆𝑆𝑜𝑜𝑛𝑛(𝑡𝑡) = 𝑠𝑠1 − 𝑎𝑎 𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡) + 𝜇𝜇𝑆𝑆𝑛𝑛𝑜𝑜(𝑡𝑡) = 𝑠𝑠2 − 𝑏𝑏
𝜇𝜇𝑆𝑆𝑛𝑛𝑜𝑜(𝑡𝑡) + 𝜇𝜇𝑆𝑆𝑛𝑛𝑛𝑛(𝑡𝑡) = 𝑦𝑦 − 𝑠𝑠1 − 𝑎𝑎 𝜇𝜇𝑆𝑆𝑜𝑜𝑛𝑛(𝑡𝑡) + 𝜇𝜇𝑆𝑆𝑛𝑛𝑛𝑛(𝑡𝑡) = 𝑦𝑦 − 𝑠𝑠2 − 𝑏𝑏
where 𝑎𝑎, 1 − 𝑦𝑦 − 𝑎𝑎, 𝑠𝑠1 − 𝑎𝑎, 𝑦𝑦 − 𝑠𝑠1 − 𝑎𝑎, 𝑏𝑏, 1 − 𝑦𝑦 − 𝑏𝑏, 𝑠𝑠2 − 𝑏𝑏, 𝑦𝑦 − 𝑠𝑠2 − 𝑏𝑏 ∈ (0,1)
Given that 𝑠𝑠1 and 𝑠𝑠2 are exogenously given parameters, and 𝑎𝑎, 𝑏𝑏, and y are constants
41
determined by other parameters in the model37, the system of eight equations (A6) has eight
unknowns. The solution, however, is not uniquely determined at this point, because (A6) does not
have full rank. However, (A6) can tell that each of eight 𝜇𝜇𝜎𝜎(𝑡𝑡)s (𝜎𝜎 ∈ 𝑇𝑇) can be expressed a
function of 𝜇𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡) and 𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡). Therefore, it is enough to show that there exist a unique steady-
state equilibrium for 𝜇𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡) and 𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡) in (0, 1). Then everything follows, because the stead-
state conditions for fractions of other agent types are automatically satisfied and the steady-state
mass of agents is uniquely determined by (A5) and (A6).
First, we show the existence of unique steady-state equilibrium for 𝜇𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡) and 𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡).
Then we show that the steady-state fractions are well-defined in (0, 1).
By combining Equations (5), (9), and (A6), the following equations hold.
𝜇𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡) = 2𝜆𝜆1(𝑠𝑠1 − 𝑎𝑎)�𝑏𝑏 − 𝜇𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡)� + 2𝜆𝜆2(𝑠𝑠2 − 𝑏𝑏)�𝑎𝑎 − 𝜇𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡)� − 𝜆𝜆𝑑𝑑𝜇𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡) + 𝜆𝜆𝑢𝑢𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡) (A7)
𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡) = −2𝜆𝜆1(1 − 𝑦𝑦 − 𝑎𝑎)𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡) − 2𝜆𝜆2(1 − 𝑦𝑦 − 𝑏𝑏)𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡) + 𝜆𝜆𝑑𝑑𝜇𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡) − 𝜆𝜆𝑢𝑢𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡)
Then, we convert (A7) to a matrix form as following.
�𝜇𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡)𝜇𝜇𝑙𝑙𝑜𝑜𝑜𝑜(𝑡𝑡)�
= A �𝜇𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡)𝜇𝜇𝑙𝑙𝑜𝑜𝑜𝑜(𝑡𝑡)� + 𝐵𝐵
where A = �−2[𝜆𝜆1(𝑠𝑠1 − 𝑎𝑎) + 𝜆𝜆2(𝑠𝑠2 − 𝑏𝑏)] − 𝜆𝜆𝑑𝑑 𝜆𝜆𝑢𝑢
𝜆𝜆𝑑𝑑 −2[𝜆𝜆1(1 − 𝑦𝑦 − 𝑎𝑎) + 𝜆𝜆2(1 − 𝑦𝑦 − 𝑏𝑏)] − 𝜆𝜆𝑢𝑢�
B = �2𝜆𝜆1(𝑠𝑠1 − 𝑎𝑎)𝑏𝑏 + 2𝜆𝜆2(𝑠𝑠2 − 𝑏𝑏)𝑎𝑎
0�
(A8)
Since (A8) is a set of linear differential equations, it has a stable equilibrium if all eigenvalues
of the Jacobian have negative real parts. In this case, the Jacobian is constant and equal to the
matrix A in (A8). To show real parts of all eigenvalues of 2 by 2 matrix A is negative, it is
enough to show that 𝑡𝑡𝑟𝑟𝑎𝑎𝑡𝑡𝑒𝑒(A) < 0 and 𝑑𝑑𝑒𝑒𝑡𝑡(A) > 0.
37 Note again that 𝑎𝑎 and 𝑏𝑏 do not depend on the initial value of agents mass.
42
By the assumption. 𝜆𝜆1, 𝜆𝜆2, 𝜆𝜆𝑑𝑑, and 𝜆𝜆𝑢𝑢 are positive. Also, by (A6), 𝑠𝑠1 − 𝑎𝑎, 𝑠𝑠2 − 𝑏𝑏, 1 −
𝑦𝑦 − 𝑎𝑎, and 1 − 𝑦𝑦 − 𝑏𝑏 are positive. Then, the following inequalities hold.
𝑡𝑡𝑟𝑟𝑎𝑎𝑡𝑡𝑒𝑒(A) = −2[𝜆𝜆1(𝑠𝑠1 − 𝑎𝑎) + 𝜆𝜆2(𝑠𝑠2 − 𝑏𝑏)] − 𝜆𝜆𝑑𝑑 − 2[𝜆𝜆1(1 − 𝑦𝑦 − 𝑎𝑎) + 𝜆𝜆2(1 − 𝑦𝑦 − 𝑏𝑏)] − 𝜆𝜆𝑢𝑢 < 0 (A9) 𝑑𝑑𝑒𝑒𝑡𝑡(A) = 4𝜆𝜆12(𝑠𝑠1 − 𝑎𝑎)(1 − 𝑦𝑦 − 𝑎𝑎) + 4𝜆𝜆1𝜆𝜆2(𝑠𝑠2 − 𝑏𝑏)(1 − 𝑦𝑦 − 𝑎𝑎) + 2𝜆𝜆𝑑𝑑𝜆𝜆1(1 − 𝑦𝑦 − 𝑎𝑎) +
4𝜆𝜆1𝜆𝜆2(𝑠𝑠1 − 𝑎𝑎)(1 − 𝑦𝑦 − 𝑏𝑏) + 4𝜆𝜆22(𝑠𝑠2 − 𝑏𝑏)(1 − 𝑦𝑦 − 𝑏𝑏) + 2𝜆𝜆𝑑𝑑𝜆𝜆2(1 − 𝑦𝑦 − 𝑏𝑏) +2𝜆𝜆𝑢𝑢[𝜆𝜆1(𝑠𝑠1 − 𝑎𝑎) + 𝜆𝜆2(𝑠𝑠2 − 𝑏𝑏)] > 0
(A10)
Now, the existence of stable equilibrium is proved. Next, we show the equilibrium solution
is unique in (0, 1).
Showing the uniqueness is simple. The steady-state solution is a constant solution of 0 =
A �𝜇𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡)𝜇𝜇𝑙𝑙𝑜𝑜𝑜𝑜(𝑡𝑡)� + 𝐵𝐵, which is just a set of two linear equations. Since 𝑑𝑑𝑒𝑒𝑡𝑡(A) ≠ 0 as shown in (A10),
𝑟𝑟𝑎𝑎𝑜𝑜𝑟𝑟(A) = 2. Hence, 𝑟𝑟𝑎𝑎𝑜𝑜𝑟𝑟([A|B]) = 2 and the system has a unique solution.
To show that the equilibrium fractions are in (0, 1), it is enough to show all eight fractions are
positive in the equilibrium because (A6) holds. By Equation (5), the following equation holds.
lim𝜇𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡)→0+
��𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡) = lim𝜇𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡)→0+
[2𝜆𝜆1[𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡) + 𝜇𝜇𝑆𝑆𝑜𝑜𝑛𝑛(𝑡𝑡)]𝜇𝜇ℎ𝑛𝑛𝑜𝑜(𝑡𝑡) + 2𝜆𝜆2[𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡) +
𝜇𝜇𝑆𝑆𝑛𝑛𝑜𝑜(𝑡𝑡)]𝜇𝜇ℎ𝑜𝑜𝑛𝑛(𝑡𝑡) + 𝜆𝜆𝑢𝑢𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡)] − lim𝜇𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡)→0+
[𝜆𝜆𝑑𝑑𝜇𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡)] (A11)
The last term of Equation (A11) converges to zero. Thus Equation (A11) show that
lim𝜇𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡)→0+
��𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡) > 0 provided that other fractions (i.e., 𝜇𝜇ℎ𝑛𝑛𝑜𝑜(𝑡𝑡), 𝜇𝜇ℎ𝑜𝑜𝑛𝑛(𝑡𝑡), 𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜(𝑡𝑡), 𝜇𝜇𝑆𝑆𝑜𝑜𝑛𝑛(𝑡𝑡),
𝜇𝜇𝑆𝑆𝑛𝑛𝑜𝑜(𝑡𝑡)) are positive. In other words, 𝜇𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡) are increasing (i.e., ��𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡) > 0) as it goes to zero
if other fractions are positive. For other fractions, similar conditions can be shown by using
Equations (6) – (12). That means, unless at least one of fractions becomes negative, all fractions
can’t be negative because they are continuous and will be increasing near zero. Thus, for any
proper initial values, that is in (0, 1) and satisfying assumptions (2) – (4), 𝜇𝜇ℎ𝑜𝑜𝑜𝑜(𝑡𝑡) and other seven
processes of agents’ fractions stay on positive. ■
43
Appendix B.
In this Appendix B, We derive the dynamics of lifetime utilities of agents in Equations (16)-
(23). First, we measure the time as the time after previous aggregate shock. In other words, from
now on, the denoted time t means time t passed after the last aggregate shock. The lifetime utility
of agents depends on the type of agents. Here, we show the utility of loo type agents.
We define the following 𝜏𝜏s are the next stopping times at which following events happen;
𝜏𝜏1=agent’s intrinsic type changes by the idiosyncratic shock, 𝜏𝜏2= trade bond 1 with hno agent,
𝜏𝜏3= trade bond 1 with hnn agent, 𝜏𝜏4= trade bond 2 with hon agent, 𝜏𝜏5= trade bond 2 with hnn
agent, 𝜏𝜏6= the aggregate shock arrived. These are six events that the utility of agent can be jumped.
For example, at 𝜏𝜏4 (if happened), the loo agent sell bond 2 to the hon agent, and the loo agent
collects the price and become a lon agent. Finally, we define 𝜏𝜏 = min (𝜏𝜏1, … , 𝜏𝜏6).
Then, the lifetime utility of loo agents satisfies the following equation.
𝑉𝑉𝑆𝑆𝑜𝑜𝑜𝑜 = 𝐸𝐸𝑡𝑡[∫ 𝑒𝑒−𝑆𝑆(𝑢𝑢−𝑡𝑡)2(1 − 𝛿𝛿)𝑑𝑑𝑢𝑢𝜏𝜏𝑡𝑡���������������
Utility gain (or loss)from owning the bonds
+ 𝑒𝑒−𝑆𝑆(𝜏𝜏1−𝑡𝑡)𝑉𝑉ℎ𝑜𝑜𝑜𝑜𝟏𝟏{𝜏𝜏1=𝜏𝜏}�������������New capital arrives
(the idiosyncratic shock)
+ 𝑒𝑒−𝑆𝑆(𝜏𝜏2−𝑡𝑡)�𝑉𝑉𝑆𝑆𝑛𝑛𝑜𝑜 + 𝑃𝑃𝑆𝑆𝑜𝑜𝑜𝑜ℎ𝑛𝑛𝑜𝑜,1�𝟏𝟏{𝜏𝜏2=𝜏𝜏}���������������������Sell bond 1 to the ℎ𝑛𝑛𝑜𝑜 agent
and recieve the price
+ 𝑒𝑒−𝑆𝑆(𝜏𝜏3−𝑡𝑡)�𝑉𝑉𝑆𝑆𝑛𝑛𝑜𝑜 + 𝑃𝑃𝑆𝑆𝑜𝑜𝑜𝑜ℎ𝑛𝑛𝑛𝑛,1�𝟏𝟏{𝜏𝜏3=𝜏𝜏}���������������������
Sell bond 1 to the ℎ𝑛𝑛𝑛𝑛 agentand recieve the price
+ 𝑒𝑒−𝑆𝑆(𝜏𝜏4−𝑡𝑡)�𝑉𝑉𝑆𝑆𝑜𝑜𝑛𝑛 + 𝑃𝑃𝑆𝑆𝑜𝑜𝑜𝑜ℎ𝑜𝑜𝑛𝑛,2�𝟏𝟏{𝜏𝜏4=𝜏𝜏}���������������������Sell bond 2 to the ℎ𝑜𝑜𝑛𝑛 agent
and recieve the price
+ 𝑒𝑒−𝑆𝑆(𝜏𝜏5−𝑡𝑡)�𝑉𝑉𝑆𝑆𝑜𝑜𝑛𝑛 + 𝑃𝑃𝑆𝑆𝑜𝑜𝑜𝑜ℎ𝑛𝑛𝑛𝑛,2�𝟏𝟏{𝜏𝜏5=𝜏𝜏}���������������������
Sell bond 2 to the ℎ𝑛𝑛𝑛𝑛 agentand recieve the price
+ 𝑒𝑒−𝑆𝑆(𝜏𝜏6−𝑡𝑡)(𝑉𝑉𝑆𝑆𝑜𝑜𝑜𝑜(0))𝟏𝟏{𝜏𝜏6=𝜏𝜏}�����������������The aggregate shock arrives
]
(B1)
where 𝟏𝟏{𝜏𝜏𝑖𝑖=𝜏𝜏} is an indicator variable that is equal to 1 if 𝜏𝜏𝑖𝑖 = 𝜏𝜏 (i.e. the event i happens first
among the six events above) and 0 otherwise.
Then by differentiating Equation (B1) with respect to t, the following equation is obtained.
��𝑉𝑆𝑆𝑜𝑜𝑜𝑜 = 𝑟𝑟𝑉𝑉𝑆𝑆𝑜𝑜𝑜𝑜 − 𝜆𝜆𝑢𝑢(𝑉𝑉ℎ𝑜𝑜𝑜𝑜 − 𝑉𝑉𝑆𝑆𝑜𝑜𝑜𝑜) − 2𝜆𝜆1𝜇𝜇ℎ𝑛𝑛𝑜𝑜�𝑉𝑉𝑆𝑆𝑛𝑛𝑜𝑜 − 𝑉𝑉𝑆𝑆𝑜𝑜𝑜𝑜 + 𝑃𝑃𝑆𝑆𝑜𝑜𝑜𝑜ℎ𝑛𝑛𝑜𝑜,1� −2𝜆𝜆1𝜇𝜇ℎ𝑛𝑛𝑛𝑛�𝑉𝑉𝑆𝑆𝑛𝑛𝑜𝑜 − 𝑉𝑉𝑆𝑆𝑜𝑜𝑜𝑜 + 𝑃𝑃𝑆𝑆𝑜𝑜𝑜𝑜ℎ𝑛𝑛𝑛𝑛,1� − 2𝜆𝜆2𝜇𝜇ℎ𝑜𝑜𝑛𝑛�𝑉𝑉𝑆𝑆𝑜𝑜𝑛𝑛 − 𝑉𝑉𝑆𝑆𝑜𝑜𝑜𝑜 + 𝑃𝑃𝑆𝑆𝑜𝑜𝑜𝑜ℎ𝑜𝑜𝑛𝑛,2� −
(B2)
44
2𝜆𝜆2𝜇𝜇ℎ𝑛𝑛𝑛𝑛�𝑉𝑉𝑆𝑆𝑜𝑜𝑛𝑛 − 𝑉𝑉𝑆𝑆𝑜𝑜𝑜𝑜 + 𝑃𝑃𝑆𝑆𝑜𝑜𝑜𝑜ℎ𝑛𝑛𝑛𝑛,2� − 𝜁𝜁(𝑉𝑉𝑆𝑆𝑜𝑜𝑜𝑜(0) − 𝑉𝑉𝑆𝑆𝑜𝑜𝑜𝑜) − 2(1 − 𝛿𝛿) The utility of other types of agent can be calculated in a similar vein. Then Equations (16)-
(23) are derived. We omit the derivation of utilities of other types to save the space.
45
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50
Table 1. Model Parameters
This table shows the model parameters used in Figure 1.
𝜆𝜆1 𝜆𝜆2 λd λu 𝜁𝜁 𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ 𝑠𝑠𝑖𝑖 𝛿𝛿 𝑞𝑞 𝑟𝑟 2500 125 0.2 2 0.05 0.7 0.75 2.5 0.5 0.1
Table 2. Distribution of Agent Masses
This table shows the distribution of agents in the steady-state equilibrium (Penal A) and right after the aggregate shock (Panel B). The results of the model using the parameters specified in Table 1 are reported below.
Panel A. Steady-state Equilibrium
��𝜇ℎ𝑜𝑜𝑜𝑜 ��𝜇ℎ𝑜𝑜𝑛𝑛 ��𝜇ℎ𝑛𝑛𝑜𝑜 ��𝜇ℎ𝑛𝑛𝑛𝑛 ��𝜇𝑆𝑆𝑜𝑜𝑜𝑜 ��𝜇𝑆𝑆𝑜𝑜𝑛𝑛 ��𝜇𝑆𝑆𝑛𝑛𝑜𝑜 ��𝜇𝑆𝑆𝑛𝑛𝑛𝑛
0.6740 0.0758 0.0725 0.0868 0.0002 0.0000 0.0033 0.0874
Panel B. Post-shock Distribution
𝜇𝜇ℎ𝑜𝑜𝑜𝑜∗ 𝜇𝜇ℎ𝑜𝑜𝑛𝑛∗ 𝜇𝜇ℎ𝑛𝑛𝑜𝑜∗ 𝜇𝜇ℎ𝑛𝑛𝑛𝑛∗ 𝜇𝜇𝑆𝑆𝑜𝑜𝑜𝑜∗ 𝜇𝜇𝑆𝑆𝑜𝑜𝑛𝑛∗ 𝜇𝜇𝑆𝑆𝑛𝑛𝑜𝑜∗ 𝜇𝜇𝑆𝑆𝑛𝑛𝑛𝑛∗
0.2022 0.0227 0.0218 0.0260 0.4719 0.0531 0.0541 0.1481
51
Table 3. Summary Statistics This table provides descriptive statistics for 1,018 unique matching pairs of liquid (young) and illiquid (old) bonds in our sample between March 2005 and March 2015. The sample consists of matching bond pairs issued by the same issuer with same credit ratings, and very similar time-to-maturities, but different ages (as a proxy of liquidity). Age (Young Bond) and Age (Old Bond) are the numbers of years passed after issuing for younger and older bonds, respectively, of a matching pair. The reported ages are the ages of the matching pair bonds when the matching pair first appeared in the sample. TTM (Young Bond) and TTM (Old Bond) are times to maturities in years of younger bonds and older bonds, respectively, of the matching pairs. The times to maturity are those when the matching pair first appeared in the sample. Rating is the credit rating score of bonds where we assign 21 to AAA rating. Offering Amounts (Young Bond) and Offering Amounts (Old Bond) are dollar offering amounts for the young bond and old bond of the matching pairs, respectively. We report the number of observations (N), mean, standard deviation (Std.), and 5%, 25%, 50% (median), 75%, and 95% quantiles.
N Mean Std. 5% 25% 50% 75% 95%
Age (Young Bond) 1,018 0.27 0.50 0.00 0.01 0.02 0.24 1.63
Age (Old Bond) 1,018 5.01 2.33 3.00 3.18 4.59 5.57 9.67 TTM (Young Bond) 1,018 5.66 3.62 2.26 4.17 5.01 6.02 10.02
TTM (Old Bond) 1,018 5.60 3.60 2.01 4.01 5.04 6.13 10.36 Rating 1,005 14.36 3.23 8 13 15 16 19 Offering Amounts (Young Bond, $M) 1,018 707,325 588,226 150,000 300,000 500,000 1,000,000 2,000,000
Offering Amounts (Old Bond, $M) 1,018 679,756 627,069 100,000 250,000 500,000 900,000 2,000,000
52
Table 4. VAR Analyses on Illiq-liq Spreads and Investor Flows This table provides relationship between illiq-liq spreads and investment cash flows by using a vector autoregression (VAR) model with one lag. We use two monthly variables in following order; corporate bond mutual fund flows, and illiq-liq spreads. Both variables are standardized to mean 0 and standard deviation 1. In Panel A, We use monthly averages of illiq-liq spreads and in Panel B, We use monthly medians of illiq-liq spreads. Each month total net assets-weighted averages of monthly flows to corporate bond mutual funds (in % of total net assets) are estimated, by using only those mutual funds that hold both liquid and illiquid bonds used in estimating illiq-liq spreads during the month. We use the holding information available at the beginning of month. The illiq-liq spread is defined as the yield of an issuer’s illiquid bond minus the yield of the same issuer’s liquid bond with the same credit rating, and very similar time-to-maturity, but different age (as a proxy of liquidity). More detailed definition of illiq-liq spreads is in Section 6. The number of observations is 117 from March 2005 to December 2014. The values in parentheses are t-statistics, and ***, **, and * denote statistical significances at the 1%, 5%, and 10% levels, respectively.
Panel A. VAR (Monthly Flows, Monthly Average of Illiq-liq Spreads)
(1) (2)
Flows Illiq-liq
Flows (-1) 0.586*** 0.099**
(7.97) (2.19)
Illiq-liq (-1) -0.100 0.870***
(-1.37) (19.55)
Constant 0.010 -0.006
(0.13) (-0.14)
R2 0.36 0.77 Log likelihood -220.23
53
Panel B. VAR (Monthly Flows, Monthly Median of Illiq-liq Spreads)
(1) (2)
Flows Illiq-liq
Flows (-1) 0.572*** 0.079***
(7.74) (2.69) Illiq-liq (-1) -0.114 0.956***
(-1.56) (32.95) Constant 0.010 -0.002
(0.14) (-0.08)
R2 0.36 0.90 Log likelihood -169.57
54
Table 5. Downgrade of Firms with Available illiq-liq Matching Bond Pairs This table provides descriptive statistics about downgrades of issuers (i.e. firms) with at least one available illiq-liq pair during the month of downgrade. Panel A provides the number of downgrade events during each year from March 2005 to December 2014. While Column 1 represents all events in our sample, Column 2, 3, and 4 represent events of downgrades from investment-grades (i.e. BBB- and above) to investment-grades (IG -> IG), from investment-grades to speculative-grades (IG -> Junk), and events of downgrade from speculative-grades to speculative-grades (Junk -> Junk), respectively. Panel B provides the number of observations (N), mean, and standard deviation (std.) of issuer credit ratings and changes in the issuer credit ratings for our sample and also for all downgrades in Compustat, where we assign 21 to AAA rating.
Panel A. Downgrade Events across Years from 2005 to 2014
Number of downgrade events of firms
with available illiq-liq matching bond pairs
(1) (2) (3) (4)
Year All IG -> IG IG -> Junk
Junk -> Junk
2005 5 3 1 1 2006 2 1 1 0 2007 6 2 1 3 2008 14 8 1 5 2009 19 9 1 9 2010 7 4 0 3 2011 6 5 0 1 2012 8 4 1 3 2013 4 3 0 1 2014 5 4 1 0 Total 76 43 7 26
55
Panel B. Downgrade Events in Our Sample vs. All Events in Compustat
N Mean Std.
Issuer Rating (Our Sample) 76 12.76 3.85
Issuer Rating (All in Compustat) 2,294 10.74 3.78 ΔIssuer Rating (Our Sample) 76 -1.43 0.96 ΔIssuer Rating (All in Compustat) 2,294 -1.41 0.86
56
Figure 1. Price of Liquid and Illiquid Bonds at the Aggregate Shock
This figure shows the prices of bond 1 and bond 2 before and after the aggregate shock. The prices are generated from the model with parameters specified in Table 1. The y-axis represents the price and x-axis represents the time in years. We assume the aggregate shock arrives once at t=0.5.
Pric
e
Time (years)
57
Figure 2. Price of Liquid and Illiquid Bonds at the Small Aggregate Shock
This figure shows the prices of bond 1 and bond 2 before and after the aggregate shock. The prices are generated from the model with parameters in Table 1, except that 𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ =10% is assumed. The y-axis represents the price and x-axis represents the time in years. We assume the aggregate shock arrives once at t=0.5.
Pric
e
Time (years)
58
Figure 3. Post-shock Distribution of Agent Fractions
This figure shows the post-shock (i.e. right after the shock arrives) distribution of agent fractions. The y-axis represents fraction of agents in percentage and the x-axis represents the assumed extent of aggregate shocks (i.e., 𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ ). The fractions are generated from the model with parameters in Table 1, except that 𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ is changed from 0% to 95%.
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% 80% 85% 90% 95%
Frac
tion
of A
gent
s
Size of the aggregate shock
hoo hon hno hnn loo lon lno lnn
59
Figure 4. Prices and Extent of the Aggregate Shock
This figure shows the prices of bond 1 and bond 2 long time after the last aggregate shock (Panel A) and right after the aggregate shock (Panel B) across the extent of shock (i.e., 𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ ). The prices are generated from the model with parameters in Table 1, except that 𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ is changed from 0% to 95%. The y-axis represents the price and x-axis represents the value of 𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ assumed.
Panel A. Prices Long Time after the Last Aggregate Shock
Panel B. Prices right after the Aggregate Shock
0
1
2
3
4
5
6
7
8
9
10
0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% 80% 85% 90% 95%
Price
Size of the aggregate shock
Bond 1 (Liquid) Bond 2 (Illiquid)
0
1
2
3
4
5
6
7
8
9
10
0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% 80% 85% 90% 95%
Price
Size of the aggregate shock
Bond 1 (Liquid) Bond 2 (Illiquid)
60
Figure 5. Prices Long Time after the Last Aggregate Shock and Different Levels of Search Intensity
This figure shows the prices of bond 1 and bond 2 long time after the last aggregate shock across different extent of liquidity differences. First, we solve model by assuming 𝜆𝜆1 (the search intensity of bond 1) = 250 and 𝜆𝜆2 (the search intensity of bond 2) = 125. In Panel A, the extent of aggregate shock is assumed to be small (𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ =10%), while it is assumed to be large (𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ =70%). Other parameters in the model is same as parameters in Table 1. While 𝜆𝜆2=125 and other parameters are fixed, we change 𝜆𝜆1 from 125 to 6250 by 125 and solve the model to get prices long time after the last aggregate shock for each pair of 𝜆𝜆1 and 𝜆𝜆2. The y-axis represents the price and x-axis represents the value of 𝜆𝜆1 assumed.
Panel A. Under the Presence of Small Aggregate Shocks (𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ =10%)
Panel B. Under the Presence of Large Aggregate Shocks (𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ =70%)
8.68.8
99.29.49.69.810
10.2
250
500
750
1000
1250
1500
1750
2000
2250
2500
2750
3000
3250
3500
3750
4000
4250
4500
4750
5000
5250
5500
5750
6000
6250
Price
Search Intensity of Bond 1
Bond 1 (Liquid) Bond 2 (Illiquid)
6.76.86.9
77.17.27.37.4
250
500
750
1000
1250
1500
1750
2000
2250
2500
2750
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Figure 6. Prices right after the Aggregate Shock and Different Levels of Search Intensity
This figure shows the prices of bond 1 and bond 2 right after the aggregate shock across different extent of liquidity differences. First, we solve model by assuming 𝜆𝜆1 (the search intensity of bond 1) = 250 and 𝜆𝜆2 (the search intensity of bond 2) = 125. In Panel A, the extent of aggregate shock is assumed to be small (𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ =10%), while it is assumed to be large (𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ =70%). Other parameters in the model is same as parameters in Table 1. While 𝜆𝜆2=125 and other parameters are fixed, we change 𝜆𝜆1 from 125 to 6250 by 125 and solve the model to get prices right after the arrival of aggregate shock for each pair of 𝜆𝜆1 and 𝜆𝜆2. The y-axis represents the price and x-axis represents the value of 𝜆𝜆1 assumed.
Penal A. Post-shock Prices under the Presence of Small Aggregate Shocks (𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ =10%)
Penal B. Post-shock Prices under the Presence of Large Aggregate Shocks (𝜋𝜋ℎ𝑥𝑥𝐵𝐵∗ =70%)
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Figure 7. Monthly Illiq-liq Spreads from 2005 to 2015
Panel A plots monthly average and 95% confidence bands of illiq-liq spreads from March 2005 through March 2015. Panel B plots monthly medians illiq-liq spreads. The illiq-liq spread is defined as the yield of liquid bond subtracted from yield of illiquid bond with same issuer, same rating, and very similar time-to-maturity, but different age (as a proxy of liquidity). The liquid bond is the youngest bond among the bonds issued by a firm. We require liquid bonds to have age younger than 2 years old and illiquid bonds to have age older than 3 years old. More detailed description of matching procedures for illiq-liq spreads is available in Section 6. Each day the illiq-liq spread is calculated for each pair of bonds. The daily yields are obtained from actual transaction data (TRACE). Each month the average (or median) of daily illiq-liq spreads is calculated. Vertical lines indicate the Lehman Brothers Bankruptcy (September 2008) and Taper Tantrum (May 2013). The recent financial crisis period (December 2007 to June 2009) is denoted by grey shades. The x-axis represents calendar dates and the y-axis represents yield spreads in percentages.
Panel A. Monthly Average of the Illiq-liq Spreads
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Panel B. Monthly Median of the Illiq-liq Spreads
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Figure 8. Differences in Time-to-maturity and Age between Liquid and Illiquid matching bonds.
This figure provides monthly average and 95% confidence bands of differences in time-to-maturity (Panel A) and age (Panel B) between the illiq-liq matching pair of bonds. Each month, we calculate the time-to-maturity (or age) of older bond minus the time-to-maturity (or age) of younger bond in a sample illiq-liq pair. Each month, we use average of age (and time-to-maturity) of a bond during the month. The x-axis represents calendar dates and the y-axis represents differences in time-to-maturity (Panel A) and age (Panel B) in years.
Panel A. Monthly Time Series of Differences in TTM
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Panel B. Monthly Time Series of Differences in Age
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Figure 9. Monthly Investor Capital Flows and Illiq-liq Spreads
This figure provides monthly investor flows (denoted as grey bars) and averages of illiq-liq spreads (denoted as a black line) from March 2005 through March 2015. The flows end December 2014 since our data for mutual funds ends 2014. The illiq-liq spread is defined as the yield of liquid bond subtracted from yield of illiquid bond with same issuer, same rating, and very similar time-to-maturity, but different age (as a proxy of liquidity). More detailed description of matching procedures for illiq-liq spreads is available in Section 6. We use capital flows to corporate bond mutual funds as a proxy of investor flows to the corporate bonds in our sample. Each month total net assets-weighted averages of monthly flows to corporate bond mutual funds (in % of total net assets) are estimated, by using only those mutual funds that hold both liquid and illiquid bonds used in estimating illiq-liq spreads during the month. We use the holding information available at the beginning of month. Vertical lines indicate the Lehman Brothers Bankruptcy (September 2008) and Taper Tantrum (May 2013). The x-axis represents calendar dates and the y-axis represents yield spreads in percentages.
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Figure 10. Impulse Responses of Illiq-liq Spreads to Investor Capital Flows
This figure plots the orthogonalized impulse responses of illiq-liq spreads to a one standard deviation shock to investor capital flows measured by net capital flows (in % of total net assets) to corporate bond mutual funds. We use a vector autoregression (VAR) model with one lag. We use two monthly variables in following order; corporate bond mutual fund flows, and illiq-liq spreads. Both variables are standardized to mean 0 and standard deviation 1. In Panel A, We use monthly averages of illiq-liq spreads and in Panel B, We use monthly medians of illiq-liq spreads. Each month total net assets-weighted averages of monthly flows to corporate bond mutual funds (in % of total net assets) are estimated, by using only those mutual funds that hold both liquid and illiquid bonds used in estimating illiq-liq spreads during the month. We use the most recent holding information available at the beginning of month. The illiq-liq spread is defined as the yield of an issuer’s illiquid bond minus the yield of the same issuer’s liquid bond with the same credit rating, and very similar time-to-maturity, but different age (as a proxy of liquidity). More detailed definition of illiq-liq spreads is in Section 6. Numbers in y-axis denote the impact on illiq-liq spreads as percentage of its one standard deviation. Numbers in x-axis denote months after the shock to fund flows. The orthogonalized impulse response function (oIRF) is denoted by a solid line. The error bands denoted by dashed lines represent the 95% confidence intervals by using a bootstrap methodology (1000 bootstraps).
Panel A. VAR (Monthly Flows, Monthly Average of Illiq-liq Spreads)
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Panel B. VAR (Monthly Flows, Monthly Median of Illiq-liq Spreads)
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Figure 11. Average Illiq-liq Spreads around Issuers’ Credit Rating Downgrades
This figure plots monthly average and 95% confidence bands of illiq-liq spreads around a month of credit rating downgrade. We use all downgrade events of issuers with at least one available illiq-liq pair during the month of downgrade between March 2005 and December 2014. We use the S&P long-term issuer level credit rating, obtained from Compustat. The illiq-liq spread is defined as the yield of liquid bond subtracted from yield of illiquid bond with same issuer, same rating, and very similar time-to-maturity, but different age (as a proxy of liquidity). More detailed description of matching procedures for illiq-liq spreads is available in Section 6. Each day the illiq-liq spread is calculated for each pair of bonds. The daily yields are obtained from actual transaction data (TRACE). Each month the average of daily illiq-liq spreads is calculated. The average of illiq-liq spread is denoted as a solid line. The error bands denoted by dashed lines represent the 95% confidence intervals on the mean. Numbers in x-axis denote months around the downgrade month (0), from three months before (-3) to nine months after (9) the event. The y-axis represents yield spreads in percentages.
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Figure 12. Subsample Analyses of Average Illiq-liq Spreads around Issuer Rating Downgrade
This figure plots monthly average and 95% confidence bands of illiq-liq spreads around a month of credit rating downgrade. This figure is a subsample analysis of Figure 11. Panel A is a subsample of events where issuer credit ratings drop from investment-grades (i.e. BBB- and above) to investment-grades. Panel B is a subsample of events where issuer credit ratings drop from investment-grades to speculative-grades (i.e. BB+ and below). Panel C is a subsample of events where issuer ratings drop from speculative-grades to speculative-grades. We use the S&P long-term issuer level credit rating, obtained from Compustat. The illiq-liq spread is defined as the yield of liquid bond subtracted from yield of illiquid bond with same issuer, same rating, and very similar time-to-maturity, but different age (as a proxy of liquidity). More detailed description of matching procedures for illiq-liq spreads is available in Section 6. Each day the illiq-liq spread is calculated for each pair of bonds. The daily yields are obtained from actual transaction data (TRACE). Each month the average of daily illiq-liq spreads is calculated. The average of illiq-liq spread is denoted as a solid line. The error bands denoted by dashed lines represent the 95% confidence intervals on the mean. Numbers in x-axis denote months around the downgrade month (0), from three months before (-3) to nine months after (9) the event. The y-axis represents yield spreads in percentages.
Panel A. Downgrades from Investment-Grades to Investment-Grades
Panel B. Downgrades from Investment-Grades to Speculative-Grades
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Panel C. Downgrades from Speculative-Grades to Speculative-Grades