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Dealer costs and customer choice

WP 23-13

Lucas B. Dyskant
Barclays Corporate &
Investment Bank
André C. Silva
Nova School of Business and
Economics
Bruno Sultanum
Federal Reserve Bank of Richmond

Dealer costs and customer choice∗
Lucas B. Dyskant

André C. Silva

Barclays Corporate & Investment Bank

Nova School of Business and Economics

Bruno Sultanum
Federal Reserve Bank of Richmond

November 2023

Abstract
We introduce a model to explain how an increase in intermediation costs leads to
structural changes in the corporate bond market. We state three facts on corporate
bond markets after the Dodd-Frank act: (1) an increase in customer liquidity provision
through prearranged matches, (2) a paradoxical decrease in measured illiquidity, and (3)
an increase in the illiquidity component on the yield spread. Investors take longer to
ﬁnish a trade and require higher illiquidity premium even though measured illiquidity
decreased. We introduce a search and matching model which explains these facts. It also
suggests the possibility of multiple equilibria and ﬁnancial instability when dealers face
high costs to intermediate transactions.
JEL classiﬁcation: D53, G12, G18.
Keywords: over-the-counter markets, intermediation costs, liquidity, corporate bonds,
Volcker rule, post-2008 regulation.
December 6, 2023. Updates in this link.

∗

The views in this paper are those of the authors and not necessarily the views of the Federal Reserve Bank
of Richmond, the Federal Reserve System, or Barclays Corporate & Investment Bank. Dyskant participated
in this work in his personal capacity and not as part of his employment with Barclays. We beneﬁted from discussions with Briana Chang, Veronica Guerrieri, Zhiguo He, Yesol Huh, Ben Lester, Dmitry Orlov, Sebastien
Plante, Erwan Quintin, Andreas Rapp, Philip Reny, Roberto Robatto, Sang Seo, Cathy Zhang and participants at several seminars. Silva thanks the hospitality of the Kenneth C. Griﬃn Department of Economics
at the University of Chicago, where part of this article was written. This work was partially funded by Fundação para a Ciência e a Tecnologia (UID/ECO/00124/2019, UIDB/00124/2020 and Social Sciences DataLab,
PINFRA/22209/2016), POR Lisboa and POR Norte (Social Sciences DataLab, PINFRA/22209/2016).

1

1

Introduction

We introduce a model to explain three observations on the corporate bond market: (1)
the increase in customer-customer prearranged matches intermediated by dealers, for which
customers provide liquidity to other customers; (2) the decrease in measured illiquidity, even
though market participants indicate more diﬃculty in trading; and (3) the increase in the
importance of illiquidity for the yield spread. To explain these observations, we introduce a
search and matching models with elements of Lagos and Rocheteau (2009) and Hugonnier,
Lester, and Weill (2022). The model generates the three stated observations. Moreover, the
model implies the possibility of multiple equilibria and ﬁnancial crises.
In the model, dealers face intermediation costs to facilitate trading. When the intermediation cost increases, customers accept to wait until they are matched with another customer.
The intermediation cost creates a customer-dealer and a customer-customer market. The
customer-dealer market represents inventory trade and the customer-customer market represents customer liquidity provision as described by Choi et al. (2023). In the customer-dealer
market, customers do not incur higher search costs. They are willing to pay a higher price
for immediacy. In the customer-customer market, customers might pay a smaller price for
the asset (or sell for a higher price), but they take longer to trade.
An important ingredient of the model is the ability of investors to choose to participate
in customer-dealer or customer-customer markets—in the spirit of Guerrieri et al. (2010),
agents direct their search in ﬁnancial markets. An increase in dealer costs makes investors
look for other investors to trade, which represents customer liquidity provision intermediated
by dealers. For a large enough increase in dealer costs, this change in composition decreases
the aggregate bid-ask spread in equilibrium. Standard indicators of illiquidity rely on observed
transactions. Therefore, a decrease in the aggregate bid-ask spread implies an improvement
in standard indicators of liquidity. The model then generates a change in the structure of
the corporate bond market together with improvement of indicators of market illiquidity.
As the arrangement of customer-customer matches takes longer to be executed, the actual
frictions in the model increase. We use the model to propose a new measure of illiquidity.
The measure obtained from the model takes into account equilibrium prices, search frictions,
and the fraction of the market that engages in inventory trade or customer liquidity provision.
2

After the 2007-2009 ﬁnancial crisis, several regulations were enacted with the objective
of avoiding future ﬁnancial crises. The US enacted the Dodd-Frank Wall Street Reform
and Consumer Protection Act in 2010. To some extent, the Dodd-Frank Act and similar
regulations in other countries accomplished their goal. This was highlighted by the ability
of the ﬁnancial sector to resist the ﬂuctuations caused by the recent pandemic. However,
the focus of academics, practitioners and government oﬃcials turned to how such regulations
aﬀect the ﬁnancial sector in normal times, when the economy is not under distress. There are
indications that the form in which trades take place in over-the-counter markets has changed
after the regulations were put in place. We propose amodel of trading in over-the-counter
markets to analyze these changes.1
The improvement in the traditional measures of market liquidity after the 2008 ﬁnancial
crisis has been documented by Bessembinder, Jacobsen, Maxwell, and Venkataraman (2018).
We conﬁrm this ﬁnding with recent data for the BPW and Amihud liquidity measures (respectively after Bao, Pan, and Wang 2011 and Amihud 2002). We focus on the corporate
bond market, a market for which over-the-counter trading is the usual method of trade. The
improvement in measured market liquidity could suggest that regulations had a minor impact on ﬁnancial market liquidity. However, we show that the impact of illiquidity on the
yield spread of corporate bonds increased. While markets seem to be more liquid, the cost
of illiquidity increased.2
In addition to the changes in the illiquidity measures, Bessembinder et al. (2018) and
Choi et al. (2023) report a decrease in dealer trade frequency. Especially, Choi et al. (2023)
indicate a change in the composition of the provision of liquidity. The provision of liquidity
has increasingly been made by customers rather than dealers. In practice, there is a perceived
movement of customers from dealers as ﬁnal trade counterparties to other customers.
The model builds on Duﬃe, Gârleanu, and Pedersen (2005), Lagos and Rocheteau (2009)
1

During a 2015 congressional hearing, for example, Rep. French Hill questioned the Federal Reserve Chair
at the time, Janet Yellen, on whether regulations were to blame for the deterioration of liquidity on diﬀerent
bond markets. Yellen replied: “I am not ruling out the possibility that regulations could play a role here, it is
simply we have not been able to understand through a lot of diﬀerent factors and we need to look at it more
to sort out just what is going on and what the diﬀerent inﬂuences are, but I am not ruling that out.”
2
We obtain similar ﬁndings on the importance of illiquidity for the yield spreads as Li and Yu (2023) and
Wu (2023), which worked in coincident and independent papers. Our empirical results are diﬀerent in some
aspects (such as our focus on the BPW and Amihud measures) and they complement their ﬁndings. However,
our focus is on the model to explain the movements in the corporate bond market.

3

(LR) and Hugonnier, Lester, and Weill (2022) (HLW). Depending on the parameters, it
implies LR or HLW as particular cases. Time is continuous. There is an asset that pays
dividends over time. There are two types of traders: customers and dealers. Dealers have
access to a competitive inter-dealer market where the asset is traded at an equilibrium price.
Customers trade in a decentralized way with dealers or with other customers. There are search
frictions when customers search for a trade counterparty. These search frictions are diﬀerent
for ﬁnding another dealer or customer. In particular, the search friction to ﬁnd a dealer is
smaller than the search friction to ﬁnd another customer. Customers are heterogeneous in
the valuation of the asset.
Each asset has a stochastic maturity date and a stochastic issue opportunity. The heterogeneity in asset valuation implies gains to trade. Assets trade hands over time. We
characterize the stationary distribution of asset holdings as well as the equilibrium bid-ask
spreads. We then show that the empirical behavior of market illiquidity and its correlation
with yield spreads can be rationalized by the model when we interpret the Dodd-Frank act
as causing an increase in intermediation costs.
The increase in costs faced by dealers increases the equilibrium bid-ask spread of dealers.
Customers that do not have the asset but have a high valuation of it still trade with dealers.
They pay a high ask price because they want to ﬁnd a trade counterparty fast. Similarly,
customers that have the asset but have a low valuation of the asset also look for dealers to
trade. They accept a lower bid price as they want to sell the asset fast. On the other hand,
customers that have intermediary valuations of the asset avoid trading with dealers. They
wait to be matched with other customers to avoid the surcharge in the form of large bid-ask
spreads. Empirically, Choi et al. (2023) ﬁnd that matched customers in fact pay lower spreads.
We show that an increase in trade costs increases the number of customer-customer trades.
As a result, the measured bid-ask spread decreases as well as measured illiquidity measures.
The paper is organized as follows. Section 2 states the observations on the corporate bond
market after 2008. Section 3 describes the model. Section 4 characterizes the equilibrium and
deﬁnes a measure of illiquidity obtained from the model. Section 5 discusses the implications
of the model for the illiquidity measures and for the liquidity premium. Section 6 concludes.

4

2

Changes in the corporate bond market

We call attention to three observations about the corporate bond market: (1) the increase in
the importance of trades customer-customer relative to trades customer-dealer (what we call
customer-customer trades are risk-free principal trades, as explained below); (2) the decrease
in the values of illiquidity measures; and (3) the increase in the liquidity premium. We
describe in detail these three observations below. Our goal with the model in section 3 is to
explain these observations.

2.1

Trade composition and perception of illiquidity

Our ﬁrst observation is the change in the composition of trades and increased perception
of illiquidity in the corporate bond market. As documented by Choi et al. (2023), after
the regulations that followed the 2008 ﬁnancial crisis, it is more common to ﬁnd trades for
which customers are matched with other customers instead of trades for which dealers use
their inventory to provide liquidity. Dealers facilitate both forms of trade. However, when
customers are matched with other customers, customers provide liquidity. In this case, the
dealer does not use its inventory of bonds. It is a customer that provides liquidity to other
customers either as a seller or a buyer.
The changes in the composition of trades have been connected with the enactment of
regulations that aﬀect depository institutions, such as banks, with access to the Federal
Reserve as a lender of last resort or to FDIC insurance (Adrian et al. 2017, Bao et al. 2018,
Choi et al. 2023; Duﬃe 2012 pointed out some risks of the new regulations). Especially,
the Volcker rule prohibits banks from engaging in proprietary trading, that is, trading that
uses the inventory of assets purchased earlier with the intention of proﬁting from a higher
sale price. The Volcker rule is part of the Dodd-Frank Wall Street Reform and Consumer
Protection Act. The Dodd-Frank act was enacted in July 2012. The Volcker rule was put
into eﬀect in July 2015 after a period of transition. The objective of the Volcker rule is to
limit risk taking of protected institutions. However, as the rule prohibits proprietary trade, it
decreases incentives of maintaining an inventory of assets and increases incentives of ﬁnding
matches between customers for direct trades without changes in inventory.

5

Choi et al. (2023) classify trades as being the result of a match customer-customer (DCDC), a match customer-dealer intermediated by an interdealer (DC-ID) and inventory trades.
They focus on over-the-counter trades in the corporate bond market. The classiﬁcation is
made with TRACE data, using dealer identiﬁers, counterparty pair types, and the time
record of the trades. DC-DC and DC-ID trades have matches identiﬁed within a period of
15 minutes. Inventory trades are not matched with the opposing side, which implies that the
asset is held after the trade as inventory. Since 2011, they ﬁnd an increase in the fraction of
customer-customer trades.
Customer-customer trades require a longer search and matching process for the execution
of the trade. Suppose that a customer contacts a dealer to sell a certain asset. This customer
demands liquidity. The conventional assumption would be that the dealer would buy the asset
and thus provide liquidity. Instead, especially for trades equal to or larger than 1 million
dollars, it is more frequent now that the dealer uses its relationships with other clients to
ﬁnd a customer willing to purchase the asset. The second customer provides liquidity. This
process can take time. The whole search process initiated when the dealer was contacted
about the intention to sell from the ﬁrst customer.
An evidence that customers indeed provide liquidity is that they are compensated for it
(Choi et al. 2023). Customers that buy the asset when the ﬁrst contact was a demand for
sale pay smaller or even negative spreads. The same happens for customers that sell the
asset when the trade was initiated by a demand to buy. In the same direction, Giannetti
et al. (2023) ﬁnd that bond mutual funds engage in more liquidity provision since 2015, and
that the performance of funds with strategies of liquidity provision has improved. Rapp and
Waibel (2023) show that regulatory costs are associated with the use of the client network for
the provision of liquidity. As we discuss below, the decrease in spreads charged to customers
implies a decrease in bid-ask spreads and an improvement of liquidity measures, even though
these DC-DC trades could take longer to be executed.
There is evidence that the search process can be costly. Transactions datasets such as
TRACE contain only the ﬁnal outcome of successful transactions. It is then not possible
to measure the duration of the whole search process. Using data from electronic platforms,
Kargar et al. (2023) ﬁnd that a substantial number of requests for quote are not promptly

6

fulﬁlled. If the quote is not fulﬁlled initially, it takes on average from 2 to 3 days for a trade
to be ﬁnalized. Another indicator of the need match customers is the advent of electronic
platforms to facilitate matching between trade counterparties (Hendershott et al. 2021).
The Volcker rule does not allow proprietary trading, but allows trading to facilitate transactions that were driven by customers. The law recognizes the role of dealers in the functioning of markets. Dealers cannot transact in a way intended to make proﬁts based on the
increase in the price of the asset, but they can proﬁt from bid-ask spreads. As a result, a
change in the market structure, with customer liquidity provision more frequent, would imply
higher transaction costs for those trades that are executed with the inventory of dealers.
In fact, Choi et al. (2023) ﬁnd that inventory trades have a transaction cost 60% higher
than before the ﬁnancial crisis. According to the classiﬁcation above, inventory trades do
not require a match of another dealer or customer to be executed. These trades are faster to
be ﬁnalized. Therefore, the higher transaction cost reﬂects a higher premium on immediacy
after the change in the market structure.
Early evidence that the new regulations aﬀected markets was shown by Adrian et al.
(2017) and Bao et al. (2018). Adrian et al. found that the ability to intermediate customer
trades of aﬀected institutions decreased. Bao et al. note that dealers aﬀected by the Volcker
rule have been the main liquidity providers. They found that the illiquidity of bonds in time
of stressed bonds has increased after the Volcker rule. As stated above, there was an increase
in the fraction of liquidity provided by customers, but only with a more costly matching
procedure. Therefore, the increase in illiquidity during stress events can be explained by a
change in the structure of markets toward costly matching.
There is therefore evidence that the structure of the corporate bond market has changed
toward the prearrangement of trades between customers. This prearrangement is made to
save on inventory of securities. As a result, the complete trade from the ﬁrst contact until
transaction becomes costly and protracted.
The changes in market structure, however, are not fully captured by standard measures of
illiquidity. These measures do not take into account the time for the arrangement of matches.
They use recorded prices at the ﬁnal moment of the trade. We next discuss the behavior of
the illiquidity measures over time.

7

2.2

Illiquidity measures

Our second observation is the improvement of the illiquidity measures since 2008. This
improvement is surprising given the changes in the market structure, as discussed above.
Trade in over-the-counter markets has moved toward prearranged matching of customers
instead of a faster trade using existing dealer inventory. It is more frequent to observe the
provision of liquidity made by customers. Given that these trades take longer to be executed,
it is surprising to observe an improvement of measured illiquidity. As we argue later, the
source of the diﬀerence is the fact that these measures use observed trading records. We later
oﬀer an alternative measure of illiquidity implied by the model in section 3.
We discuss the behavior of two measures of illiquidity: the γ measure, proposed by BPW,
and the Amihud measure, proposed by (Amihud, 2002). Figure 1 shows the evolution of the
measures over time.
The γ measure (BPW) is given by the covariance of subsequent price changes. The γi
measure for bond i is deﬁned as
γi = −Cov(∆pit , ∆pit+1 ),

(1)

where ∆pit = pit − pit−1 and pit is the logarithm of the clean price Pit of bond i on trade t.
The clean price is the bond price minus accrued interest since the last coupon payment. We
require a bond to have at least ten pairs of consecutive annualized-returns to estimate γi .
The objective of the measure is to extract a transitory component from observed prices.
This transitory component is interpreted as the impact of illiquidity, as eﬃcient markets with
no trading frictions imply uncorrelated returns. Let pt = ft + ut , where ft corresponds to
a fundamental component, equal to the value of the asset with no market frictions, and ut
corresponds to the transitory component, uncorrelated with the fundamental value. If the
fundamental value ft follows a random-walk process, then (1) implies that γ depends only
on the transitory component ut .
In addition to γ, we estimate the Amihud measure (Amihud 2002). The Amihud measure

8

for each bond is given by the average of absolute returns divided by the volume of trades,
Nid
1 X
|rij |
AMDid =
,
Nid j=1 Vid

(2)

where Nid is the number of available returns rij of bond i on day d, and Vid is the volume of
trade of bond i on day d in millions of dollars. We require at least two trades on each day to
estimate AMDid .
High Amihud measure means high price change per unit of volume, that is, high impact
or order ﬂow. Liquid markets should not show large changes in price relative to volume.
Therefore, a high Amihud measure is interpreted as lack of market liquidity. Table 1 reports
the correlations between γ, Amihud and other variables.3
Table 1: Correlations between illiquidity measures and other variables
γ
γ
1.00
AMD
.466
Spread
.385
CDS
.290
Volume
−.002
Frequency .047
Maturity
.163
Age
.017
Turnover
.013
ZTD
−.051

AMD
1.00
.444
.347
−.055
.196
.149
.109
−.008
−.198

Spread

1.00
.816
.040
.146
.092
.079
.125
−.080

CDS

1.00
.056
.140
−.026
.056
.127
−.088

Volume

1.00
.420
.097
−.202
.588
−.199

Frequency

1.00
−.052
−.001
.303
−.356

Maturity

1.00
−.075
.110
.084

Age

1.00
−.209
.016

Turnover

1.00
−.034

ZTD

1.00

Correlations between our main illiquidity measures, γ and AMD, and other commonly-used liquidity metrics,
the spread, and the CDS. Data description in appendix B. Spread is the corporate bond yield spread with
respect to the US Treasury with the same maturity (appendix B). Maturity is the issue’s time to maturity.
Maturity and age are calculated in years at the last business day of each month. Turnover is the traded volume
divided by the amount outstanding. ZTD is the percentage of zero-trading days.

We deﬁne a measure of aggregate market level illiquidity over time by taking the median,
mean or volume-weighted average of bond level measures in each cross-section. γ and AMD
increase when liquidity worsens. Both illiquidity measures strongly increased during the
ﬁnancial crisis up until the ﬁrst half of 2009. After the crisis, liquidity in the corporate bond
market gradually improved. The covid shock was large but brief and did not aﬀect the trend.
Figure 1 show the aggregate measures for the corporate bond market γ and AMD over time.
3

Additional measures of illiquidity are given, among others, by Mahanti et al. (2008) and Dick-Nielsen
et al. (2012). Mahanti et al. present a liquidity measure based on the accessibility of the issues. Dick-Nielsen
et al. introduce a measure computed by an average of diﬀerent illiquidity measures.

9

18

5
May 2010

Mean
Median

16

4.5

Aug 2011

4

14

Dec 2015
3.5

12

3

10
2.5

8

Mean
Median

2

6
1.5

4

1

2
0
2008

0.5

2010

2012

2014

2016

2018

0
2010

2020

2012

(a) γ, complete period

2014

2016

2018

2020

(b) γ, after 2010

14

4
May 2010

Mean
Median

3.5

12

Mean
Median

Aug 2011
Dec 2015

3

10

2.5

8
2

6
1.5

4
1

2

0
2008

0.5

2010

2012

2014

2016

2018

0
2010

2020

(c) AMD, complete period

2012

2014

2016

2018

2020

(d) AMD, after 2010

Figure 1: Time series of γ and AMD illiquidity measures. Periods: (1) May 2010, when the
Dodd-Frank bill passed the U.S. Senate, and the European debt crisis deepened with the
ECB announcement of the Securities Market Programme and Greece’s bailout; (2) August
2011, when the U.S. had its credit rating downgraded; and (3) December 2015, when the Fed
raised interest rates for the ﬁrst time since the ﬁnancial crisis.

10

It is possible to identify signiﬁcant events in the corporate debt market that create local
peaks in the time-series of the illiquidity measures. Two of these local peaks can be associated
with events related to credit and regulatory changes reminiscent of the ﬁnancial crisis. The
ﬁrst local peak can be associated with the passing of the Dodd-Frank bill by the US Senate
and the deepening of the European debt crisis around May 2010, when Greece agreed to a
bailout and the European Central Bank announced the Securities Market Programme. The
second local peak occurs around the downgrade of the US credit rating in August 2011. The
third peak in illiquidity follows the decision of the Federal Reserve in December 2015 to raise
interest rates. This increase in interest rates was the ﬁrst increase after a period of 7 years
of low interest rates close to zero.
Diﬀerent from the previous events, the decision of the FOMC to raise rates is not a credit
event. Its eﬀects on corporate bond liquidity can be understood by the microstructure of
the corporate bond market. The predominance of over-the-counter trading in the corporate
bond market provides dealers with an important role in supplying liquidity. To pursue their
activity, dealers must hold inventory, thereby incurring costs and risk. The ability with
which liquidity suppliers can manage their inventories aﬀects market liquidity (ComertonForde et al. 2010). The increase in illiquidity in this period is also consistent with the
changes in liquidity provision discussed above and the funding liquidity channel discussed in
Brunnermeier and Pedersen (2009) and in Boudt et al. (2017).
As interest rates rise, it becomes more costly and riskier for dealers to maintain corporate bonds in their inventories. Fleming and Remolona (1999) state that the release of major
macroeconomic changes by the FOMC worsens liquidity as it aﬀects inventory controls. Chordia et al. (2001) show that an increase in short term interest rate negatively impacts liquidity
because of the increase in inventory costs. Anderson and Stulz (2017) show that bond illiquidity is higher around extreme VIX changes more recently than it was before the crisis. These
results indicate that an increase in interest rates should have a higher impact on illiquidity
for riskier bonds. In accordance with this view, ﬁgure 2 shows that the Fed tightening had a
higher impact on illiquidity for high yield bonds.
According to our ﬁrst point, liquidity provision by dealers has been replaced by customer
liquidity provision. On the other hand, rather than a measured decrease in liquidity, the

11

6
May 2010
5

4.5

Inv Grade
High Yield

May 2010

Inv Grade
High Yield

Aug 2011

4

Aug 2011

3.5

Dec 2015
4

Dec 2015

3
2.5

3

2

2

1.5
1

1
0.5

0
2010

2012

2014

2016

2018

0
2010

2020

2012

(a) γ

2014

2016

2018

2020

(b) AMD

Figure 2: γ and AMD illiquidity measures after the 2008 ﬁnancial crisis according to the
corporate bond rating. Description of the selected periods in Fig 1.

140

2.5
May 2010
Aug 2011

120
2
Dec 2015
100
1.5

80

60

1

40
0.5
20

0
2010

0
2012

2014

2016

2018

2020

Figure 3: Median γ (left) and median Primary Dealers monthly net positions in corporate
debt instruments (right). Primary Dealers net positions in billions of U.S. Dollars; data from
the New York Fed. Description of the selected periods in Fig 1.

12

indicators or illiquidity show a declining trend. Figure 3 shows the decrease in inventories together with a decrease in the liquidity measures. The ﬁgure shows the net position of Primary
Dealers in corporate debt instruments (commercial papers, bonds, notes and debentures) and
illiquidity over time. We explain this apparent paradox with the model of section 3.
Also used as a measure of liquidity, turnover has declined after the 2008 ﬁnancial crisis.
We calculate daily turnover for an individual bond by dividing the amount traded in each
day by the amount outstanding at the end of the corresponding month. We then deﬁne the
monthly turnover measure for an individual bond by the median of its daily turnover. Figure
4 shows the median turnover of all bonds as an aggregate measure (the behavior looks similar
for the mean turnover). The ﬁgure also shows the moving average of 12 months.
The turnover rate decreased consistently after August 2009. The turnover 12-month
average decreased from 13.8% in August 2009 to 7.8% in August 2017. This decline is
consistent with the decrease in inventories as discussed above. As we show later, the decline
in turnover is consistent with our results in Proposition 3, which states that an increase
in intermediation costs, such as the one induced by the Dodd-Frank regulations, leads to a
decrease in turnover.
18
Turnover
Moving average (12 months)

16

14

12

10

8

6
2008

2010

2012

2014

2016

2018

2020

Figure 4: Monthly turnover (mean from daily values). Turnover has decreased since 2009.

13

2.3

Liquidity premium

Our third observation is the increase in the premium on illiquidity. We measure illiquidity
with monthly data on γ and AMD. We regress yield spreads on the illiquidity measures and
bond characteristics as controls such as risk (CDS), volume, and maturity. We ﬁnd that
the coeﬃcient on γ increased 5.7 times from 2007 to 2021 and that the coeﬃcient on AMD
increased 4.6 times in the same period (table 2).4
We run a Fama-MacBeth regression (Fama and MacBeth 1973) of corporate yield spreads
on γ and AMD. In the ﬁrst step, we estimate N cross-sectional regressions for each month,
where N is the number of issues. In the second step, we average the coeﬃcients over the
T periods in the sample. The t-statistics are calculated with standard errors corrected for
serial correlation by Newey-West (Newey and West 1987). Our sample implies 115 monthly
cross-sections from a sample of 3,073 bonds and 139,168 bond-month observations, as stated
in appendix B. We estimate the coeﬃcients for the complete period December 2007–June
2021 and the following four sub-periods: (1) the ﬁnancial crisis period from December 2007
to December 2009; (2) the post-crisis period from January 2010 to November 2015; (3) the
rate normalization period from December 2015 to February 2020; and (4) the COVID-19
pandemic period from March 2020 to June 2021. Table 2 summarizes our results. As the
results with γ and AMD are consistent with each other, we focus our discussion on the
illiquidity measure γ.
We start our analysis by looking at the full sample period. Taken individually, the coeﬃcients on the illiquidity measures are smaller only to the variables reﬂecting the issuer’s
coupon and credit quality (CDS and credit rating). The ﬁrst three lines show the results
with the illiquidity measures and CDS taken individually. Illiquidity is an important element
for customers to consider in the corporate bond market.
The inclusion of controls maintains the economic and statistical signiﬁcance of illiquidity.
The coeﬃcient on AMD for the regression with all controls (line 5 of table 2) implies that an
increase of one standard deviation of the AMD of an issue increases corporate yield spreads
4
Li and Yu (2023) and Wu (2023), in independent work, also ﬁnd an increase in the coeﬃcients related to
illiquidity. They use the bid-ask spread as a measure of illiquidity. Wu (2023) use monthly data, as we do. Li
and Yu (2023) uses quarterly data; they ﬁnd an increase of four times of the coeﬃcient on the bid-ask spread.
We ﬁnd a larger increase in the coeﬃcient on illiquidity using γ and AMD and monthly data.

14

Table 2: Corporate Yield Spreads on γ, AMD and controls
γ

AMD

CDS EqVol. Cpn

IG

Call

Volume Freq. Maturity

Age

Turnover ZTD Constant Adj.R2

Obs.

Panel A: Complete Period, December 2007–June 2021
.429
[11.01]

.181

196, 345

.149

196, 345

.646

196, 345

.370
[6.40]
.603
[6.66]

1.41
[7.86]
1.37
[8.09]
.664
[8.32]
.297
[3.04]
.237
[2.49]

.791

196, 345

.792

196, 345

.085

16, 687

.052

16, 687

.744

16, 687

.720
[3.74]
1.28
[3.56]

3.33
[6.19]
3.28
[7.58]
1.48
[5.75]
.638
[1.28]
.544
[1.40]

.791

16, 687

.792

16, 687

.169

90, 727

.120

90, 727

.668

90, 727

.157
[3.00]
.317
[6.78]

1.20
[18.13]
1.21
[17.22]
.446
[18.52]
.228
[3.85]
.187
[3.12]

.801

90, 727

.800

90, 727

.225

68, 997

.192

68, 997

.584

68, 997

.390
[8.58]
.585
[10.27]

.858
[23.32]
.818
[22.42]
.592
[25.85]
.199
[2.57]
.132
[1.74]

.776

68, 997

.778

68, 997

.278
[9.58]
.787
[32.11]
.143
.535
.548
.175
[9.61]
[20.38] [9.12] [11.93]
.098
.537
.579
.179
[8.70] [19.59] [9.71] [11.33]

−.851
[−9.20]
−.862
[−9.26]

−.042
[−1.82]
−.058
[−2.11]

−.467
.423
[−4.39] [8.68]
−.410
.281
[−3.96] [6.12]

.010
[2.01]
.015
[2.39]

.004
[.70]
−.005
[−1.05]

.010
[5.73]
.012
[6.79]

Panel B: Crisis, December 2007–December 2009
.151
[4.56]
.102
[3.96]
.935
[20.33]
.042
.758
.877
[4.37]
[16.19] [5.86]
.036
.764
.876
[3.62] [10.42] [6.26]

.290
−1.32 −.188
[4.97] [−3.07] [−2.16]
.300
−1.32 −.233
[5.77] [−3.89] [−2.42]

−.295
.009
−.046
[−1.02] [.18]
[−3.13]
−.196 −.062 −.048
[−.56] [−1.60] [−2.88]

.058
[3.09]
.045
[4.27]

.009
[1.84]
.010
[3.06]

Panel C: Post-Crisis, January 2010–November 2015
.327
[10.37]
.211
[9.85]
.827
[67.56]
.098
.541
.392
.146
[9.50]
[23.55] [5.97] [16.05]
.069
.547
.419
.148
[8.64] [22.93] [6.19] [16.83]

−.692
[−14.86]
−.709
[−15.27]

−.049
[−2.01]
−.065
[−2.86]

−.424
.460
[−4.51] [13.67]
−.411
.377
[−4.66] [14.99]

.018
−.003
[7.43] [−1.81]
.023
−.010
[10.38] [−4.22]

.008
[6.26]
.009
[8.30]

Panel D: Rate Normalization, December 2015–February 2020
.606
[16.40]
.368
[10.20]
.644
[19.13]
.225
.440
.569
.150
[13.06]
[16.52] [5.63] [19.55]
.150
.432
.622
.154
[9.06] [16.38] [6.22] [18.94]

−.738
.010
−.393
.535
[−16.23] [0.97] [−1.75] [6.15]
−.754
.008
−.328
.326
[−16.70] [0.76] [−1.45] [3.43]

.026
−.011
[23.51] [−8.08]
.033
−.021
[36.09] [−7.77]

.011
[2.58]
.013
[3.19]

Individual bond yield spreads regressed on the illiquidity measures γ and AMD and other variables for different periods. Coeﬃcients estimated using Fama-MacBeth and standard errors corrected by Newey-West.
T-statistics reported in square brackets. γ and AMD are the illiquidity measures detailed in section B. AMD
multiplied by 103 . EqVol. is the annualized volatility of the issuer’s equity returns and Cpn is the issue’s
coupon. IG is 1 if the bond is Investment Grade and 0 otherwise. Call is 1 if the bond is callable and 0
otherwise. Volume is calculated as the total $ amount traded ×10−11 . Frequency in thousands of trades.
Maturity and Age calculated in years at the last business day of the month. Turnover is the monthly median
of daily volume divided by amount outstanding and ZTD is the percentage of zero-trading days. Adj. R2 is
the time series average of cross-sectional adjusted-R2 ’s.

15

Table 2: Corporate Yield Spreads on γ, AMD and controls (continued)
γ

AMD

CDS EqVol. Cpn

IG

Call Volume Freq. Maturity

Age

Turnover ZTD Constant Adj.R2 Obs.

Panel E: COVID-19 Pandemic, March 2020–June 2021
.748
[11.38]
.566
[13.53]
.832
[14.88]
.239
.463
.658 .208
[8.62]
[6.10] [5.49] [8.47]
.165
.474
.686 .203
[4.54] [4.99] [5.32] [7.82]

−1.19
[−10.97]
−1.16
[−9.49]

.049
[1.30]
.038
[.89]

−1.17
[−5.50]
−.999
[−4.58]

.544
[4.53]
.246
[2.29]

.015
[2.65]
.020
[4.99]

−.003
[−.45]
−.010
[−1.82]

.021
[6.75]
.022
[8.00]

.708
[4.03]
.870
[5.22]

.1.11
[5.18]
.878
[5.49]
.592
[4.22]
.383
[5.50]
.315
[3.66]

.242

19,934

.293

19,934

.591

19,934

.799

19,934

.801

19,934

Individual bond yield spreads regressed on the illiquidity measures γ and AMD and other variables for different periods. Coeﬃcients estimated using Fama-MacBeth and standard errors corrected by Newey-West.
T-statistics reported in square brackets. γ and AMD are the illiquidity measures detailed in section B. AMD
multiplied by 103 . EqVol. is the annualized volatility of the issuer’s equity returns and Cpn is the issue’s
coupon. IG is 1 if the bond is Investment Grade and 0 otherwise. Call is 1 if the bond is callable and 0
otherwise. Volume is calculated as the total $ amount traded ×10−11 . Frequency in thousands of trades.
Maturity and Age calculated in years at the last business day of the month. Turnover is the monthly median
of daily volume divided by amount outstanding and ZTD is the percentage of zero-trading days. Adj. R2 is
the time series average of cross-sectional adjusted-R2 ’s.

28 basis points. This increase is equivalent to 13% of the average yield spread. The results
are even more substantial for γ. An increase of one standard deviation in the γ of an issue is
associated with an increase in the yield spread of 40 basis points. This increase corresponds
to 18% of the average yield spread.5
Turnover also shows a robust, positive coeﬃcient, contrasting with that of volume. In
some sense this result is intriguing given that volume is an element of the turnover and that
the two measures, as a result, have a strong correlation. As shown in table 1, volume and
turnover have a correlation of 0.727. One explanation, which is consistent with our model,
is that turnover better captures search frictions since trade volume can increase artiﬁcially
with the amount of bonds outstanding while turnover corrects for such changes.
Credit risk, captured by the CDS spread, is the most relevant pricing factor of corporate
spreads. An increase of 100 basis points to an issuer’s CDS is associated with an increase
in yield spreads of 59 basis points. The credit quality of an issuer is not exclusively priced
by its CDS. The credit rating also plays an important role in the pricing of yield spreads.
Investment grade bonds have spreads substantially lower, by 85 basis points, than their lower
5
We consider the one standard deviation change to be the time series average of cross-sectional standard
deviations within each interval. According to table 3, for the complete period, one standard deviation of γ
corresponds to 3.931. Similarly, we consider the average yield spread to be the time series mean of crosssectional averages within that period. These and other metrics are detailed in table 3.

16

grade peers.6
Among the additional controls, trade frequency, in particular, yields a positive coeﬃcient.
That is, bonds that trade more frequently have on average higher spreads. In contrast, volume
traded has a negative slope. These results combined indicate that bonds with larger average
size per trade have on average tighter spreads.7
We now analyze yield spreads and illiquidity over time. As shown in table 2, the coeﬃcient
on illiquidity has been increasing over time. So, although the illiquidity measures have been
decreasing over time, their importance on the credit spread has been increasing. We relate
this observation to the changes in regulations with the Dodd-Frank act. Illiquidity is most
relevant both in its ability to explain the variation of yield spreads and in its contribution to
basis points in relative terms.
The explanatory power of illiquidity is larger for later periods. In terms of absolute
basis points, a one standard deviation increase to an issue’s illiquidity is associated with an
increase of about 30 basis points to yield spreads during the monetary tightening period.
This value compares to 50 basis-points during the ﬁnancial crisis and 18 basis-points during
the post-crisis period.
Although the impact of a one standard deviation increase in illiquidity is smaller in
absolute terms after the rate normalization (30 versus 50 basis points during the ﬁnancial
crisis), the impact is higher in relative terms. The higher impact in relative terms occurs
because the corporate yield spreads decreased during the period. A one standard deviation
increase in bond illiquidity is associated with an increase in the average yield spread of 12%
during the ﬁnancial crisis and 11% in the post-crisis periods. After the normalization, an
increase of one standard deviation in bond illiquidity is associated with an increase in the
average yield spread of 18%.
6

See, for example, Forte and Pena (2009) for the role of CDS for corporate bonds.
In the same direction, Chordia et al. (2000) ﬁnd a positive correlation between spreads and the number
of individual transactions, and a negative correlation between spreads and individual volume.
7

17

3
3.1

Model
Environment

Our model describes over-the-counter markets as an economy in which agents take decisions
under search frictions. It builds on the literature initiated by Duﬃe et al. (2005).
Agents, time, goods and assets

There are two types of agents in the economy: a

measure one of inﬁnitely-lived customers and a measure one of inﬁnitely-lived dealers. Time
is continuous and inﬁnite. All agents discount the future at rate r > 0, and have access to a
transferable utility technology. There is an endogenous supply s ≥ 0 of assets. A unit of the
asset pays a unit ﬂow of dividend goods, which cannot be traded—that is, the agent holding
an asset consumes its dividend good. Customers can hold either zero or one unit of the asset,
and dealers do not hold assets. We refer to customers holding an asset as owners, and to
those not holding an asset as non-owners.
Preferences

Customers are heterogeneous in the utility ν that they derive from consuming

the dividend ﬂow of the asset. We refer to ν as the customer utility type. Types are ﬁxed
over time, common knowledge, independent across customers, and initially drawn from the
cumulative distribution F . The distribution F has support R, and a continuous density
f (·) > 0. Moreover, we assume that

R 2
ν f (ν)dν < ∞ and that there is no free disposal of

assets. The assumption that the distribution of types has unbounded support is convenient
because we do not have to consider corner solutions. However, we can obtain our main
results also with a bounded support [ν, ν̄] if we assume that the density is suﬃcient low at
¯
the extremes. Similarly, the assumption of no free disposal can be replace with the assumption
that the measure of agents with ν < 0 is suﬃciently low.
Decentralized market There is a decentralized asset market in the style of Duﬃe et al.
(2005), where trade occurs in one of two ways: customers can chose to search for a dealer,
as in LR, or to search for another customer, as in HLW. Customers cannot search for both
simultaneously. The contact rate is λD when searching for a dealer, and λC /2 > 0 when
searching for another customer. Hence, customers searching for a dealer will meet one at
18

rate λD , whereas those searching for another customer will meet one at rate λC (with arrival
rate λC /2 > 0, the customer ﬁnds another customer, and with arrival rate λC /2 > 0 another
customer ﬁnds him). Customers searching for other customers do not meet a customer
searching for dealers. We refer to trades between customers without dealer intermediation as
CC trades, and trades intermediated by dealers as DC trades.
The inter-dealer market Dealers have access to a competitive inter-dealer market where
assets are traded at an endogenous price p. When a dealer meets a customer, the dealer
bargains over the possible gains from trade coming from either buying an asset from the
customer, and reselling it at price p, or buying it at the price p, and selling it to the customer.
However, there is an intermediation cost τ ≥ 0 that has to be paid by the dealer if the dealer
buys or sells that asset, even if the asset only stays in the balance sheet for an inﬁnitesimal
amount of time. As a result, the net price by which the dealer sells an asset is p − τ , and the
net price by which the dealer buys an asset is p + τ .
Bargaining

Customers trade with dealers using a Nash bargaining protocol where the

bargaining power of the customer is θD ∈ [0, 1], and customers trade with other customers
n ∈ [0, 1],
using a Nash bargaining protocol where the bargaining power of the non-owner is θC
o = 1 − θn .
and the bargaining power of the asset owner is θC
C

We assume the following relation between search and bargaining parameters:
o , λ θ n }.
Assumption 1. λD θD > max{λC θC
C C

Assumption 1 guarantees that customers are better oﬀ searching for dealers than searching
for other customers when the intermediation cost τ is equal to zero. That is, in the absence
of intermediation costs, the dealers have a superior technology for trade.
Asset supply

Assets mature and customers produce new assets following two Poisson

distributions. With Poisson arrival rate µ > 0, the asset matures. Which means that the
asset disappears from the economy. With Poisson arrival rate η > 0, a customer can issue a
new asset at no cost. Similar to Bethune et al. (2022), the existence of asset maturity and
issuance implies that a steady state with positive trade emerges even without time-varying

19

types. Adding time-varying types in our model is not as straightforward as it would be in
their model, however, and we discuss this later in the paper.

3.2

Value functions and reservation value

We now describe how agents evaluate future payoﬀs and assets. We focus on steady-state
equilibria and omit time subscripts. Let Φo (ν) and Φn (ν) denote the cumulative distribution
of owners and non-owners. Since each owner holds exactly one unit of the asset, the measure
of assets is s =

R

dΦo . Let {ΩoD , ΩoC } and {ΩnD , ΩnC } be two partitions of R. The set ΩoD

represents the owners that search for dealers and ΩoC represents the set of owners that search
for non-owners. Analogously, ΩnD represents the set of non-owners that search for dealers and
ΩnC represents the set of non-owners that search for owners. We denote the search partitions
of customers by P = {ΩoD , ΩoC , ΩnD , ΩnC }, and assume that customers of the same type make
the same search decisions.
Denote the value functions of owners and non-owners searching for a dealer by VDo (ν)
and VDn (ν) and the value functions of owners and non-owners searching for other customers
by VCo (ν) and VCn (ν). The value function of customers is the maximum between the searching choices. That is, V o (ν) = max{VDo (ν), VCo (ν)} and V n (ν) = max{VDn (ν), VCn (ν)}. The
reservation value of a customer, ∆(ν), is the compensation that makes a customer indiﬀerent
between holding and not holding an asset. That is, ∆(ν) = V o (ν) − V n (ν).
Searching for dealers

The value function of an owner of type ν searching for dealers is

rVDo (ν) = ν − µ∆(ν) + λD θD max{(p − τ ) − ∆(ν), 0}.

(3)

The ﬁrst term of the value function, ν, is the utility ﬂow of holding the asset. The second
term, −µ∆(ν), is the loss of the reservation value due to asset maturity. The third term,
λD θD max{(p − τ ) − ∆(ν), 0}, is the proﬁt of an owner when meeting a dealer. When trading
with a dealer, an owner sells the asset if the inter-dealer price for the asset minus the intermediation cost cost τ is larger than the reservation value of the owner. If the owner sells the
asset, the gains from trade are (p − τ ) − ∆(ν) and the owner keeps a share θD of it. If the
owner does not sell the asset, the gains from trade are zero.

20

The value function of a non-owner of type ν searching for dealers is
rVDn (ν) = η∆(ν) + λD θD max{∆(ν) − (p + τ ), 0}.

(4)

The ﬁrst term of the value function, η∆(ν), is the gain of the reservation value obtained
from an issuance opportunity. The second term, λD θD max{∆(ν) − (p + τ ), 0}, is the proﬁt
of an non-owner when meeting a dealer. An non-owner buys an asset from a dealer if the
inter-dealer asset price plus the intermediation cost is smaller than the reservation value of
the non-owner. If the non-owner buys the asset, the gains from trade are ∆(ν) − (p + τ ) and
the non-owner keeps a share θD of it. If the non-owner does not buy the asset, the gains from
trade are zero.
Searching for customers

The value function of an owner of type ν searching for a non-

owner is
rVCo (ν)

= ν − µ∆(ν) +

Z
o
λC θC

Ωn
C

[∆(ν̃) − ∆(ν)]1{∆(ν̃)>∆(ν)} dΦn (ν̃).

(5)

The ﬁrst term of the value function, ν, is the utility ﬂow of holding the asset. The second
term, −µ∆(ν), is the loss of the reservation value because of asset maturity. The third term is
the expected proﬁts of an owner when meeting a non-owner. When trading with a non-owner
of type ν̃, an owner of type ν sells the asset if the reservation value of his counterparty, ∆(ν̃),
is higher than the reservation value of the owner. That is, if ∆(ν̃) > ∆(ν). The gains from
o of it. We obtain the expected value of
trade are ∆(ν̃) − ∆(ν) and the owner keeps a share θC

the gains from trade by integrating it in ν̃ over ΩnC using the distribution of owners Φn (ν̃).
The value function of a non-owner of type ν searching for an owner is
Z

rVCn (ν)

= η∆(ν) +

n
λ C θC

ΩoC

[∆(ν) − ∆(ν̃)]1{∆(ν)>∆(ν̃)} dΦo (ν̃).

(6)

The ﬁrst term of the value function, η∆(ν), is the expected gain of reservation value because
of an asset issuance. The second term is the expected proﬁt of a non-owner when searching
for an owner. When trading with an owner of type ν̃, a non-owner of type ν buys the asset
if his reservation value, ∆(ν), is higher than the reservation value of the owner. That is, if
21

n of
∆(ν) > ∆(ν̃). The gains from trade are ∆(ν) − ∆(ν̃) and the non-owner keeps a share θC

it. We obtain the expected value of the gains from trade by integrating it in ν̃ over ΩoC using
the distribution of owners Φo (ν̃).
Value functions, reservation value and the optimal searching choice The value
functions V o and V n of a customer type ν satisfy
V o (ν) = max{VDo (ν), VCo (ν)}

and V n (ν) = max{VDn (ν), VCn (ν)},

(7)

and the reservation value function satisﬁes
∆(ν) = V o (ν) − V n (ν).

(8)

We characterize the search partition P = {ΩoD , ΩoC , ΩnD , ΩnC } in the following way. For ΩoD ,
an owner searches for a dealer if it yields higher value than searching for a non-owner. In the
same way, for ΩnD , a non-owner searches for a dealer if it yields higher value then searching
for an owner. We have
ΩoD = {ν ∈ R; VDo (ν) ≥ VCo (ν)}

and ΩnD = {ν ∈ R; VDn (ν) ≥ VCn (ν)}.

(9)

and ΩnC = {ν ∈ R; VCn (ν) > VDn (ν)},

(10)

Analogously,
ΩoC = {ν ∈ R; VCo (ν) > VDo (ν)}

where we assume that customers search for dealers when indiﬀerent between searching for a
dealer or other customers, and search for other customers only when they are strictly better
oﬀ doing so than searching for dealers. For the equilibrium class that we consider, this
assumption is without loss of generality because there is a measure zero of customers that
are indiﬀerent between the two and trade in equilibrium.

22

3.3

Inter-dealer market clearing

The interdealer market clears when the measure of owners ﬁnding dealers to sell an asset is
equal to the measure of non-owners ﬁnding dealers to buy an asset. That is,
Z

λD

1{∆(ν)<p−τ } dΦ (ν) = λD

Z

o

ΩoD

Ωn
D

1{∆(ν)>p+τ } dΦn (ν).

(11)

The left-hand side describes the measure of owners that want to sell the asset at the interdealer market price net of the intermediation cost, p − τ , and ﬁnd a dealer. The right-hand
side have an analogous description for the sellers that want to buy an asset at the inter-dealer
market price added the intermediation cost, p + τ . As the search intensity parameter is the
same for potential sellers and buyers, this parameter cancels out from the formula. We will
use this equation to ﬁnd the equilibrium price p∗ in this market.

3.4

The distribution of assets

The cumulative distribution of owners is given by Φo and the cumulative distribution of
non-ownersis given by Φn . The change over time of the distribution of owners Φo satisﬁes
Φ̇o (ν) = ηΦn (ν) − µΦo (ν)
− λD
− λC

Z ν

−∞

1{ν̃∈ΩoD ,∆(ν̃)<p−τ } dΦo (ν̃) + λD

Z ν Z ∞
−∞ ν

Z ν Z ∞

+ λC

−∞ ν

Z ν
−∞

1{ν̃∈ΩnD ,∆(ν̃)>p+τ } dΦn (ν̃)

1{ν̃∈ΩoC ,ν̂∈ΩnC ,∆(ν̂)>∆(ν̃)} dΦ (ν̂)dΦo (ν̃)
n

1{ν̃∈ΩoC ,ν̂∈ΩnC ,∆(ν̃)>∆(ν̂)} dΦo (ν̂)dΦn (ν̃),

(12)

where Φ̇o (ν) = 0 for all ν in an steady-state equilibrium. On the right-hand side of (12), the
ﬁrst term accounts for the inﬂow of owners that issue an asset. The second term accounts
for the outﬂow of owners because of asset maturity. The third and fourth terms account for
owners searching for dealers. The third term for the outﬂow of owners with type below ν
searching for dealers and that sell their asset and the fourth for the inﬂow of non-owners with
type below ν searching for dealers and that buy an asset. The ﬁfth and sixth terms account
for customers searching for other customers. The ﬁfth term for the outﬂow of owners with
type below ν searching for other costumers, and that sell their asset to non-owners of type
23

above ν. The sixth term for the inﬂow of non-owners with type below ν searching for other
costumers, and that buy an asset from owners of type above ν.
We will see that the last term in equation (12) will be equal to zero in equilibrium. The
reason is that, in equilibrium, the reservation value function ∆ is monotonically increasing
in the utility type ν. As a result, the measure of non-owners with type ν̃ and owners with
type ν̂ such that ∆(ν̃) > ∆(ν̂) and ν̃ < ν̂ is equal to zero.
As the measure of customers, F , is exogenous, the measures of owners and non-owners
satisfy the equilibrium condition
Φo (ν) + Φn (ν) = F (ν).

(13)

All assets in the economy are held by owners. So, the stock of assets is s = Φo (∞).

3.5

Equilibrium

We deﬁne a symmetric stationary equilibrium in the following way.
Deﬁnition 1. An equilibrium is a family of value functions, reservations value, price, distributions and partitions, {V o , V n , ∆, p, Φo , Φn , P} satisfying equations (3)–(13).
An equilibrium can be a complicated object. To simplify it further, let ΩC = ΩoC = ΩnC =
(νl , νh ) and ΩD = ΩoD = ΩnD = (−∞, νl ] ∪ [νh , ∞). Notice that we can have ΩoC = ΩnC as we
have some customers of type ν holding the asset and other customers of the same type that
do not hold the asset. Deﬁne the following class of equilibrium.
Deﬁnition 2. A symmetric stationary equilibrium {V o , V n , ∆, p, Φo , Φn , P} is regular if
ΩC = ΩoC = ΩnC = (νl , νh ) and ΩD = ΩoD = ΩnD = (−∞, νl ] ∪ [νh , ∞) for some νl , νh ∈ R
satisfying νl ≤ νh with strict inequality if τ > 0, and the reservation value ∆ is continuous
and strictly increasing.
We use the notation {V o , V n , ∆, p, Φo , Φn , νl , νh } instead of {V o , V n , ∆, p, Φo , Φn , P} when
referring to a regular equilibrium since νl and νh characterize P. Figure 5 illustrates the
partition of a regular equilibrium.
What motivates looking for an equilibrium with the characteristics of a regular equilibrium
is the following. Customers with type close to each other, inside ΩC = (νl , νh ), choose to
24

Trade with other customers

ΩC

νl

νh

...

ν
...

ΩD
Trade with dealers

Figure 5: Partition {ΩD , ΩC } in a regular equilibrium.
trade among themselves to avoid the cost τ that has to be paid when trading with dealers
because they do not gain much from trading to justify the cost. Customers with extreme
types, that is, outside ΩC = (νl , νh ), are in a hurry to trade and they are willing to cover
higher dealer cost. These customers have very low or very high values for ν. Customers of
type ν ≤ νl that hold the asset search for dealers to sell their asset. Customers of type ν ≥ νh
that do not have the asset search for dealers to buy an asset.
We also impose that νl < νh when the intermediation cost is strictly positive, τ > 0.
The reason is that we can always build an equilibrium where customers do not search for
costumers because they expect other customers to do the same. In this case, the probability
of ﬁnding a customer is zero so customers may as well search for dealers. The assumption
that νl < νh if τ > 0 rules out equilibria built on this sort of weak inequality. In the next
section, we characterize a regular equilibrium and provide conditions that it exists.

4

Equilibrium

A regular equilibrium has two blocks. Given νl and νh , customers type ν ∈ ΩD = (−∞, νl ] ∪
[νh , ∞) act as in LR. Customers type ν ∈ ΩC = (νl , νh ) act as in HLW. We can solve these
two blocks separately using the tools developed in these papers.
The challenge is to characterize νl and νh that are consistent with the equilibrium equations (9) and (10). That is, to ﬁnd νl and νh such that customers searching for dealers, with
25

low types ν ≤ νl or high types ν ≥ νh , are not better oﬀ searching for other customers.
Analogously, that customers searching for customers, with intermediary types νl < ν < νh ,
are not better oﬀ searching for dealers.

4.1

Solving the LR block

The reservation value of a type-ν customer searching for a dealer is ∆(ν) = VDo (ν) − VDn (ν).
The value functions, VDo (ν) and VDn (ν), of a type-ν customer searching for a dealer when
holding and not holding an asset are stated in equations (3) and (4). Taking the diﬀerence
between the two equations to isolate ∆(ν) yields the following lemma.
Lemma 1 (Reservation value, dealer market). Consider a regular equilibrium {V o , V n , ∆,
p, Φo , Φn , νl , νh } and the set of utility types ΩD = (−∞, νl ] ∪ [νh , ∞). Then, the reservation
value ∆(ν) satisﬁes
∆(ν) =




σD [ν + λD θD (p − τ )],

ν ≤ νl



σD [ν − λD θD (p + τ )],

ν ≥ νh

(14)

where
σD =

1
.
r + η + µ + λD θ D

(15)

Moreover, ∆(νl ) ≤ p − τ and ∆(νh ) ≥ p + τ .
The derivative of the reservation value with respect to ν is given by σD . As HLW, we can
interpret σD as the local surplus at ν. It captures the trade surplus generated if the asset of
an agent type ν is transferred to an agent type ν + dν. The local surplus is independent of
ν for all ν ∈
/ (νl , νh ). That is, because all agents that search for a dealer face the same price
after bargaining and intermediation costs, the trade surplus in this case is constant.
In addition to the interpretation above, we interpret σD as an indicator of market friction.
To see this, suppose that there are no search frictions when the agent searches for a dealer.
In the model, when λD → ∞. In this case, ∆(ν) is constant in ν at p − τ or p + τ and so
σD = 0. Higher values of σD are associated with higher search frictions. We will later ﬁnd
an analogous value of search frictions for the customer-customer market.
We now turn to the distributions of owners and non-owners Φo and Φn . From Lemma 1,
owners of type ν ≤ νl always sell to dealers whereas non-owners of type ν ≥ νh always buy
26

from dealers. Moreover, non-owners of type ν ≤ νl are inactive. They have a small reservation
value ∆(ν). It neither compensates for them paying the price asked by dealers nor searching
for other customers. It does not compensate searching for other customers because they
would have to ﬁnd a customer with the asset and with an even smaller reservation value so
that this other customer would like to sell the asset. In the same way, owners of type ν ≥ νh
are inactive. Their high reservation value does not compensate the bid price made by dealers.
It also unlikely that this owner could ﬁnd another customer with even higher reservation value
that would like to purchase the asset. So, they do not search for other customers. Therefore,
non-owners of type ν ≤ νl and owners of type ν ≥ νh are inactive.
Lemma 2 (Distributions, dealer market). A regular equilibrium {V o , V n , ∆, p, Φo , Φn , νl ,
νh } is such that the cumulative distribution of owners satisﬁes
ηF (ν)
,
η + µ + λD

(16)

(η + λD )[1 − F (ν)]
η
−
η+µ
η + µ + λD

(17)

Φo (ν) = F (ν) − Φn (ν) =
for all ν ≤ νl , and
Φo (ν) = F (ν) − Φn (ν) =
for all ν ≥ νh . Moreover, νl and νh satisfy
ηF (νl ) = µ[1 − F (νh )]


and

Φo (νl ) + Φn (νh ) =

µ
.
η+µ

(18)



It is useful to deﬁne ν ∗ = F −1 µ/(µ + η) . When νl = νh = ν ∗ , all customers trade with
dealers and ν ∗ is the marginal customer—customers type ν > ν ∗ buy assets and customers
type ν < ν ∗ sell assets. When we decrease νl below ν ∗ , reducing the measure of customers
that sell assets, we have also to increase νh in order to keep the market clearing in the
inter-dealer market. That is, the asset inﬂow into the market,
λD Φo (νl ) = λD

27

ηF (νl )
,
η + µ + λD

equals the asset outﬂow from the market,


λD [Φ (∞) − Φ (νh )] = λD
n

4.2

n



µ[F (∞) − F (νh )]
µ
µ[1 − F (νh )]
− Φn (νh ) = λD
= λD
.
η+µ
η + µ + λD
η + µ + λD

Solving the HLW block

For customers searching for other customers, the value functions and reservation values are
obtained in the following way. For the value functions, from equations (5) and (6), the value
function of a type ν ∈ ΩC customer holding an asset satisﬁes
rVCo (ν)

= ν − µ∆(ν) + λC

rVCn (ν) = η∆(ν) + λC

Z ν
νl

Z νh
ν

o
θC
[∆(ν̃) − ∆(ν)]dΦn (ν̃), and

(19)

n
θC
[∆(ν) − ∆(ν̃)]dΦo (ν̃).

(20)

Combined with the deﬁnition of reservation value, ∆(ν) = VCo (ν) − VCn (ν), given in equation
(8), the equations above imply the following lemma.
Lemma 3 (Reservation value, customer-customer market). A regular equilibrium {V o , V n ,
∆, p, Φo , Φn , νl , νh } satisﬁes
Z ν

∆(ν) = ∆(νl ) +

σC (ν̃)dν̃

(21)

νl

for all ν ∈ (νl , νh ), and
σC (ν) =

n

1


o



o .



n Φo (ν) − Φo (ν )
r + µ + η + λC θC Φn (νh ) − Φn (ν) + θC
l

(22)

for almost all ν ∈ (νl , νh ).
The trade surplus between a seller of type ν and buyer of type ν + dν is approximately
equal to σC (ν)dν, thus providing us with the interpretation discussed in HLW of σC (ν)dν
as the local surplus. The function σC (ν) discounts the additional utility dν by the discount
rate r, the likelihood that the asset will mature µ, the loss in the likelihood of issuing an
asset η, and the loss in option value from either meeting another buyer with higher valuation








o Φn (ν ) − Φn (ν) , or ﬁnding another seller with lower valuation, λθ n Φo (ν) − Φo (ν ) .
λC θC
h
l
C

28

We now turn to the distributions Φo , Φn among customers searching other customers.
Lemma 4 (Distributions, customer-customer market). A regular equilibrium {V o , V n , ∆, p,
Φo , Φn , νl , νh } is such that the cumulative distribution of owners satisﬁes
Φ̃o (ν) = F (ν) − F (νl ) − Φ̃n (ν)
−

µ+η+λC [F (νh )−F (ν)−sC ]
2λC

q

+





{µ+η+λC [F (νh )−F (ν)−sC ]}2 +4λC η F (ν)−F (νl )
2λC

,

(23)

ν ∈ (νl , νh ), where Φ̃o (ν) ≡ Φo (ν) − Φo (νl ) and Φ̃n (ν) ≡ Φn (ν) − Φn (νl ), and
sC ≡ Φo (νh ) − Φo (νl ) =


η 
F (νh ) − F (νl ) .
µ+η

(24)

Figure 6 shows a representation of the reservation value as function of the utility type
ν. Lemma 1 implies that ∆(ν) is linear for ν ≤ νl and ν ≥ νh . Moreover, ∆(νl ) ≤ p − τ
and ∆(νh ) ≥ p + τ . That is, owners with ν ≤ νl choose sell to dealers and non-owners with
ν ≥ νh choose to buy from dealers. Lemmas 3 and 4 imply the nonlinear shape of ∆ in
(νl , νh ). Customers that trade with dealers have ν ≤ νl . Customers that trade with other
customers have ν ∈ (νl , νh ).

∆(ν)
∆(νh )

∆(νl )

νl

νh

ν

Figure 6: Reservation value as function of the customer type, ∆(ν). Customers that trade
with dealers have ν ≤ νl . Customers that trade with other customers have ν ∈ (νl , νh ).

29

4.3

Solving for νl and νh

The results in sections 4.1 and 4.2 establish necessary conditions for the equilibrium objects
V o , V n , ∆, p, Φo , Φn , s, νl and νh —that is, equations (4)–(24). In those equations, the equilibrium objects can all be written as functions of νl and νh . In this section, we provide
necessary conditions on νl and νh and also show that these conditions, combined with equations (4)–(24), are not only necessary but also suﬃcient for a regular equilibria, and therefore
it provides a full characterization.
Lemma 5. A regular equilibrium {V o , V n , ∆, p, Φo , Φn , νl , νh } satisﬁes
2τ λD θD =

Z νh
σC (ν) − σD

σD

νl

dν.

(25)

Moreover,
Z

Z

o
νh
ν
λC θ C
σC (ν̃)dν̃dΦn (ν)
λD θD νl νl
n Z νh Z νh
λ C θC
σC (ν̃)dν̃dΦo (ν).
= ∆(νh ) − τ −
λ D θD νl ν

p = ∆(νl ) + τ +

(26)

Lemmas 2 to 5 establish necessary conditions that are satisﬁed in all regular equilibria.
In the proposition below, we show that these conditions are not only necessary but suﬃcient.
Therefore they fully characterize a regular equilibrium.
Proposition 1. If a family {V o , V n , ∆, p, Φo , Φn , νl , νh } is a regular equilibrium, the family
{∆, p, Φo , Φn , νl , νh } satisﬁes equations (14)–(26). Reversely, if {∆, p, Φo , Φn , νl , νh } satisﬁes
equations (14)–(26), then {V o , V n , ∆, p, Φo , Φn , νl , νh } is a regular equilibrium where the value
functions V o and V n are constructed using equations (3)–(7).
Proposition 1 gives us an easy way to solve for an equilibrium. Given νl and νh , we can
substitute our solutions for Φo , Φn and σC into equation (25) to write then as functions of νl
and νh . We then deﬁne the function g : (−∞, ν ∗ ] → [ν ∗ , ∞) as g(νl ) = F −1 [(µ − ηF (νl ))/µ],
which uses the results in Lemma 2, to deﬁne implicitly νh as a function of νl . That is,
νh = g(νl ). An equilibrium νl then solves
G(νl ) = τ,
30

(27)

where G : (−∞, ν ∗ ] → R is given by
G(νl ) ≡

1
2λD θD

Z g(νl )
σC (ν; νl , νh ) − σD

σD

νl

dν,

(28)

Note that the function G satisﬁes G(ν ∗ ) = 0 and G(νl ) > 0 for νl suﬃciently small.
After obtaining a νl that satisﬁes G(νl ) = τ , we can then obtain all the other equilibrium
objects using equations (3)–(7) and (14)–(26). Moreover, as the ﬁrst part of proposition 1
establishes suﬃciency, all regular equilibrium can be obtained in this way.
An important result is that proposition 1 does not imply uniqueness. We can have
multiple equilibria because G may not be monotone. The intuition is the following. When
many customers decide to search for customers instead of for dealers, then the probability of
matching is higher and the gain of searching for customers increases. More customers search
for customers if they are convinced that others will follow this strategy. This behavior can
lead to multiple equilibria. But we can show that it only happens when intermediation costs
are suﬃciently high.
Proposition 2. There exists τ̄ > 0 such that a regular equilibrium is unique for τ ∈ [0, τ̄ ).
Proposition 2 establishes that strategic complementarity is not strong enough to generate
multiplicity when the intermediation cost τ is close to zero. Assumption 1 implies that
searching for dealers is in general preferable than searching for customers. If τ is small
enough, no matter how many customers search for customers, it is still preferable to search
for dealers. Multiplicity only happens if τ is large enough so that the measure of customers
searching for customers aﬀects the decision to search for dealers or for customers.
Figure 7 shows the diﬀerent types of equilibria. The ﬁgure shows the results from simulan = θ o = 0.5, λ = 3 and λ = 1. The distribution
tions with r = 0.05, µ = η = 0.3, θD = θC
D
C
C

of utility types, in panel 7a, is a combination of three normal distributions.8 Panel 7b shows
the function G and characterizes the equilibrium νl for given values of τ . The equilibrium is
unique around τ1 and τ3 . For τ = τ1 , the equilibrium νl is unique and approximately equal
to 4.6. For τ = τ3 , it is for νl around 0.7. For τ = τ2 , we have three equilibria associated
with νl approximately equal to 1.5, 2 and 2.5.
8

f (ν) = 0.45N (2, 0.25) + 0.1N (5, 0.25) + 0.45N (8, 0.25).

31

(a) Density of types f

(b) Function G

Figure 7: Economy with diﬀerent equilibrium patterns according to the intermediation cost τ .
Small customer-customer market for τ1 . Large customer-customer market for τ3 . Multiplicity
of equilibria for τ2 .
An interpretation of G is that it is a proxy for the expected diﬀerence in valuation of an
owner with νl that wants to sell the asset and a non-owner with νh that wants buy the asset,
both using CC trades. A large diﬀerence in valuation, high G, implies that a non-owner might
need to pay a substantial amount to the owner to acquire the asset. If G is high relative to
τ , it is better to switch from CC to DC trade. A buyer might pay p + τ net of bargaining in
a DC trade, but it would still be smaller than the expected price to pay in a CC trade. This
case is such that G(νl ) > τ . A switch from CC to DC implies an increase in νl and a smaller
interval (νl , νh ).
G(νl ) decreases with νl if the valuation of agents that engage in CC trades gets closer to
each other as νl increases. This is the case in ﬁgure 7 around τ3 . Consider an increase in τ .
It would decrease the gain of DC trades. The equilibrium νl would decrease and the set of
CC trades would increase. Similarly, an increase in λC would make CC trades more eﬀective.
It would imply a downward shift in G. For τ = τ3 , it would decrease νl and increase the set
of CC trades. We can apply the same reasoning to changes around τ1 .
The G function in this example, however, increases for certain values of νl as (νl , νh )
includes the higher density of utility types around 2 and 8 in the limits of the interval.
Surprisingly, an increase in τ2 would increase the equilibrium around νl∗ = 2. The measure of
customers trading with dealers would increase with the intermediation cost τ . Similarly, an

32

increase in λC , which makes CC trades more eﬀective, would shift G downward and decrease
the set of CC trades.
These eﬀects are related with the instability of equilibrium for νl = 2. For this equilibrium,
suppose that a small set of agents to the left of νl = 2 switch their decisions from DC trades to
CC trades. The set of agents in CC trades would increase to (νl − ϵ, νh′ ), where νh′ = g(νl − ϵ).
We would then have G(νl − ϵ) < G(νl ) < τ2 , which implies that it is beneﬁcial for an agent
to the left of νl − ϵ also to switch from DC to CC trades. All agents to the left would
behave in the same way, which would increase further the set of agents in CC trades, until
the equilibrium with νl around 0.7 is reached.
At τ = τ2 and νl around 1.5, the equilibrium is stable. A switch from DC to CC trades
of a small set of agents to the left of νl = 1.5 would increase G. It would be better to return
to DC trades. The same reasoning can be applied to a switch from CC to DC trades to the
right of νl = 1.5 and also to the other stable equilibrium for τ = τ2 around νl = 2.5.
A stable equilibrium is therefore associated with a region in which G is decreasing and
an unstable equilibrium with a region in which G is increasing. A small perturbation in the
set of agents in CC or DC trades in regions where G is decreasing would make agents return
to the previous decision of the counterparty. However, in regions where G is increasing, such
perturbation would make agents switch the trading counterparty permanently toward a new
equilibrium. For these reasons, when necessary to derive results, we focus on the region where
G is decreasing and so of a stable equilibrium.

4.4

A measure of illiquidity

The interpretation of σD and σC as measures of trade distortions leads to a natural deﬁnition
of a measure of illiquidity. σD and σC are the derivatives of the reservation value ∆ with
respect to ν. As discussed above, σD and σC given in (15) and (22) increase if the trade
distortions increase. If λD or λC go to inﬁnity, which means that there are no search frictions,
then σD or σC go to zero.
A natural measure of illiquidity obtained from the model is therefore given by a weighted
average of the frictions faced by all investors in the market. That is, deﬁne a measure of

33

illiquidity by

Z ∞

σ=

−∞

σi (ν) dF (ν),

(29)

where σi (ν) = σD (ν) when ν ∈ (−∞, νl ] ∩ [νh , +∞) and σi (ν) = σC (ν) when ν ∈ (νl , νh ).
Given the expressions of σD and σC in (15) and (22), we see that the measure of illiquidity
depends on the interest rate and characteristics of the asset (µ and η) as well as on the
measure of investors that engage in diﬀerent forms of trades.

5

Turnover, illiquidity, and liquidity premium

Several indicators are used to measure illiquidity in ﬁnancial markets. Here we focus on
turnover and the bid-ask spread. We also discuss the liquidity premium—that is, the compensation required to induce investors to purchase an asset that is less liquid.
Turnover is deﬁned by the ratio of the volume of assets traded over a certain period to
the amount outstanding of the asset. High turnover ratio indicates high trading activity and
is associated with liquidity. Denote turnover by T . The value of turnover implied by the
model is given by
R

λD {Φo (νl ) + [Φn (∞) − Φn (νh )]} + λC ννlh
R
T ≡
Φo (ν) dν
R
R
ν
ν
2λD Φo (νl ) + λC νlh ν h dΦn (ν̃) dΦo (ν)
.
=
η/(η + µ)

R νh
ν

dΦn (ν̃) dΦo (ν)

(30)

The volume of assets sold by customers to dealers is λD Φo (νl ) and the volume of assets
bought by customers from dealers is λD [Φn (∞) − Φn (νh )]. As market clearing implies that
λD Φo (νl ) = λD [Φn (∞) − Φn (νh )], we have that the total volume of bonds traded between
customers and dealers is 2λD Φo (νl ). The total volume of bonds traded between customers is
λC

R νh R νh
νl

ν

dΦn (ν̃)dΦo (ν). Finally, the amount of bonds outstanding is s =

R

dΦo (ν) =

η
η+µ .

Another measure of liquidity is the diﬀerence between the bid and ask prices. That is,
the diﬀerence between the bid on the asset made by the dealer when the customer wants to
sell an asset and the price asked by the dealer when the customer wants to buy an asset.
In our model, we deﬁne the bid-ask price as |pbuy − psell |, where psell is the value received
by the customer when the customer sells the asset, and pbuy is the value paid by the customer

34

when the customer purchases the asset. Usually, psell < pbuy .9 A common interpretation is
that the market is less liquid when |pbuy − psell | increases. We now calculate the value of
|pbuy − psell | as implied by the model, and show that this is not always the case.
In a regular equilibrium, the average price paid by customers buying from dealers is
R∞

pbuy
D

=

νh

[θD (p + τ ) + (1 − θD )∆(ν)]dΦn (ν)
Φn (∞) − Φn (νh )

R ∞ (1−θD )ν+(r+η+µ+λD θD )(p+τ )

=

νh

r+η+µ+λD θD
Φn (∞) − Φn (νh )

dΦn (ν)

R∞

=p+τ +

(1−θD )ν
n
νh r+η+µ+λD θD dΦ (ν)
,
Φn (∞) − Φn (νh )

(31)

and the average price paid by customers buying from customers is
R νh R ν

pbuy
C

=

νl

n
o
o
n
νl [θC ∆(ν) + θC ∆(ν̃)]dΦ (ν̃)dΦ (ν)
R νh R ν
.
o
n
νl νl dΦ (ν̃)dΦ (ν)

(32)

So the average price paid by customers is
buy
pbuy = P[buy from dealer]pbuy
D + P[buy from customer]pC

R

R

buy
νh ν
o
n
λD [Φn (∞) − Φn (νh )]pbuy
D + λC νl νl dΦ (ν̃)dΦ (ν)pC
R νh R ν
=
λD [Φn (∞) − Φn (νh )] + λC νl νl dΦo (ν̃)dΦn (ν)

(33)

Similarly, the average price received by customers selling to dealers is
R νl

psell
D

=

−∞ [θD (p

R νl

=

−∞

− τ ) + (1 − θD )∆(ν)]dΦo (ν)
Φo (νl )

(1−θD )ν+(r+η+µ+λD θD )(p−τ )
dΦo (ν)
r+η+µ+λD θD
Φo (νl )

R νl

=p−τ +

(1−θD )ν
o
−∞ r+η+µ+λD θD dΦ (ν)
,
Φo (νl )

(34)

and the average price received by customers selling to customers is
R νh R νh n
o
n
o
νl ν [θC ∆(ν̃) + θC ∆(ν)]dΦ (ν̃)dΦ (ν)
R νh R νh
.
psell
C =
n
o
νl

ν

dΦ (ν̃)dΦ (ν)

(35)

For example, if the customer sells the asset to a dealer, psell is the bid made by the dealer to buy the
asset. If the customer purchases the asset from a dealer, pbuy is the price asked by the dealer for the asset.
9

35

So the average price received by customers selling assets is
sell
psell = P[sell to dealer]psell
D + P[sell to customer]pC

R

R

νh νh
n
o
sell
λD Φo (νl )psell
D + λC νl ν dΦ (ν̃)dΦ (ν)pC
R νh R ν
=
.
λD Φo (νl ) + λC νl νl dΦn (ν̃)dΦo (ν)

(36)

The bid-ask spread is then given by
BA ≡ pbuy − psell =

λD Φo (νl )(pbuy
− psell
D )
R νhDR ν
o
n
λD Φ (νl ) + λC νl νl dΦ (ν̃)dΦo (ν)
R∞

=

2λD Φo (νl )τ

+ λD (1 − θD )

λD Φo (νl ) + λC

νh

R νh R ν
νl

R νl

νdΦn (ν)−

−∞

νdΦo (ν)

r+η+µ+λD θD

n
o
νl dΦ (ν̃)dΦ (ν)

.

(37)

The second and third equalities above are obtained using the inter-dealer market clearing
condition and the structure of the customer-customer market. The inter-dealer market clearing condition implies that Φo (νl ) = Φn (∞) − Φn (νh ). As every asset sold by a customer is
buy
bought by another customer in the customer-customer market, we have that psell
C = pC .

5.1

The impact of τ on turnover

In regions with a unique equilibrium, an increase in intermediation cost decreases the measure
of customers trading with dealers. As dealers under our assumptions have a more eﬃcient
matching technology, this force tends to decrease turnover when intermediation costs increase.
We call this the dealer-eﬃciency impact on turnover. However, there is another force. When
an asset is intermediated by dealers, there is not a long chain of trades to allocate the asset.
The asset goes from an investor with ν ≤ νl to a dealer, then another dealer, and then to
an an investor with ν ≥ νh . In customer-to-customer trades, the asset can potentially be
allocated by a long chain of customers until it gets to an investor with very high νh and ceases
to be traded. The length of this chain tends to increase turnover when intermediation costs
increase and more investors chose customer-customer trade. We call this the customer-chain
impact on turnover.
Proposition 3. The turnover is always decreasing in τ in a neighborhood of τ = 0. Moreover,
if the search technologies satisfy λC ≤ λD ≤

√
2−1
√
(η
2

36

+ µ), then the turnover is decreasing in

τ in all regions with a unique equilibrium.
For low intermediation costs (τ close to zero), the dealer-eﬃciency impact on turnover
dominates the customer-chain impact. Turnover decreases when the intermediation cost
increases. In this case, the measure of investors choosing customer-customer trade is small.
It takes too long to form a chain of trades and the asset is likely to mature before it is
formed. For τ not close to zero, it is possible that the customer-chain impact on turnover
dominates the dealer-eﬃciency impact. For this to happen, the search technology has to
be suﬃciently eﬃcient so that assets do not mature, or possible buyers get an issuance
opportunity, before the asset is traded and the chain is formed. Formally, we show that
whenever λC ≤ λD ≤

√
2−1
√
(η + µ)
2

the dealer-eﬃciency impact dominates the customer-chain

impact on turnover.

5.2

The impact of τ on bid-ask spreads

There are two ways in which an increase in the intermediation cost of dealers, τ , impacts the
bid-ask spread: the intensive margin and the extensive margin.
On the intensive margin, customers have to compensate dealers for their costs to intermediate the trade. As a result, an increase in τ implies an increase in the bid-ask spread charged
by dealers. On the extensive margin, the higher bid-ask spread charged by dealers drive customers to trade with other customers instead of trading with dealers (customer matching).
Since we assume that trading with customers is slow but has zero intermediation cost for
customers, the extensive margin reduces the average bid-ask spread.
These two forces move the bid-ask spread in opposite directions and therefore the impact
of τ on the average bid-ask spread is ambiguous. However, we ﬁnd that an increase in τ
from τ = 0 implies an initial increase in the bid-ask spread, which could be interpreted as a
decrease in liquidity. The intensive margin dominates in this case. On the other hand, for
a high enough value of τ , the bid-ask spread decreases. In this case, the extensive margin
dominates. The bid-ask spread would decrease, which could be interpreted as an improvement
in liquidity.
Proposition 4. The following holds regarding the behavior of bid-ask spreads.
i. There exists τ̃ > 0 such that the bid-ask spread is increasing in τ for all τ ∈ [0, τ̃ ).
37

ii. There exist τ0 , τ1 > 0, with τ0 < τ1 , and associated regular equilibria E0 and E1 , such
that the bid-ask spread in E0 is strictly bigger than the bid-ask spread in E1 .
Proposition 4 has two parts. Part 1 asserts that, for small intermediation costs, the bidask spread has the expected behavior—a higher costs for dealers to intermediate transactions
increases the bid-ask spread in the market. Part 2 shows that, on the other hand, the bid-ask
spread is not monotone in τ . This result is illustrated in Figure 8.
The intuition for Proposition 4 is the following. When τ is not very high, most trades are
DC trades. In this case, an increase in τ increases the trading cost for most customers. This
force increases the average bid-ask spread, which is the ﬁrst part of the proposition. However,
as stated in the second part of the proposition, if τ increases enough, many customers stop
trading with dealers and the bid-ask spread actually decreases. In fact, the bid-ask spread
converges to zero as τ goes to inﬁnity.

BA(τ )

0

τ1

τ2 τ3

τ

Figure 8: An increase in the intermediation cost parameter τ1 increases the bid-ask spread.
τ2 implies multiplicity of equilibrium. An increase in τ3 implies a decrease in the bid-ask.
Figure 8 shows a possible relation between the bid-ask spread and τ . It is compatible
with the function G in ﬁgure 7b. For small values of τ , the bid-ask spread is increasing in
τ . For intermediary values of τ , there can be multiplicity of equilibrium and corresponding
multiplicity of bid-ask spreads. For large values of τ , the bid-ask spread is decreasing τ .
A decreasing bid-ask spread as the intermediation cost τ increases is a result of the higher
number of customer-customer transactions.

38

5.3

The impact of τ on liquidity premium

The presence of bid-ask spreads in ﬁnancial markets raises concerns about the potential for
increased customer trading costs, which could lead to lower prices or wider spreads. We refer
to the increase in prices implied by higher illiquidity as the liquidity premium.
In our model, the liquidity premium is best measured as a function of σD and σC of
customers, as discussed in section 4.4. However, these parameters are not directly observable
in practice. Instead, we employ empirical measures of illiquidity, such as those proposed by
Bao et al. (2011).
One challenge with these empirical measures is that they capture the overall bid-ask
spread in the market. As discussed in the previous section, this spread can either narrow
or widen due to a composition eﬀect when intermediation costs increase. We show here
that increases in dealer intermediation costs can mechanically inﬂate or deﬂate the liquidity
premium due to their reliance on the bid-ask spread.
To understand this non-monotonic relationship, consider two economies, A and B, charo = θn = θ = 1 ,
acterized by identical parameters θD , λC , λD , µ, η, and r. Assume that θC
C
C
2

µ = η, τ A = τ , and τ B = τ + dτ , where dτ ≥ 0 is a small perturbation and ν̄ A > ν̄ B .
Furthermore, suppose that both economies possess a unique equilibrium, implying the existence of strictly decreasing functions GA (νl ) and GB (νl ) associated with economies A and
B, respectively. This uniqueness of equilibrium allows us to deﬁne the bid-ask spread as a
function of the intermediation cost, resulting in BAA = BA(τ ) and BAB = BA(τ + dτ ).
As discussed previously, the empirical measures of illiquidity employed in practice partially
capture the bid-ask spread. Let’s assume that the illiquidity measure is given by the following
equation:
L(τ ) = L̄(τ ) + αBA(τ )
where L̄(τ ) represents other aspects of illiquidity that increase with intermediation costs,
implying L̄′ (τ ) > 0, and α > 0 determines the proportion of the bid-ask spread captured by
the illiquidity measure. Consequently, the liquidity premium can be expressed as
LP (τ ) =

pA (τ ) − pB (τ + dτ )
,
L̄(τ + dτ ) + BA(τ + dτ ) − BA(τ ) − L̄(τ )

39

(38)

where pA (τ ) and pB (τ + dτ ) denote equilibrium prices in economies A and B.
Proposition 5. Assume that L̄′ (τ ) = dL̄ > 0 and dτ > 0 is suﬃciently small. Then, the
liquidity premium is non-monotone in τ . Moreover, the liquidity premium for low τ is lower
than the liquidity premium for high τ . That is, LP (0) < limτ →∞ LP (τ ) =

ν̄ A −ν̄ A
.
dL̄ dτ

Proposition 5 underscores the inherent non-monotonic relationship between liquidity premium and intermediation costs, which extends to any measure that captures bid-ask spreads.
This proposition also yields a key prediction of the model: as an economy transitions from low
to high intermediation costs, the liquidity premium tends to increase. This non-monotonic
behavior in the liquidity premium stems from its dependence on measures that capture the
bid-ask spread.

6

Conclusions

We propose a model to explain structural changes observed in the corporate bond market
since the 2008 ﬁnancial crisis. It has been identiﬁed that a larger fraction of trades executed
in a way that dealers do not need to maintain asset inventory. In these trades, customers
provide liquidity to other customers. These trades happen more frequently for larger amounts.
They take longer to be executed. At the same time, standard measures of illiquidity show a
decrease in illiquidity since 2008 whereas investors require a higher illiquidity premium. The
model explains these at ﬁrst sight conﬂicting observations.
The model combines Lagos and Rocheteau (2009) and Hugonnier, Lester, and Weill (2022).
Lagos and Rocheteau study trades between customers and dealers. Hugonnier, Lester, and
Weill study trades between customers and customers. We combine the two models two include
the decision of a customer to trade with a dealer with another costumers. Both models study
decisions on OTC markets with search frictions, as in Duﬃe et al. (2005).
In the model, τ represents the intermediation cost paid by dealers. We interpret the
regulations in Dodd-Frank, which include increased capital requirements, increased reporting requirements, and increased restrictions on trading activities, as an increase in τ . The
regulations made it more expensive for dealers to provide liquidity. We then examine the
equilibrium outcomes from the model.

40

When the intermediation cost of dealers increases, customers seek liquidity from other
customers, increasing the number of customer-customer trades and decreasing the number
of customer-dealer trades. In the context of the model, the measure of customers in the
customer-customer market increases. The average bid-ask spread, which considers the ﬁnal
transaction prices, decreases. However, the average trade becomes more costly. A measure of
illiquidity based on ﬁnal prices would imply a decrease in illiquidity, as we ﬁnd empirically.
The model allows us to propose a new measure of illiquidity. This measure takes into
account the distortions caused by the search frictions, as well as the bargaining power, number
of customers engaged in customer-customer or customer-dealer trades and other variables. An
increase in the measure of agents that engage in customer-customer trades increases the value
of this comprehensive measure of illiquidity.
The model implies the possibility of multiple equilibria and ﬁnancial crises. Depending
on τ , there can be an equilibrium with a small number of customer-customer trades and an
another one with a large number of customer-customer trades. This is so because the decision
to direct search on one market or the other depends on the expected number of agents that
engage in the same activity. Both equilibria are stable. A movement from one equilibrium to
another would cause abrupt changes in the market that can be perceived as ﬁnancial crises.
The 2008 ﬁnancial crisis generated a strong response in ﬁnancial regulations. Our results
indicate a way to connect the changes in regulations with changes in the structure of ﬁnancial
markets. Especially, in the structure of a market heavily based on over-the-counter trades
such as the corporate bond market.

References
Adrian, Tobias, Nina Boyarchenko, and Or Shachar, 2017, Dealer balance sheets and bond
liquidity provision, Journal of Monetary Economics 89, 92–109. (pp. 5 and 7.)
Amihud, Yakov, 2002, Illiquidity and stock returns: cross-section and time-series eﬀects,
Journal of Financial Markets 5, 31–56. (pp. 3 and 8.)
Anderson, Mike, and René M. Stulz, 2017, Is post-crisis bond liquidity lower?, Working Paper
23317, National Bureau of Economic Research. (pp. 11 and 64.)

41

Bao, Jack, Maureen O’Hara, and Xing (Alex) Zhou, 2018, The Volcker rule and corporate
bond market making in times of stress, Journal of Financial Economics 130, 95–113. (pp. 5
and 7.)
Bao, Jack, Jun Pan, and Jiang Wang, 2011, The illiquidity of corporate bonds, The Journal
of Finance 66, 911–946. (pp. 3, 8, 39, 64, and 68.)
Bessembinder, Hendrik, Stacey Jacobsen, William Maxwell, and Kumar Venkataraman, 2018,
Capital commitment and illiquidity in corporate bonds, The Journal of Finance 73, 1615–
1661. (p. 3.)
Bethune, Zachary, Bruno Sultanum, and Nicholas Trachter, 2022, An information-based theory of ﬁnancial intermediation, The Review of Economic Studies 89, 2381–2444. (p. 19.)
Boudt, Kris, Ellen C.S. Paulus, and Dale W.R. Rosenthal, 2017, Funding liquidity, market
liquidity and TED spread: A two-regime model, Journal of Empirical Finance 43, 143–158.
(p. 11.)
Brunnermeier, Markus K., and Lasse Heje Pedersen, 2009, Market liquidity and funding
liquidity, The Review of Financial Studies 22, 2201–2238. (p. 11.)
Choi, Jaewon, Yesol Huh, and Sean Seunghun Shin, 2023, Customer liquidity provision:
Implications for corporate bond transaction costs, Management Science 0, xxx–xxx. (pp. 2,
3, 4, 5, 6, and 7.)
Chordia, Tarun, Richard Roll, and Avanidhar Subrahmanyam, 2000, Commonality in liquidity, Journal of Financial Economics 56, 3–28. (p. 17.)
Chordia, Tarun, Richard Roll, and Avanidhar Subrahmanyam, 2001, Market liquidity and
trading activity, The Journal of Finance 56, 501–530. (p. 11.)
Comerton-Forde, Carole, Terrence Hendershott, Charles M. Jones, Pamela C. Moulton, and
Mark S. Seasholes, 2010, Time variation in liquidity: The role of marketmaker inventories
and revenues, The Journal of Finance 65, 295–331. (p. 11.)
Dick-Nielsen, Jens, 2009, Liquidity biases in TRACE, The Journal of Fixed Income 19, 43–55.
(p. 63.)
42

Dick-Nielsen, Jens, 2014, How to clean Enhanced TRACE data, Working Paper. (p. 63.)
Dick-Nielsen, Jens, Peter Feldhütter, and David Lando, 2012, Corporate bond liquidity before
and after the onset of the subprime crisis, Journal of Financial Economics 103, 471–492.
(pp. 9 and 64.)
Duﬃe, Darrell, 2012, Market making under the proposed Volcker rule, Report to the Securities Industry and Financial Markets Association, Stanford University. (p. 5.)
Duﬃe, Darrell, Nicolae Gârleanu, and Lasse Heje Pedersen, 2005, Over-the-counter markets,
Econometrica 73, 1815–1847. (pp. 3, 18, and 40.)
Fama, Eugene F., and James D. MacBeth, 1973, Risk, return, and equilibrium: Empirical
tests, Journal of Political Economy 81, 607–636. (p. 14.)
Fleming, Michael J., and Eli M. Remolona, 1999, Price formation and liquidity in the U.S.
treasury market: The response to public information, The Journal of Finance 54, 1901–
1915. (p. 11.)
Forte, Santiago, and Juan Ignacio Pena, 2009, Credit spreads: An empirical analysis on the
informational content of stocks, bonds, and CDS, Journal of Banking and Finance 33,
2013–2025. (p. 17.)
Giannetti, Mariassunta, Chotibhak Jotikasthira, Andreas C. Rapp, and Martin Waibel, 2023,
Intermediary balance sheet constraints, bond mutual funds’ strategies, and bond returns,
Working Paper. (p. 6.)
Guerrieri, Veronica, Robert Shimer, and Randall Wright, 2010, Adverse selection in competitive search equilibrium, Econometrica 78, 1823–1862. (p. 2.)
Hendershott, Terrence, Dmitry Livdan, and Norman Schurhoﬀ, 2021, All-to-all liquidity in
corporate bonds, Working Paper. (p. 7.)
Hugonnier, Julien, Benjamin Lester, and Pierre-Olivier Weill, 2022, Heterogeneity in decentralized asset markets, Theoretical Economics 17, 1313–1356. (pp. 2, 4, 18, 25, 26, 28,
and 40.)

43

Kargar, Mahyar, Benjamin Lester, Sébastien Plante, and Pierre-Olivier Weill, 2023, Sequential search for corporate bonds, Working Paper. (p. 6.)
Krishnamurthy, Arvind, 2002, The bond/old-bond spread, Journal of Financial Economics
66, 463–506. (p. 68.)
Lagos, Ricardo, and Guillaume Rocheteau, 2009, Liquidity in asset markets with search
frictions, Econometrica 77, 403–426. (pp. 2, 3, 18, 25, and 40.)
Li, Jian, and Haiyue Yu, 2023, Investor composition and the liquidity component in the U.S.
corporate bond market, Working Paper. (pp. 3 and 14.)
Mahanti, Sriketan, Amrut Nashikkar, Marti Subrahmanyam, George Chacko, and Gaurav
Mallik, 2008, Latent liquidity: A new measure of liquidity, with an application to corporate
bonds, Journal of Financial Economics 88, 272–298. (p. 9.)
Newey, Whitney K., and Kenneth D. West, 1987, A simple, positive semi-deﬁnite, heteroskedasticity and autocorrelation consistent covariance matrix, Econometrica 55, 703–
708. (p. 14.)
Rapp, Andreas C., and Martin Waibel, 2023, Managing regulatory pressure: Bank regulation
and its impact on corporate bond intermediation, Research Paper 23-12, Swedish House
of Finance. (p. 6.)
Wu, Botao, 2023, Post-crisis regulations, trading delays, and increasing corporate bond liquidity premium, Working Paper. (pp. 3 and 14.)

A

Proofs

Proof of Lemma 1
Proof. Take the diﬀerence between equations (3) and (4) to obtain
r∆(ν) = ν − µ∆(ν) − η∆(ν)
+ λD θD max{p − τ − ∆(ν), 0} − λD θD max{∆(ν) − p − τ, 0},
44

(39)

which implies that
=⇒ ∆(ν) =

ν + λD θD [p + τ + max{∆(ν), p − τ } − max{∆(ν), p + τ }]
r + ν + µ + λD θD

(40)

for all types ν ∈ (−∞, νl ] ∪ [νh , ∞). Equation (40) is associated with a functional operator
satisfying all Blackwell’s conditions for a contraction. Then, by the contraction mapping
theorem, there is a unique function ∆ satisfying the equation (40). Also note that if τ = 0,
then the results follow directly from equation (40). So we focus on the case with τ > 0.
Since ∆ is strictly increasing and continuous, we must have that ∆(νl ) ≤ p − τ . To see
this, notice ﬁrst that, if p − τ < ∆(νl ) < p + τ , then the customer would not trade with
a dealer because of transaction costs, as the reservation value of a potential seller is higher
than the highest bid price of a dealer, p − τ , and the reservation value of a potential buyer
is smaller than the lowest ask price of a dealer, p + τ . The last terms in equations (3) and
(4) would be zero. Therefore, searching for a dealer is equivalent to be inactive. In this case,
the customer would be better oﬀ searching for customers type ν ∈ (νl , νh ) to obtain a share
of the gains from trade. This implies that νl ∈
/ ΩD , which is a contradiction. Implicit in this
argument is the fact that the densities of Φo and Φn are bounded away from zero in the set
(νl , νh ) because of issuance and maturity (see proof of Lemma 2), and νl ̸= νh (which holds
by assumption on a regular equilibrium with τ > 0).
Moreover, if ∆(νl ) ≥ p + τ , then either p − τ < ∆(ν) < p + τ for some customer type
ν ∈ ΩD or ∆(ν) ≥ p + τ for all customer type ν ∈ ΩD . The ﬁrst cannot hold because again
it would imply ν ∈
/ ΩD . The second would be inconsistent with inter-dealer market clearing
because all customers searching for a dealer would want to buy assets as their reservation
value would be greater than or equal to the highest ask price.
Therefore, we must have ∆(νl ) ≤ p − τ . An analogous argument applies for νh in the
opposite direction. That is, ∆(νh ) ≥ p + τ . With ∆(νl ) ≤ p − τ and ∆(νh ) ≥ p + τ , we can
solve for the max relations in equation (40), which then implies equation (14).
Proof of Lemma 2

45

■

Proof. First note that equation (12) implies
Φ̇o (∞) = ηΦn (∞) − µΦo (∞)

(41)

as, when ν goes to inﬁnity, both the inﬂow and outﬂow from trading goes to zero. Then,
from Φ̇o (∞) = 0 and equation (13), we have that
η[F (∞) − Φo (∞)] − µΦo (∞) = 0 ⇐⇒ Φo (∞) =

η
,
η+µ

(42)

which characterizes the total supply of assets s = Φo (∞), equal, by deﬁnition, to the measure
of owners. This also establishes that the measure of non-owners is given by
Φn (∞) = F (∞) − Φo (∞) = 1 − Φo (∞) =⇒ Φn (∞) =

µ
.
η+µ

(43)

Consider now the case ν ≤ νl . According the law of motion for Φo , given by equation
(12), we have
Φ̇o (ν) = ηΦn (ν) − µΦo (ν) − λD Φo (ν),

(44)

as no customer with ν̃ ≤ ν ≤ νl will either search for other customers or purchase the asset
from a dealer. Substituting Φn (ν) = F (ν) − Φo (ν) and setting Φ̇o (ν) = 0 implies
Φo (ν) =

ηF (ν)
,
η + µ + λD

ν ≤ νl .

(45)

Consider now the case ν ≥ νh . In this case, it is useful to work with the measure of
non-owners of type above ν, Φn (∞) − Φn (ν). Using equations (13) and (12), we have
0 = Φ̇n (∞) − Φ̇n (ν)

(46)

= −η[Φn (∞) − Φn (ν)] − λD [Φn (∞) − Φn (ν)] + µ[Φo (∞) − Φo (ν)]

(47)

= −η[1 − s − F (ν) + Φo (ν)] − λD [1 − s − F (ν) + Φo (ν)] + µ[s − Φo (ν)]

(48)

= −(η + λD )[1 − F (ν)] + (η + µ + λD )[s − Φo (ν)]

(49)

=⇒ s − Φo (ν) =

(η + λD )[1 − F (ν)]
.
η + µ + λD

46

(50)

As s =

η
η+µ ,

we have
Φo (ν) =

(η + λD )[1 − F (ν)]
η
−
,
η+µ
η + µ + λD

ν ≥ νh .

(51)

Now let us show that ηF (νl ) = µ[1 − F (νh )]. According to the market clearing condition
(11) and Lemma 1,
o

Z ∞

Φ (νl ) =

−∞

We know that Φo (νl ) =

1{ν∈ΩnD ,∆(ν)>p+τ } dΦn (ν) =⇒ Φo (νl ) = Φn (∞) − Φn (νh ).
ηF (νl )
η+µ+λD .

(52)

Moreover,

Φn (∞) − Φn (νh ) = F (∞) − Φo (∞) − [F (νh ) − Φo (νh )]


= 1 − s − F (νh ) −



η
(η + λD )[1 − F (νh )]
µ[1 − F (νh )]
+
. (53)
=
η+µ
η + µ + λD
η + µ + λD
µ
n
η+µ − Φ (νh )
η
µ
η+µ = η+µ .

Thus, ηF (νl ) = µ[1 − F (νh )]. Finally, the result that Φo (νl ) =
equation (52) and the fact that Φn (∞) = F (∞) − Φo (∞) = 1 −

comes from
■

Proof of Lemma 3
Proof. By taking the diﬀerence between equations (19) and (20), we know that the reservation
value satisﬁes
∆(ν) =

ν + λC

R νh o
Rν n
n
o
ν θC [∆(ν̃) − ∆(ν)]dΦ (ν̃) − λC νl θC [∆(ν) − ∆(ν̃)]dΦ (ν̃)

r+µ+η

.

(54)

Moreover, because ∆ is continuous and monotone, equation (54) implies that ∆ is Lipschitz
continuous in the interval (νl , νh ). To see this, note that we can rearrange equation (54) to
show that
∆(ν + t) − ∆(ν)
1 + 2λC supx f (x)[∆(νh ) − ∆(νl )]
≤
t
r+µ+η
for all ν and ν + t in the interval (νl , νh ), where f is the density of the distribution F . Given
that ∆ is Lipschitz continuous in the interval (νl , νh ), ∆ is diﬀerentiable almost everywhere
in the interval (νl , νh ) and satisﬁes ∆(ν) = ∆(νl ) +

47

Rν

νl

σC (ν̃)dν̃, where σC (ν) denote the

derivative of ∆. Using this result, take the derivative on both sides of equation (54) to obtain
σC (ν) =

o [Φn (ν ) − Φn (ν)]σ (ν) − λ θ n [Φo (ν) − Φo (ν )]σ (ν)
1 − λC θC
C
C C
C
h
l
.
r+µ+η

(55)
■

We then obtain σC (ν) by rearranging the equation above.
Proof of Lemma 4
˙ o (ν) = Φ̇o (ν) − Φ̇o (ν ). From equation (12), we have
Proof. We have Φ̃
l
˙ o (ν) = η Φ̃n (ν) − µΦ̃o (ν) − λ
Φ̃
C

Z ν Z νh
νl

dΦn (ν̂)dΦo (ν̃)

ν



= η Φ̃n (ν) − µΦ̃o (ν) − λC Φ̃o (ν) Φn (νh ) − Φn (ν)










= η Φ̃n (ν) − µΦ̃o (ν) − λC Φ̃o (ν) F (νh ) − F (ν) + λC Φ̃o (ν) Φ̃o (νh ) − Φ̃o (ν)




n



= η F (ν) − F (νl ) − Φ̃o (ν) µ + η + λC F (νh ) − F (ν) − Φ̃o (νh )

o

− λC Φ̃o (ν)2 . (56)

˙ o (ν) = 0 to obtain equation (23).
We can then solve the quadratic equation above with Φ̃
Equation (24) is obtained by the solution of the quadratic equation for ν = νh .

■

Proof of Lemma 5
Proof. In a regular equilibrium, customers of type νl are indiﬀerent between searching for
dealers or customers. The reason is that equations (3)–(6) and the continuity of ∆ imply the
continuity of VCo , VCn , VDo and VDn . As a result,
rVCo (νl )

= νl − µ∆(νl ) + λC
= νl − µ∆(νl ) +

Z νh
νl

o
λC θC

o
θC
[∆(ν) − ∆(νl )]dΦn (ν)

Z νh Z ν
νl

σC (ν̃)dν̃dΦn (ν)

νl

= νl − µ∆(νl ) + λD θD [p − τ − ∆(νl )] = rVDo (νl ).

(57)

Which implies that
p = ∆(νl ) + τ +

o
λC θC
λD θD

Z νh Z ν
νl

48

νl

σC (ν̃)dν̃dΦn (ν).

(58)

And similarly,
rVCn (νh ) = η∆(νh ) + λC

Z νh
νl

n
= η∆(νh ) + λC θC

n
θC
[∆(νh ) − ∆(ν)]dΦo (ν)

Z νh Z νh
νl

σC (ν̃)dν̃dΦo (ν)

ν

= η∆(νh ) + λD θD [∆(νh ) − p − τ ] = rVDn (νh ).

(59)

Which implies that
n
λ C θC
p = ∆(νh ) − τ −
λ D θD

Z νh Z νh
νl

σC (ν̃)dν̃dΦo (ν).

(60)

ν

Equalizing the above two price equations and using lemma 3 we obtain
Z νh

2τ λD θD = λD θD
n
− λC θC

σC (ν)dν
ν
Z νlh

Z νh

νl

ν

o
σC (ν̃)dν̃dΦo (ν) − λC θC

Z νh Z ν
νl

σC (ν̃)dν̃dΦn (ν).

(61)

νl

Applying integration by parts in the last two terms we obtain
Z νh

2τ λD θD = λD θD

σC (ν)dν
ν

− λC

Z νh nl
νl





o



n
o
θC
Φo (ν) − Φo (νl ) + θC
Φn (νh ) − Φn (ν) σC (ν)dν.







(62)


o Φn (ν ) − Φn (ν) + θ n Φo (ν) − Φo (ν ) } =
From the deﬁnition of σC (ν), we have λC {θC
h
l
C
1
σC (ν)

− (r + µ + η). Substituting above and rearranging implies
Z νh

2τ λD θD =
νl

[(r + µ + η + λD θD )σC (ν) − 1]dν.

(63)

As r + µ + η + λD θD = 1/σD , we obtain
2τ λD θD =

Z νh
σC (ν) − σD

σD

νl

dν.

(64)
■

This concludes the proof.

49

Proof of Proposition 1
Proof. The necessity of equations (14)–(26) are established in Lemmas 2–5. So let us focus on the suﬃciency. Consider a family {∆, p, Φo , Φn , νl , νh } satisfying equations (14)–
(26) and value functions V o and V n constructed using equations (3)–(7) given the family
{∆, p, Φo , Φn , νl , νh }. Let us show that the family {V o , V n , ∆, p, Φo , Φn , νl , νh } is a regular
equilibrium—that is, it satisﬁes equations (3)–(13) and deﬁnition 2.
Equations (3)–(7):
Equations (9)–(10):

These equations are satisﬁed by the construction of V o and V n .
First let us show that VDo (ν) ≥ VCo (ν) for all ν ≤ νl .

o
VDo (ν) ≥ VCo (ν) ⇐⇒ λD θD [(p − τ ) − ∆(ν)] ≥ λC θC

⇐⇒ (p − τ ) − ∆(ν) ≥

o
λC θ C
λD θ D

Z νh
νl

Z νh
νl

[∆(ν̃) − ∆(ν)]dΦn (ν̃)

[∆(ν̃) − ∆(νl )]dΦn (ν̃)

λ θo

C C
n
n
+ λD
θD [Φ (νh ) − Φ (νl )][∆(νl ) − ∆(ν)].

From equation (26) we know that

o
λC θC
λD θD

R νh
νl

[∆(ν̃) − ∆(νl )]dΦn (ν̃) = (p − τ ) − ∆(νl ), therefore

VDo (ν) ≥ VCo (ν) ⇐⇒ ∆(νl ) − ∆(ν) ≥

o
λC θ C
[Φn (νh ) − Φn (νl )][∆(νl ) − ∆(ν)].
λD θ D

o
λC θ C
λD θD ∈ (0, 1). From the equations
µ[F (νh )−F (νl )]
∈ [0, 1). Using equation (14) in
η+µ

Assumption 1 implies that

(16) and (17) we have that

Φn (νh ) − Φn (νl ) =

Lemma 1, we have that

VDo (ν) ≥ VCo (ν) ⇐⇒ νl − ν ≥

o
λC θC
[Φn (νh ) − Φn (νl )][νl − ν].
λD θD

We can then see that VDo (ν) ≥ VCo (ν) holds. Moreover, it holds with strictly inequality for all
ν < νl . The proofs that VDn (ν) ≥ VCn (ν) for all ν ≤ νl ; VDo (ν) ≤ VCo (ν) for all ν ∈ (νl , νh ); and
VDn (ν) ≤ VCn (ν) for all ν ∈ (νl , νh ) are analogous.

50

Equation (8):

Let us start with ν ≤ νl . In this case we have that V o (ν) = VDo (ν) and

V n (ν) = VDn (ν) based on equation (9). Then, from equations (3) and (4) we have that
ν + λD θD (p − τ ) − (η + µ + λD θD )∆(ν)
r
(r + η + µ + λD θD )∆(ν) − (η + µ + λD θD )∆(ν)
=
= ∆(ν).
r

V o (ν) − V n (ν) =

The result for ν ≥ νh is analogous. For ν ∈ (νl , νh ) we have that
r[V o (ν) − V n (ν)] = ν − (η + µ)∆(ν) + λC
− λC

Z ν
νl

Z νh
ν

o
θC
[∆(ν̃) − ∆(ν)]dΦn (ν̃)

n
θC
[∆(ν) − ∆(ν̃)]dΦo (ν̃)

Replacing equation (21) and applying integration by parts we get
r[V o (ν) − V n (ν)] = ν − (η + µ)∆(νl ) − (η + µ)
Z νh

+ λC
ν

o
θC
[Φn (νh )

−

νl

σC (ν̃)dν̃
νl

− Φ (ν̃)]σC (ν̃)dν̃ − λC
n

= ν − (η + µ)∆(νl ) + λC
Z ν

Z ν

Z νh
ν

o
θC
[Φn (νh )

Z ν
νl

n
θC
[Φo (ν̃) − Φo (νl )]σC (ν̃)dν̃

− Φ (ν̃)]σC (ν̃)dν̃
n

n
{η + µ + λC θC
[Φo (ν̃) − Φo (νl )]} σC (ν̃)dν̃

= ν − (η + µ)∆(νl ) + λC
− ν + νl −

Z ν
νl

{r +

Z νh
ν

o
θC
[Φn (νh ) − Φn (ν̃)]σC (ν̃)dν̃

o
λ C θC
[Φn (νh )

= νl − (r + η + µ)∆(νl ) + λC

− Φn (ν̃)]} σC (ν̃)dν̃

Z νh
νl

o
θC
[Φn (νh ) − Φn (ν̃)]σC (ν̃)dν̃ + r∆(ν).

Now we can replace equation (26) to obtain
o
r[V o (ν) − V n (ν)] = νl − (r + η + µ)∆(νl ) + λC θC
[p − τ − ∆(νl )] + r∆(ν)
o
o
= νl + λC θC
(p − τ ) − (r + η + µ + λC θC
)∆(νl ) + r∆(ν) = r∆(ν),

where the last equality we obtained using equation (14) applied to ∆(νl ).

51

Equation (11):

The left-hand side of Equation (11) is given by
Z

λD

ΩoD

1{∆(ν)<p−τ } dΦo (ν) = λD

Z νl
−∞

dΦo (ν) = λD Φo (νl ).

The right-hand side is
Z

λD

1{∆(ν)>p+τ } dΦn (ν) = λD

Ωn
D

Z ∞
νh

dΦn (ν) = λD [Φn (∞) − Φn (νh )] .

Therefore, we have market clearing if, and only if, Φo (νl ) = Φn (∞) − Φn (νh ). This equation
holds because, from equation (17), Φo (∞) =
equation (18),

µ
η+µ

Equation (12):

η
η+µ

=⇒ Φn (∞) = 1 − Φo (∞) =

µ
η+µ ,

and, from

− Φn (νh ) = Φo (νl ).

First, consider ν ≤ νl . Then, equation (12) is given by

Φ̇o (ν) = ηΦn (ν) − µΦo (ν) − λD Φo (ν) = ηF (ν) − (η − µ − λD )Φo (ν).
Equation (16) states that Φo (ν) =

ηF (ν)
η−µ−λD .

Thus, Φ̇o (ν) = ηF (ν) − ηF (ν) = 0. Consider

now ν ≥ νh . Then, equation (12) is given by
Φ̇o (ν) = ηΦn (ν) − µΦo (ν) − λD Φo (νl ) + λD [Φn (ν) − Φn (νh )]
= (η + λD )F (ν) − (η + µ + λD )Φo (ν) − λD [Φo (νl ) + Φn (νh )].
Using equations (17) and (18) we then have
Φ̇o (ν) = (η + λD )F (ν) + (η + λD )[1 − F (ν)] −
= η + λD −

η(η + µ + λD )
λD µ
−
η+µ
η+µ

η(η + µ) + λD (η + µ)
= η + λD − (η + λD ) = 0.
η+µ

Finally, let us consider ν ∈ (νl , νh ). In this case we have
Φ̇o (ν) = ηΦn (ν) − µΦo (ν) − λD Φo (νl ) − λC [Φo (ν) − Φo (νl )][Φn (νh ) − Φn (ν)]
= η[Φn (ν) − Φn (νl )] − µ[Φo (ν) − Φ0 (νl )] + ηΦn (νl ) − µΦo (νl ) − λD Φo (νl )
− λC [Φo (ν) − Φo (νl )][F (νh ) − F (ν)] + λC [Φo (ν) − Φo (νl )][Φo (νh ) − Φo (ν)].
52

We have shown that ηΦn (νl ) − µΦo (νl ) − λD Φo (νl ) = 0 when considering the case ν ≤ νl . By
using this result and the notation Φ̃o (ν) = Φo (ν) − Φ0 (νl ) and sC = Φ̃o (νh ), we obtain
Φ̇o (ν) = η[F (ν) − F (νl )] − (η + µ)Φ̃o (ν)
− λC Φ̃o (ν)[F (νh ) − F (ν)] + λC Φ̃o (ν)Φ̃o (νh ) − λC Φ̃o (ν)2
= η[F (ν) − F (νl )] − (η + µ)Φ̃o (ν)
− Φ̃o (ν) {η + µ + λC [F (νh ) − F (ν) − sC ]} − λC Φ̃o (ν)2 .
The distribution Φ̃o (ν), as deﬁned in equation (23), is the positive root of the equation above.
Therefore, Φ̇o (ν) = 0.
Equation (13):

This is directly stated in equations (16), (17) and (23).

We showed that the family {V o , V n , ∆, p, Φo , Φn , νl , νh } is an equilibrium. That is, that
it satisﬁes equations (3)–(13). It is easy to see that it must also be a regular equilibrium
because equation (26) implies that νl ≤ νh with strict inequality if τ > 0, and equations (14)
■

and (21) imply that ∆ is continuous and strictly increasing.
Proof of Proposition 2

Proof. First note that equation (26) is necessarily satisﬁed by all regular equilibrium. Therefore, it suﬃces to show that in a neighborhood of τ = 0, there is a unique pair (νl , νh )
satisfying equation (26) for all τ in this neighborhood. Equation (26) can be rewritten as
G(νl ) =

Z g(νl ) 
σC (ν; νl , g(νl ))

1
2λD θD

σD

νl



− 1 dν = τ.

When τ = 0, then G(νl ) = τ implies that νl = g(νl ) = νh = ν ∗ . That is because
σC (ν;νl ,g(νl ))
σD

− 1 is bounded away from zero. To see this notice that

⇔

σC (ν; νl , g(νl ))
−1>0
σD
r + µ + η + λD θ D

n







o > 1

n







o

o Φn (ν ) − Φn (ν) + θ n Φo (ν) − Φo (ν )
r + µ + η + λC θC
h
l
C

o
n
⇔ λD θD > λC θC
Φn (νh ) − Φn (ν) + θC
Φo (ν) − Φo (νl ) .

53

n







o Φn (ν ) − Φn (ν) + θ n Φo (ν) − Φo (ν )
But note that λC θC
h
l
C

which implies that

σC (ν;νl ,g(νl ))
σD

o

o , θn } < λ θ ,
< λC max{θC
D D
C

− 1 is bounded away from zero. As a result, we can only

have G(νl ) = 0 if the limits in the integral are the same. Then, since G(·) is continuous, it
suﬃces to show that it is strictly monotone in a neighborhood (ν̄l , ν ∗ ]. Note also that G(·) is
diﬀerentiable and that


1









σC (g(νl ); νl , g(νl ))
σC (νl ; νl , g(νl ))
G (νl ) =
g (νl )
−1 −
−1
2λD θD
σD
σD


Z g(νl )
1
1 ∂σC (ν; νl , g(νl ))
∂σC (ν; νl , g(νl ))
+
+ g ′ (νl )
dν.
2λD θD νl
σD
∂νl
∂νh
′

′

(νl )
l ,g(νl ))
− 1 is
The ﬁrst term on the right-hand is negative since g ′ (νl ) = − µfηf(g(ν
, and σC (ν;νσD
l ))

bounded away from zero. Moreover, using the deﬁnition of σC , we can bound it above by
−

1

"

2λD θD

#

r + η + µ + λD θD
o , θn } − 1 .
r + η + µ + λC max{θD
D

Therefore, to establish that G′ (νl ) < 0 in a neighborhood of (ν̄l , ν ∗ ] we just have to show that
the second term converges to zero when νl ↗ ν ∗ . Since g(νl ) → ν ∗ when νl ↗ ν ∗ , it suﬃces
to show that the terms inside the integral,

∂σC (ν;νl ,g(νl ))
∂νl

and

∂σC (ν;νl ,g(νl ))
,
∂νh

are bounded. We

can write the ﬁrst term as below.
n



n





o



o Φn (ν ) − Φn (ν) + θ n Φo (ν) − Φo (ν )
∂ θC
h
l
C
∂σC (ν; νl , g(νl ))
2
= −σC (ν; νl , g(νl )) λC
∂νl
∂νl

= −σC (ν; νl , g(νl ))2 λC



o Φn (ν ) − Φn (ν ) + Φn (ν ) − Φn (ν) + θ n Φ̃o (ν)
∂ θC
h
l
l
C

∂νl

n

= −σC (ν; νl , g(νl ))2 λC

o µ[F (νh )−F (νl )] − θ o [F (ν) − F (ν )] + Φ̃o (ν)
∂ θC
l
C
η+µ

(
o
= −σC (ν; νl , g(νl ))2 λC θC

ηf (νl )
+
η+µ

∂ Φ̃o (ν)

o

o

∂ν
) l
.

∂νl

The ﬁrst term in parenthesis is bounded. The second term is obtained by applying the implicit
function theorem to equation (56) and it yields
h

∂ Φ̃o (ν)
∂νl

ηf (νl ) 1 +

λC Φ̃o (ν)
η+µ

i

,
= −n

o
µ + η + λC F (νh ) − F (ν) − Φ̃o (νh ) + 2λC Φ̃o (ν)
54

l ,g(νl ))
which is also bounded. Similarly for ∂σC (ν;ν
,
∂νh

n

∂σC (ν; νl , g(νl ))
= −σC (ν; νl , g(νl ))2 λC
∂νh

o

o µ[F (νh )−F (νl )] − θ o [F (ν) − F (ν )] + Φ̃o (ν)
∂ θC
l
C
η+µ

(

o
= −σC (ν; νl , g(νl ))2 λC θC

µf (νh )
+
η+µ

∂ Φ̃o (ν)
∂νh

∂ν
)h
.

Again, the ﬁrst term in parenthesis is bounded. The second term is
o

Φ̃ (ν)
µf (νh ) λCη+µ
∂ Φ̃o (ν)
= −n
,

o
∂νh
µ + η + λC F (νh ) − F (ν) − Φ̃o (νh ) + 2λC Φ̃o (ν)

which is also bounded. Therefore, G(·) is strictly monotone in a neighborhood (ν̄l , ν ∗ ] with
G′ (ν) < 0 in this neighborhood. If we then deﬁne the neighborhood [0, τ̄ ), where τ̄ = G(ν̄l ),
we can conclude that there is a unique regular equilibrium for any τ ∈ [0, τ̄ ).

■

Proof of Proposition 3
Proof. We know that there exists neighborhood [0, τ̄ ) of τ = 0 that regular equilibrium is
unique. Moreover, because G(ν ∗ ) = 0, we must have G′ (νl ) ≤ 0 for any νl < ν ∗ with an unique
equilibrium in a neighborhood around it. To see this note that if G′ (νl ) > 0 for an νl with
G(νl ) = τ , then there exists νl′ > νl such that G(νl′ ) > G(νl ). But then G(νl′ ) > G(νl ) ≥ G(0)
and we can conclude by continuity that there must be another νl′′ with G(νl′′ ) = τ . This is a
contradiction since we started assuming that there is an unique regular equilibrium at νl .
Consider then any (τ 0 , τ 1 ) and (νl0 , νl1 ) such that all τ ∈ (τ 0 , τ 1 ) is associated with a
unique regular equilibrium at some νl (τ ) ∈ (νl0 , νl1 ). Since these regular equilibrium are
characterized by G(νl ) = τ and G′ (νl ) ≤ 0, we must then have that νl (τ ) is decreasing
in τ for all τ ∈ (τ 0 , τ 1 ). Therefore, to obtain that the turnover is decreasing in τ in the
neighborhood (τ 0 , τ 1 ), it suﬃces to show that it is increasing in νl .
From equation (30) we have that

T =

2λD Φo (νl ) + λC

R νh R νh
νl ν
η
η+µ

dΦn (ν̃)dΦo (ν)

From equation (16) we have Φo (νl ) =

ηF (νl )
η+µ+λD ,

55

=

2λD Φo (νl ) + λC

R νh
νl

η
η+µ

Φ̃o (ν)dΦ̃n (ν)

.

which is increasing in νl with its derivative

given by

∂Φo (νl )
∂νl

ηf (νl )
η+µ+λD .

=
∂

R νh
νl

For the second term we have that

Φ̃o (ν)dΦ̃n (ν)
′
o
n
n
= −
Φ̃o
(ν
l )ϕ (νl ) + g (νl )Φ̃ (νh )ϕ (νh )
∂νl
Z νh
dϕn (ν) o
dΦ̃o (ν) n
ϕ (ν) +
Φ̃ (ν)dν.
+
dνl
dνl
νl

Again for the ﬁrst term inside the integral
∂ Φ̃o (ν)
∂ Φ̃o (ν)
−ηf (νl )
dΦ̃o (ν)
=
+ g ′ (νl )
=n
.

o
dνl
∂νl
∂νh
µ + η + λC F (νh ) − F (ν) − Φ̃o (νh ) + 2λC Φ̃o (ν)
Now we can apply the implicit function theorem to equation (56) and obtain that
[η + λC Φ̃o (ν)]f (ν)
ϕo (ν) = n
.
o

µ + η + λC F (νh ) − F (ν) − Φ̃o (νh ) + 2λC Φ̃o (ν)
Thus,
dΦ̃o (ν)

2λC ηf (ν) dν
dϕo (ν)
l
i2
= − hn

o
dνl
µ+η+λC F (νh )−F (ν)−Φ̃o (νh ) +2λC Φ̃o (ν)

+ λC f (ν)

dΦ̃o (ν)
dνl

hn

hn

i

o



+2λC Φ̃o (ν)

µ+η+λC F (νh )−F (ν)−Φ̃o (νh )



µ+η+λC F (νh )−F (ν)−Φ̃o (νh )

o

i2

+2λC Φ̃o (ν)

o

2λC Φ̃o (ν) dΦ̃dν(ν)
l

− λC f (ν) hn
i2

o
µ + η + λC F (νh ) − F (ν) − Φ̃o (νh ) + 2λC Φ̃o (ν)
=n

λC f (ν)



dΦ̃o (ν)
dΦ̃o (ν)
o
dνl −2λC ϕ (ν) odνl



µ+η+λC F (νh )−F (ν)−Φ̃o (νh )

λC [ϕn (ν)−ϕo (ν)]

=n

+2λC Φ̃o (ν)

Also note that from the above we obtain ϕn (ν) and



dΦ̃o (ν)
dνl o



µ+η+λC F (νh )−F (ν)−Φ̃o (νh )

dϕn (ν)
dνl

from the identity f (ν) = ϕo (ν) +

ϕn (ν). Then we obtain that
[η + λC Φ̃o (ν)]f (ν)
ϕn (ν) = f (ν) − n

o
µ + η + λC F (νh ) − F (ν) − Φ̃o (νh ) + 2λC Φ̃o (ν)
h





i

{µ + λC F (νh ) − F (ν) − Φ̃o (νh ) } + λC Φ̃o (ν) f (ν)

= n
.

o
µ + η + λC F (νh ) − F (ν) − Φ̃o (νh ) + 2λC Φ̃o (ν)

56

.

+2λC Φ̃o (ν)

Now we can write that
∂

R νh
νl

Z νh
Φ̃o (ν)dΦ̃n (ν)
dϕn (ν) o
dΦ̃o (ν) n
Φ̃ (ν)dν
= g ′ (νl )Φ̃o (νh )ϕn (νh ) +
ϕ (ν) +
∂νl
dνl
dνl
νl
ηf (νl )
= −Φ̃o (νh )
µ + η + λC Φ̃o (νh )

− ηf (νl )

Z νh

λ [ϕo (ν)−ϕn (ν)]Φ̃o (ν)
ϕn (ν)+ n
 C
o
µ+η+λC F (νh )−F (ν)−Φ̃o (νh ) +2λC Φ̃o (ν)

n

o



µ+η+λC F (νh )−F (ν)−Φ̃o (νh )

νl

With this equation in hand we now can show that

dT
dνl

dν.

+2λC Φ̃o (ν)

≥ 0, which is equivalent to show that

2λD ηf (νl )
≥
η + µ + λD
λC Φ̃o (νh )ηf (νl )
+ λC ηf (νl )
µ + η + λC Φ̃o (νh )

Z νh

λ [ϕo (ν)−ϕn (ν)]Φ̃o (ν)
ϕn (ν)+ n
 C
o
µ+η+λC F (νh )−F (ν)−Φ̃o (νh ) +2λC Φ̃o (ν)

n



µ+η+λC F (νh )−F (ν)−Φ̃o (νh )

νl

o

dν,

+2λC Φ̃o (ν)

which happens if, and only if,
2λD
≥
η + µ + λD
λC Φ̃o (νh )
+ λC
µ + η + λC Φ̃o (νh )

Z νh

λ [ϕo (ν)−ϕn (ν)]Φ̃o (ν)
ϕn (ν)+ n
 C
o
µ+η+λC F (νh )−F (ν)−Φ̃o (νh ) +2λC Φ̃o (ν)

n



o

µ+η+λC F (νh )−F (ν)−Φ̃o (νh )

νl

dν.

+2λC Φ̃o (ν)

Note that
n



µ + η + λC F (νh ) − F (ν) − Φ̃o (νh )

o





+ 2λC Φ̃o (ν) = µ + η + λC Φ̃n (νh ) − Φ̃o (ν) + Φ̃o (ν) .

57

Then we have that
o

n

o

λC [ϕ (ν)−ϕ (ν)]Φ̃ (ν)


Z νh ϕn (ν)+
µ+η+λC Φ̃n (νh )−Φ̃n (ν)+Φ̃o (ν)


dν
νl

Z νh

=
νl

µ+η+λC Φ̃n (νh )−Φ̃o (ν)+Φ̃o (ν)



f (ν)

 dν

µ+η+λC Φ̃n (νh )−Φ̃n (ν)+Φ̃o (ν)

n
o
Z νh ϕo (ν) µ+η+λC Φ̃n (νh )−Φ̃n (ν)+Φ̃o (ν) −λC [ϕo (ν)−ϕn (ν)]Φ̃o (ν)
n
−
dν
o2

νl

µ+η+λC Φ̃n (νh )−Φ̃n (ν)+Φ̃o (ν)

Z νh

=
νl



νl

 dν −

µ+η+λC Φ̃n (νh )−Φ̃n (ν)+Φ̃o (ν)

Z νh

=

f (ν)



f (ν)

 dν −

µ+η+λC Φ̃n (νh )−Φ̃n (ν)+Φ̃o (ν)

"
Z νh
d
νl

dν

#
Φ̃o (ν)



 dν

µ+η+λC Φ̃n (νh )−Φ̃n (ν)+Φ̃o (ν)

Φ̃o (νh )
.
µ + η + λC Φ̃o (νh )

So again it suﬃces to show that
2λD
≥
η + µ + λD
λC Φ̃o (νh )
+ λC
µ + η + λC Φ̃o (νh )
λC Φ̃o (νh )
+ λC
µ + η + λC Φ̃o (νh )

⇐⇒

Z νh "
νl

Z νh

λ [ϕo (ν)−ϕn (ν)]Φ̃o (ν)
ϕn (ν)+ n
 C
o
µ+η+λC F (νh )−F (ν)−Φ̃o (νh ) +2λC Φ̃o (ν)

n

νl

Z νh

o



µ+η+λC F (νh )−F (ν)−Φ̃o (νh )

dν =

+2λC Φ̃o (ν)

f (ν)
λC Φ̃o (νh )
 dν −
Φ̃n (νh ) − Φ̃n (ν) + Φ̃o (ν)
µ + η + λC Φ̃o (νh )



µ + η + λC
Z νh
2λD
≥
⇐⇒
η + µ + λD
νl
νl



λC f (ν)

 dν

µ+η+λC Φ̃n (νh )−Φ̃n (ν)+Φ̃o (ν)

#

2λD
λC [F (νh ) − F (νl )]
−

 dF (ν) ≥ 0.
η + µ + λD
µ + η + λC Φ̃n (νh ) − Φ̃n (ν) + Φ̃o (ν)

First note that the above inequality holds in a neighborhood of τ = 0 because it implies that
νl ≈ νh . Moreover, if λC ≤ λD ≤

√
2−1
√
(η
2

+ µ), we must have that

2λD
λC [F (νh ) − F (νl )]
≥


η + µ + λD
µ + η + λC Φ̃n (νh ) − Φ̃n (ν) + Φ̃o (ν)
holds. We can see this by comparing the two functions

2x
η+µ+x

and

ax
η+µ+bx ,

where a =

F (νh ) − F (νl ) and b = Φ̃n (νh ) − Φ̃n (ν) + Φ̃o (ν). Both functions are strictly increasing in x,
and equal zero at x = 0. Moreover, the derivative of the ﬁrst function is strictly greater than

58

√
2−1
√
(η
2

the derivative of the second one for x ≤

+ µ), which implies that

λC [F (νh ) − F (νl )]
2λD
λD [F (νh ) − F (νl )]
>

 ≥

.
n
n
o
η + µ + λD
µ + η + λD Φ̃ (νh ) − Φ̃ (ν) + Φ̃ (ν)
µ + η + λC Φ̃n (νh ) − Φ̃n (ν) + Φ̃o (ν)
■

This concludes the proof.
Proof of Proposition 4
Proof. The bid-ask spread, deﬁned in equation (37), is
R∞

BA =

2λD Φo (νl )τ

+ λD (1 − θD )

λD Φo (νl ) + λC

νh

R νl

νdΦn (ν)−

νdΦo (ν)

r+η+µ+λD θD

R νh R ν
νl

−∞

.

n
o
νl dΦ (ν̃)dΦ (ν)

To see the ﬁrst part note that for τ = 0 we know that νl = νh = ν ∗ and the equilibrium is
unique. So we can take the derivative of BA in equation (37) with respect to τ evaluated
at τ = 0 and show that it has to be strictly positive. Given that all the functions are
diﬀerentiable, we must have this derivative strictly positive in a neighborhood of τ = 0.
We have that
∂BA ∂BA ∂νl
dBA
=
+
×
.
dτ
∂τ
∂νl
∂τ
For the ﬁrst term in the right-hand side we have
∂BA
2λD Φo (νl )
R R
,
=
∂τ
λD Φo (νl ) + λC ννlh ννl dΦn (ν̃)dΦo (ν)
which implies that

∂BA
∂τ

= 2 when evaluated at τ = 0 since in this case νl = νh = ν ∗ . For the

second term we have that
"R ∞

∂BA
∂νl

d
dνl

τ =0,
νl =ν ∗

νh

R νl

νdΦn (ν)−

−∞

νdΦo (ν)

#

r+η+µ+λD θD

λD Φo (νl )

= λD (1 − θD ) h
i2
R R
λD Φo (νl ) + λC ννlh ννl dΦn (ν̃)dΦo (ν)
R∞
νh

− λD (1 − θD )

νdΦn (ν)−

R νl

−∞

νdΦo (ν)

r+η+µ+λD θD

h

d
dνl

h

λD Φo (νl ) + λC

λD Φo (νl ) + λC

59

R νh R ν
νl

νl

i

R νh R ν
νl

n
o
νl dΦ (ν̃)dΦ (ν)

dΦn (ν̃)dΦo (ν)

i2

.

Note that
d
dνl

Z ∞
νh

Z



Z νl



Z

∞
νl
νµf (ν)
d
νηf (ν)
νdΦ (ν) −
νdΦ (ν) =
dν −
dν
dνl νh η + µ + λD
−∞
−∞ η + µ + λD
[νh − νl ]ηf (νl )
−νh µf (νh )g ′ (νl ) − νl ηf (νl )
=
,
=
η + µ + λD
η + µ + λD
n

o

which is zero when evaluated at νl = νh = ν ∗ . Moreover,


d
λD Φo (νl ) + λC
dνl



Z νh Z ν

n

o

dΦ (ν̃)dΦ (ν)
νl

νl

is positive in a neighborhood of τ = 0 since this is basically turnover, which we showed in
the previous proof that is increasing in νl in a neighborhood of τ = 0. Thus,
We also have shown that
dBA
dτ

∂νl
∂τ

τ =0,
νl =ν ∗

∂BA
∂νl τ =0,
νl =ν ∗

≤ 0.

< 0 in a neighborhood of τ = 0. Therefore, we have that

=

∂BA
∂τ
|

τ =0,
νl =ν ∗

{z

=2+

}

∂BA
∂νl

|

τ =0,
νl =ν ∗

{z

≤0

}

∂νl
∂τ

×

|

τ =0,
νl =ν ∗

{z

≤0

> 0.

}

This proves the ﬁrst part of the proposition. Namely, that the bid-ask spread is increasing in
τ in a neighborhood of τ = 0.
In order to show the second part of the proposition, it suﬃces to show that BA converges
to zero as τ converges to inﬁnity. First lets us show that νl converges to −∞ when τ converges
to inﬁnity.
In equilibrium we must have that
G(νl ) =

The term

σC (ν;νl ,νh )−σD
σD

1
2λD θD

Z g(νl )
σC (ν; νl , νh ) − σD

σD

νl

is bounded below by

dν = τ.

n ,θ o }
λD θD −λC max{θC
C
,
η+µ+λD θD

and above by

λD θ D
η+µ+λD θD .

Therefore, as τ converges to inﬁnity, in order to obtain an equilibrium we must have νl
converging to −∞, and νh = g(νl ) converging to ∞.

60

Consider now the formula for the bid-ask spread,
R∞

BA =

We can see that

R∞
νh

2λD Φo (νl )τ

λD Φo (νl ) + λC

νdΦn (ν) −

To show that Φo (νl )τ =

+ λD (1 − θD )

R νl

o
−∞ νdΦ (ν)

ηF (νl )τ
η+µ+λD

νh

R νl

νdΦn (ν)−

νdΦo (ν)

r+η+µ+λD θD

R νh R ν
νl

−∞

.

n
o
νl dΦ (ν̃)dΦ (ν)

converge to zero since

R∞

−∞ ν

2 f (ν)dν

is bounded.

converges to zero is not as simple because F (νl ) converges

to zero and τ converges to inﬁnity. But note that
G′ (νl )F (νl )2
τ
1
= lim
= lim
.
dν
νl ↘−∞
τ ↗∞ 1/F (νl )
τ ↗∞ f (νl ) l /F (νl )2
f
(ν
)
l
dτ

lim F (νl )τ = lim

τ ↗∞

And, as we have established in the of Proposition 2,










σC (νl ; νl , g(νl ))
σC (g(νl ); νl , g(νl ))
−1 −
−1
G (νl ) =
g (νl )
2λD θD
σD
σD


Z g(νl )
1 ∂σC (ν; νl , g(νl ))
∂σC (ν; νl , g(νl ))
1
+ g ′ (νl )
dν.
+
2λD θD νl
σD
∂νl
∂νh
1

′

′

So we only need to show that each of the terms above, when multiplied by
to zero as νl converges to −∞. Let us start showing that
0 ≤ lim −νl F (νl ) = lim

Z νl

νl ↘−∞ −∞

νl ↘−∞

)2

F (νl
f (νl )

|νl |f (ν)dν ≤ lim

converges

converges to zero. Note that
Z νl

νl ↘−∞ −∞

where the equality in the end comes from the fact that

F (νl )2
f (νl ) ,

|ν|f (ν)dν = 0,

R 2
ν f (ν)dν is ﬁnite. Therefore we can

conclude that limνl ↘−∞ νl F (νl ) = 0. But then
−νl
νl ↘−∞ 1/F (νl )

0 = lim −νl F (νl ) = lim
νl ↘−∞

L’Hôpital

=

−1
F (νl )2
=
lim
.
νl ↘−∞ −f (νl )/F (νl )2
νl ↘−∞ f (νl )
lim

Now let us look the individual terms of G′ (νl ). We have that




F (νl )2 σC (νl ; νl , g(νl ))
−1 =0
νl ↘−∞ f (νl )
σD
lim

61

h

because

σC (νl ;νl ,g(νl ))
σD

i

− 1 is bounded. We have that




F (νl )2 ′
σC (g(νl ); νl , g(νl ))
lim
g (νl )
−1
νl ↘−∞ f (νl )
σD
"

#

[1 − F (νh )]2 σC (νh ; g −1 (νh ), νh )
= lim
−1 =0
νh ↗−∞
f (νh )
σD
h

because again

σC (νh ;g −1 (νh ),νh )
σD

i

in the same fashion that we showed that limνl ↘−∞
0≤

Z g(νl )
∂σC (ν; νl , g(νl ))

∂νl

νl


Z g(νl ) 


Z g(νl )

dν =
νl

λ ησ (ν; ν , g(ν

[1−F (νh )]2
f (νh )

− 1 is bounded and we can show that limνh ↗−∞
)2

F (νl
f (νl )

= 0. Moreover,
(

σC (ν; νl , g(νl ))2 λC

∂ Φ̃o (ν)
o ηf (νl )
θC
−
η+µ
∂νl
h

λC ησC (ν; νl , g(νl ))2 1 +

))2

=0

λC Φ̃o (ν)
η+µ

)

i

dν




C
l
l
o C
θC
f (νl )dν
+n

o

η+µ
νl


µ + η + λC F (νh ) − F (ν) − Φ̃o (νh ) + 2λC Φ̃o (ν) 


h
i
λC Φ̃o (ν)

2
Z g(νl ) 
 λ ησ (ν; ν , g(ν ))2

λC ησC (ν; νl , g(νl )) 1 + η+µ
C
l
l
o C
o
θC
≤
+n
f (ν)dν



η+µ
νl


µ + η + λC F (νh ) − F (ν) − Φ̃o (νh ) + 2λC Φ̃o (ν) 

=

which is bounded because the term in brakets is bounded and

R g(νl )
νl

f (ν)dν ≤ 1. Therefore,

we have that
F (νl )2
νl ↘−∞ f (νl )
lim

Z g(νl )
∂σC (ν; νl , g(νl ))

∂νl

νl

dν = 0.

The proof that
F (νl )2
lim
νl ↘−∞ f (νl )

Z g(νl )
νl

g ′ (νl )

∂σC (ν; νl , g(νl ))
dν = 0.
∂νh

is analogous. With that, we can conclude that limτ ↗∞ BA = 0, which implies the second
part of Proposition 4, and concludes the proof.

■

Proof of Proposition 5
Proof. First not that, due to the assumed symmetry in the parameters, we have that pA (τ ) =

62

σD ν̄ A and pB (τ + dτ ) = σD ν̄ A . Therefore, the liquidity premium can be written as
LP (τ ) =

ν̄ A − ν̄ B
.
dL̄ + BA(τ + dτ ) − BA(τ )

We have shown in the proof of Proposition 4 that

dBA
dτ

which implies that BA(τ + dτ ) − BA(τ ) > 0 and LP (τ ) <

> 0 in a neighborhood of τ = 0,
ν̄ A −ν̄ B
.
dL̄

We have also show that limτ ↗∞ BA = 0. Since we know that BA(τ ) > 0 in a neighborhood of τ = 0, this implies that
BA(τ ) < 0 and LP (τ ) >

ν̄ A −ν̄ B
dL̄

dBA
dτ

< 0 in for some τ , which implies that BA(τ + dτ ) −

. Moreover, because

lim BA(τ + dτ ) − BA(τ ) = lim BA(τ + dτ ) − lim BA(τ ) = 0,

τ ↗∞

we have that limτ ↗∞ LP (τ ) =

B

τ ↗∞

ν̄ A −ν̄ B
.
dL̄

τ ↗∞

This concludes te proof.

■

Data

We use corporate bonds transactions data from the TRACE Enhanced (ETRACE) database
from January 2005 to June 2021. This initial data set provides us with a total of 171,140,493
trades as well as with 283,250 uniquely-identiﬁable bonds.10
We use a procedure based in Dick-Nielsen (2009) and Dick-Nielsen (2014) to ﬁlter out
errors, cancellations, reversals and double counting as well as transactions missing individual
CUSIP identiﬁcation. We subsequently drop trades missing yield information and trades
that are either on a when-issued basis, in a non-secondary Market, with a special condition,
automatic give-ups, or in equity-linked notes.11
10

The Trade Reporting and Compliance Engine (TRACE) is the “FINRA-developed vehicle that facilitates
the mandatory reporting of over-the-counter secondary market transactions in eligible ﬁxed income securities.”
The bond transactions report was implemented in diﬀerent phases. It started with Phase I, on July 2002, for
investment grade bonds and with issue size greater than or equal to $1 bi, and it continued later with the
requirements expanded in Phase II in 2003. The complete implementation occurred in 2005, with Phase III.
The report of corporate bond transactions is mandatory for all broker-dealers FINRA members. Therefore,
Phase III virtually contains complete coverage of all public transactions. For consistency of the selection into
the dataset, our dataset focus on Phase III. The Enhanced TRACE diﬀers from the Standard TRACE in that
it discloses more detailed information in individual transactions, e.g., actual trade size.
11
To remove any potentially erroneous trades still remaining in the database, we also add a price ﬁlter for
trades with prices deviating more than 25% from the daily average. This procedure cleans only about 0.1%
of the trades.

63

To avoid having many bonds in our sample that trade only momentarily, we add the following two conditions: (1) the bond must have existed in ETRACE for at least one complete
year; and (2) the bond must have traded at least 75% of its relevant trading days (BPW and
Anderson and Stulz 2017). Bonds must also have suﬃcient trades to satisfy the conditions,
as deﬁned in the following section, necessary to calculate their individual illiquidity measures.
Having applied all these trade-based criteria, we are left with 55,753,160 transactions in 5,410
unique issues.
We use Bloomberg to collect bond information on issuance and maturity dates, provisions,
coupons, currency denomination, amount outstanding, and ratings. We use the amount
outstanding of each issue at the last business day of each month. A bond is deﬁned as
investment grade if its rating is greater than or equal to BBB– from S&P and Fitch or Baa3
from Moody’s. We ﬁrst use the rating from Standard & Poor’s; if this rating is unavailable, we
use the rating from Fitch; and if this rating is unavailable, we use the rating from Moody’s.12
We exclude trades that took place outside the range of issuance and maturity dates of an
issue, and bonds for which the outstanding amount at the last business day of that speciﬁc
month was zero. Defaulting bonds are eliminated from the sample for as long as they are
considered in default, and so are bonds with missing information. We only keep in our sample
callable or non-provisional, ﬁxed-rate bonds issued in the US. Callable bonds comprise a
signiﬁcant portion of our sample. Removing these bonds would negatively impact the quality
of our results. Instead, we control our results for callability by introducing a dummy to our
model. At this stage, our sample consists of 45,026,565 trades in 4,255 individual bonds.
We calculate the individual yield spread as the diﬀerence between the yield of the corporate bond and the yield of the government bond with the same maturity, as in BPW. The
constant maturity yield curve is obtained from the Federal Reserve Bank of St. Louis FRED
dataset. We use linear interpolation to calculate the yield of the government bond matching
the exact maturity of the corporate bond. The monthly cross-sectional yield spread of a
corporate bond is then calculated as the average daily spread in the month.
We use the Eikon dataset to collect each issuer’s daily 5-year Credit Default Swap (CDS)
quotes, which we use to proxy for the issuer’s credit risk. Our measure of credit risk for each
12

Although we use a diﬀerent order based on data availability, this process is similar to Dick-Nielsen et al.
(2012).

64

9
600
Mean
Median

Mean
Median

8

500

7
6

400

5
300

4
3

200

2
100

1
0
2008

2010

2012

2014

2016

2018

0
2008

2020

(a) CDS

2010

2012

2014

2016

2018

2020

(b) Yield spreads

Figure 9: 5-Year Credit Default Swaps and yield spreads for individual corporate bond issues
in the sample.
monthly cross-section is the average of the issuer’s end-of-day CDS spreads. As this data is
suﬃciently large for the bonds in our database from December 2007, we redeﬁne our sample
period to begin in December 2007. We use stock prices to calculate the annualized equity
return volatility of each issuer. Bonds missing CDS and equity volatility data are excluded
from our dataset. We collect the daily stock prices of the issuers from CRSP.
Our ﬁnal bond sample consists of 32,435,392 trades in 3,073 unique issues, which are
distributed over a period of 115 months starting from December 2007. In total, we have
139,168 combinations of bond-month observations. The number of observations varies between monthly cross-sections depending on, among other things, newly-issued and matured
bonds, trade frequency, and issues satisfying our selection criteria in the observed crosssection. Our ﬁnal sample is predominantly composed by investment grade bonds.
We separate our sample period in three time intervals characterized by diﬀerent macroeconomic conditions: (1) the ﬁnancial crisis period from December 2007 to December 2009;
(2) the post-crisis period with historically low interest rates from January 2010 to November
2015; and (3) the monetary tightening period from December 2015 to June 2017. Figure 9
shows the CDS and the yield spreads over time of the bonds in the sample. Table 3 presents
a summary of our data together with the illiquidity measures described in the next section.

65

Table 3: Summary statistics

Observations
Investment Grade
N. of Bonds
Callable
N. of Firms
N. of Trades
γ
AMD (×103 )
Spread
CDS (×10−2 )

Complete Period
Dec ‘07–June ‘17

Crisis
Dec ‘07–Dec ‘09

Post-Crisis
Jan ‘10–Nov ‘15

Monetary Tightening
Dec ‘15–June ‘17

139,168
87%
3,073
33%
416
32,435,392

17,165
86%
1,134
33%
227
6,078,875

91,424
87%
2,688
33%
388
20,192,250

30,579
87%
1,905
31%
310
6,164,267

Mean Median SD Mean Median SD Mean Median SD Mean Median SD
2.225 1.044 3.931 5.776 2.623 11.648 1.284 .639 1.842 1.067 .480
1.586
2.933 1.755 3.945 6.497 3.789 9.603 2.003 1.255 2.386 1.717 .947
2.325
2.163 1.543 2.179 4.160 2.856 4.634 1.598 1.202 1.387 1.646 1.090 1.909
1.674 1.063 2.008 2.712 1.529 3.837 1.385 .972 1.375 1.390 .789
1.969

This table reports a summary of our sample variables together with a summary of the main variables calculated.
The observations are the bond-month combinations. The mean, median and standard deviation are the timeseries averages of the respective cross-sectional measures within each sub-period. Spread is the corporate bond
yield spread detailed in section B. γ and AMD are the illiquidity measures detailed in section B.

C

Additional empirical results

C.1

The determinants of bond illiquidity

Given the importance of bond-level illiquidity for the yield spread, we now study the determinants of illiquidity for a particular bond. We regress our illiquidity measures on a list of
varying characteristics such as CDS, time to maturity, age, volume, frequency and issuer’s
equity return volatility and credit rating. We also make use the intrinsic characteristics of a
bond such as coupon paid, issuance size and whether the bond is callable or not. Our model
consists in regressing pooled OLS estimations with two-dimensional clustered standard errors.
Our results have strong statistical and economical signiﬁcance as we discuss next. Results
are reported in table 4.
We ﬁnd that the credit risk of the issuer and the time to maturity are the most important
characteristics of a particular bond to explain variations in illiquidity of an issue. Illiquidity
is positively correlated to both credit risk and time to maturity.
When the CDS of an issuer widens by 100 basis points, we ﬁnd that the γ’s of its bonds
increases by .901. This increase corresponds to about 40% of the average γ reported in table
3. Similarly, we ﬁnd that one additional year in time to maturity increases γ by .165, which
corresponds to 7% of the average γ.
66

Table 4: γ and AMD on bond characteristics
γ
CDS

.817
[6.75]

Maturity

AMD
.901
[6.50]

.161
[11.02]

.697
[7.89]

.165
[11.84]

.719
[6.93]
.128
[7.87]

.139
[9.29]

Age

−.002
[−.12]

.117
[5.42]

Coupon

.078
[2.14]

.079
[2.17]

Volume

−.750
[−3.78]

−2.01
[−7.32]

.510
[2.15]

3.24
[12.82]

−.787
[−8.31]

−1.17
[−8.59]

EqVol.

.481
[1.38]

.308
[1.70]

IG

2.18
[5.05]

2.19
[6.78]

−.318
[−3.08]

−.053
[−.50]

Frequency

ln(Issuance Size)

Call

Constant

Adj.R2
Obs.

.526
[4.26]

.666
[6.15]

2.27
[3.28]

1.39
[10.17]

1.57
[13.06]

5.01
[5.94]

.089
139, 168

.032
139, 168

.142
139, 168

.111
139, 168

.035
139, 168

.226
139, 168

Bond-level illiquidity measures regressed on bond characteristics. We run a pooled OLS regression with
standard-errors clustered by bond and month. T-statistics in square brackets. γ and AMD are the illiquidity
measures detailed in section B. AMD is multiplied by 103 . Maturity is the issue’s time to maturity. Maturity
and age are calculated in years at the last business-day of each month. Volume is calculated as the total $
amount traded ×10−11 and frequency is in thousands of trades. Issuance size is in $ millions. EqVol. is the
issuer’s annualized equity return volatility. IG is 1 if the bond is Investment Grade and 0 if otherwise. Call is
1 if the bond is callable and 0 if otherwise.

67

The variables representing trading activity, that is, volume and frequency, show contrasting results. Bonds with greater volume are more liquid, but so are bonds that trade less
frequently. These results support the view that bonds with larger average size per trade are
potentially more liquid. Callable bonds show as more liquid for both γ and AMD than issues
with no embedded options.
The results for the credit rating and coupon are surprising. We ﬁnd that investment-grade
bonds have substantially higher γ’s. This contrasts with the time-series measures observed
in ﬁgure 2, which shows, instead, that investment-grade bonds have smaller γ’s. Once CDS
and credit quality are considered simultaneously, as in table 4, we obtain that the eﬀect of
credit quality on γ captured more strongly by the CDS than by the credit rating. This eﬀect
occurs because high-yield bonds have, on average, signiﬁcantly higher CDS spreads than
investment-grade issues. The coupon paid also presents a non-intuitive positive slope, which
implies that bonds that pay higher coupons are less liquid.
The results for γ and AMD point in general to the same direction. An exception is
the coeﬃcient for age, negative, not statistically signiﬁcant, for γ, but positive and strongly
signiﬁcant for AMD. A positive coeﬃcient for age indicates a market preference for on-the-run
securities as oppose to older, oﬀ-the-run issues from the same ﬁrm. This ﬁnding is consistent
with the on-the-run and oﬀ-the-run spread (see, for example, Krishnamurthy 2002).
In summary, our results indicate that the main determinants of illiquidity of a particular
bond are credit risk and time to maturity. We address in the next section whether corporate
bond liquidity can be aﬀected by aggregate factors, common to all bonds.

C.2

Commonality in liquidity

We now investigate which factors aﬀect aggregate liquidity of corporate bonds. We analyze
factors such as ﬁnancial stability, economic conditions, the term structure, and market volatility. To avoid having our results driven by few issues with exceptionally high illiquidity, we
focus on market illiquidity as measured by the median γ and AMD of individual bonds.13
We proxy ﬁnancial conditions by the NFCI Index, for which historical data is retrieved
from the Chicago Fed database.14 The volatility measures for the equities market and the
13
14

BPW also consider the median as aggregate market liquidity for similar reasons.
The National Financial Conditions Index (NFCI) captures “ﬁnancial conditions in money markets, debt

68

benchmark 10-year Treasury interest rate are respectively given by the Cboe’s VIX and
TYVIX indices. We retrieve historical data for both indices from FRED.15 Additionally,
we retrieve from FRED historical the diﬀerence between the 10-year and 2-year constant
maturity Treasury rates as a measure of the slope of the yield curve. The corporate debt
net position data of Primary Dealers is retrieved from the NY Fed and monthly medians are
used as observations. We use the sum of individual bonds total volume traded and number of
transactions as an indication of market volume and frequency. We regress monthly changes in
aggregate market illiquidity on monthly changes in ﬁnancial, economic and market measures.
When ﬁnancial conditions tighten (NFCI increases), we ﬁnd that corporate bond market
liquidity decreases. Aggregate ﬁnancial conditions are closely related with corporate bond
market liquidity. Table 5 reports the results across diﬀerent market illiquidity measures.
About changes in the term structure of interest rates. We ﬁnd that a higher slope of the
yield curve is associated with an increase in corporate bond market illiquidity. When the
yield curve steepens, corporate bond liquidity lowers.
Another important element contributing to corporate bond market illiquidity is the balance sheet of Primary Dealers. The net position of Primary Dealers in corporate debt instruments is positively correlated to market illiquidity. The results are similar for γ and AMD.
This result supports the evidence that the post-crisis liquidity provision in corporate debt
instruments has been shifting from traditional Primary Dealers to other market players.
Volatility in equity markets yields a positive slope, as expected. However, its statistical
signiﬁcance weakens in the presence of control variables. We ﬁnd that illiquidity increases
with the VIX. Alternatively, the volatility of interest rates, proxied by the benchmark 10year Treasury note, has a negative slope. When TYVIX is regressed alone, the coeﬃcient is
positive but with a small t-statistics of .13 and a nearly null adjusted-R2 .
We observe a relation between trading activity measures and illiquidity at the aggregate
level similar to the observed at the individual bond level. Illiquidity decreases with volume
and increases with frequency. Although presenting a negative sign, volume, however, is not
statistically signiﬁcant for γ as it is for AMD.

and equity markets and the traditional and ‘shadow’ banking systems.”
15
In FRED, the data for the Cboe’s 10-Year US Treasury Note Volatility Index, TYVIX, is referred to by
its legacy ticker, VXTYN.

69

Table 5: Monthly changes in aggregate market illiquidity on monthly changes in macro variables
∆γ
Aggr. by
∆NFCI

Median
2.14
[4.32]

∆ AMD

Mean

1.73 5.48
[8.10] [6.11]

Vol. Weight
4.82 3.30
[6.97] [1.92]

Median

2.57 1.49
[1.59] [7.98]

Mean

1.73 2.99
[8.62] [7.20]

Vol. Weight
2.69 2.07
[6.78] [8.56]

1.69
[5.38]

∆VIX

.051 .012
[2.51] [1.97]

.113 .008
[2.16] [.62]

.062 .004
[1.44] [.15]

.031 .012
[2.45] [1.91]

.062 .006
[2.32] [1.01]

.045 .009
[2.15] [.93]

∆YCurve

1.00 .622
[1.95] [3.74]

1.00 .468
[1.08] [.85]

.451
[.63]

.420 .622
[2.24] [4.72]

.463
[.94]

.206
[.66]

.576 .463
[2.92] [2.58]

.015
[2.49]

.012
[1.57]

−.022
[−1.08]

.015
[2.61]

.006
[1.18]

0
[.00]

∆TYVIX

−.091
[−4.83]

−.043
[−.28]

.420
[1.16]

−.091
[−4.82]

−.154
[−3.81]

−.025
[−.45]

∆Volume

−.190
[−1.71]

−.626
[−1.19]

.374
[.44]

−.190
[−1.64]

−.838
[−2.37]

−.480
[−2.84]

.434
[2.41]

1.04
[1.65]

−.143
[−.21]

.434
[2.28]

.970
[2.56]

.517
[2.24]

∆PrimaryDealers

70
∆Frequency

.578
[.77]

Constant

.012 −.003 −.031 .036 −.005 .037
[.66] [−.09] [1.98] [.91] [−.07] [.98]

.019 −.006 −.019 .004 −.007 .031 .008 −.015 −.002 .002 −.013 −.009
[.25] [−.08] [−.22] [.24] [−.22] [2.17] [.33] [−.25] [−.08] [.18] [−.42] [−.47]

Adj.R2

.401 .394

.111 .034

.734

.471 .236

.540

.182

.471 .283

.678

.478 .234

.732

.422 .265

.534

Results of changes in aggregate market illiquidity on changes in macro variables. The market illiquidity can be aggregated by the median, mean or volumeweighted average of individual illiquidity measures within each of the 115 monthly cross-sections covered in our sample. Therefore, we have 114 observations
of monthly changes in aggregate market liquidity. Coeﬃcients are estimated using OLS and standard errors are corrected by Newey-West. T-statistics are
reported in square brackets. γ and AMD are the illiquidity measures detailed in section B. AMD is multiplied by 103 . NFCI is the Chicago Fed’s National
Financial Conditions Index. VIX and TYVIX are respectively the Cboe’s volatility indices for the equities market and the benchmark 10-year Treasury
Note. YCurve is the yield curve slope measured as the spread between the 10-year and the 2-year constant maturity yields. PrimaryDealers is the monthly
median of corporate debt net positions in billions of U.S. dollars of Primary Dealers reported by the NY Fed. Volume is the total $ amount traded in a
month ×10−13 and Frequency the total number of trades in a month ×10−5 .
Full text of Working Papers (Federal Reserve Bank of Richmond) : Dealer Costs and Customer Choice, Working Paper 23-13

FRASER
