Post-Crisis Regulations, Market Making, and Liquidity in the … · 2019. 9. 27. · Post-Crisis Regulations, Market Making, and Liquidity in the Corporate Bond *Market Xinjie Wang

Post-Crisis Regulations, Market Making, and Liquidity in the Corporate

Bond Market*

Xinjie Wang and Zhaodong (Ken) Zhong†

May 12, 2019

Keywords: Basel III, capital requirement, market making, liquidity, inventory, corporate bond

JEL classifications: G14, G21, G23, G24, G28

* We are grateful for the valuable comments from participants at the Sixth Asian Quantitative Finance Conference,

the Fourth PKU-NUS Annual International Conference on Quantitative Finance and Economics, and 2019 China International Conference in Finance. All remaining errors are our own. Xinjie Wang acknowledges financial support

from Southern University of Science and Technology (Grant No. Y01246210, Y01246110). † Xinjie Wang is from the Department of Finance, Southern University of Science and Technology, Shenzhen, China.

Email: [email protected]; Zhaodong (Ken) Zhong is from the Department of Finance and Economics,

Rutgers Business School, Rutgers University, Newark and New Brunswick, NJ, USA. Email:

[email protected].

mailto:[email protected]

mailto:[email protected]

Post-Crisis Regulations, Market Making, and Liquidity in the Corporate

Bond Market

Abstract

We theoretically and empirically study the effects of Basel III capital requirements on the

liquidity of corporate bonds. We first use a simple model to provide a unified explanation of

several puzzling empirical findings after the recent financial crisis: (i) narrowing bid-ask spreads,

(ii) a reduction in dealers’ inventory, (iii) increased price impact during periods of stress, and (iv)

increased difficulty in executing large trades. Next, we empirically test two new predictions of

the model: we find that bonds with higher risk weights in risk-weighted assets experience larger

drops in trade size and higher increases in price impact in the post-regulation period.

1

1 Introduction

Even though a decade has passed since the 2008 financial crisis, market practitioners are still

worried about the fading liquidity in the corporate bond market.1 They contend that bond dealers

no longer provide as much liquidity as they did before the crisis because they are constrained by

post-crisis financial regulations (e.g., the capital and liquidity requirements in Basel III). As

anecdotal evidence, Figure 1 shows that primary dealers’ aggregate inventory of corporate

securities declines significantly after the 2008 financial crisis, from over $250 billion in 2007 to

less than $25 billion in 2015. However, recent empirical studies show that some price-based

liquidity measures, such as bid-ask spreads, on corporate bonds after the 2008 financial crisis are

actually not worse than those before the crisis (e.g., Anderson and Stulz, 2017; Bessembinder,

Jacobsen, Maxwell, and Venkataraman 2018; Trebbi and Xiao 2018).2 At the same time, there is

evidence showing a higher price impact in periods of market stress after the crisis (Anderson and

Stulz, 2017; Bao, O’Hara, and Zhou, 2018), increased difficulty in the execution of large trades

(Anderson and Stulz, 2017), and a reduction in dealers’ capital commitment (Bessembinder et al.,

2018). Why do price-based liquidity measures not reflect the widely held view among market

participants that liquidity deteriorated after the crisis? How can the observed puzzling liquidity

patterns in the corporate bond market after the crisis be explained?

In this paper, we attempt to address these questions by first studying the impact of Basel III

capital requirements on bond liquidity in a simple inventory model and then empirically testing

some new predictions based on our model. The post-crisis regulations can affect the market-

making activities of bank-affiliated dealers through at least three potential channels. The first

1 See, for example, “Corporate bond market structure: The time for reform is now”, BlackRock Viewpoint, 2014,

and “Letter to JP Morgan shareholders”, JP Morgan, April 2016. 2 See also “Bond liquidity metrics: Reading between the lines”, TABB group, 2016.

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channel is related to inventory capacity. Banks raise capital from shareholders and allocate

capital internally across different business units. The higher capital requirements in Basel III

force banks to reserve more capital for assets held in their inventory.3 Therefore, given the same

amount of capital, this effectively reduces the maximum inventory capacity of banks. Once the

maximum capacity of inventory is reached, a dealer is no longer able to accommodate trades that

will increase its inventory.

The second channel is through the total capital cost (which is equal to the product of the

cost of capital and the required capital for holding the trading assets). Capital is costly to banks

as investors require a certain rate of return for holding banks’ equities. Given the same required

rate of return, the higher capital requirements in Basel III force banks to hold more capital

against the same amount of trading assets and thus effectively increase the total capital cost.

When the total capital cost is higher, it will be less profitable for bank-affiliated dealers to

maintain a large inventory. In other words, dealers are less willing to hold a large inventory even

it is still within their capacity limits.

The third channel is through price elasticity. The post-crisis regulations can change the

competitive advantages between bank-affiliated dealers and non-bank dealers. Since non-bank

dealers are not subject to the same set of capital requirements against their inventory as their

bank-affiliated counterparts, they have more freedom to trade in terms of trade sizes, riskiness of

assets, and trading motives. In addition, many of the electronic trading platforms developed post

crisis allow easier access for non-bank dealers to compete with bank-affiliated market makers.

All these developments have led to bank-affiliated dealers facing a greater price elasticity of

demand.

3 Basel III requires banks to maintain 4.5% of “Common Equity Tier 1 (CET1)” of risk-weighted assets (RWAs) at

all times. In addition, a minimum “Tier 1 capital” of 6% (4.5% of CET1 + 1.5% additional Tier 1) of RWAs is

required in Basel III. Basel III also requires minimum total capital to be 8% (Tier 1 + Tier 2) of RWAs.

3

Our model is based on the inventory model in Amihud and Mendelson (1980), with added

features relevant to the abovementioned channels through which the post-crisis regulations affect

market making.4 We begin our analysis with an examination of the impact of each of the three

channels on market liquidity and trading activities using Monte Carlo (MC) simulations. First,

the post-crisis regulations tighten the limit on inventory capacity, which forces the market maker

to manage the inventory more prudently. Simulation results show that smaller inventory capacity

results in a lower average inventory position and a higher price impact, while the effect of

inventory capacity on average bid-ask spread is insignificant.

Second, we examine the impact of total capital cost on liquidity and market-making

activities. We assume that trading assets held in the inventory incur a cost of capital per unit time.

The total capital cost of market making is therefore proportional to average inventory position.

Since this total capital cost reduces the profit of the market maker, the market maker would

naturally prefer a smaller inventory to reduce the capital cost and increase its profit. Our

simulation results show that higher capital cost (a result of a higher required capital ratio for

holding) indeed leads to a reduction of average inventory position and an increase in price

impact.

Third, we assess the impact of price elasticity on liquidity and market-making activities.

When the market maker increases the ask price or decreases the bid price, bond traders can turn

to non-bank market makers or electronic trading platforms. Thus, the price elasticity of the buy

and sell orders of the market maker captures the level of competition between bank-affiliated

4 We consider a bank-affiliated market maker that is subject to the post-crisis regulations. Overall, the amount of capital for market making determines the maximum capacity of the market maker’s inventory. In addition, any unit

in the bank must generate a required rate of return on capital. Therefore, any capital held for trading assets in the

inventory incurs a capital cost to the market maker. The market maker sets bid and ask prices which determine the

arrival intensities of sell and buy orders from investors. A higher bid price increases the intensity of sell orders, and

a higher ask price reduces the intensity of buy orders. The sensitivity of the intensities to bid/ask prices is

determined by the price elasticity. More details of the model are provided in Section 3.

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market makers and alternative trading options. We simulate different levels of competition by

changing the price elasticity in our model. Higher price elasticity is expected to be associated

with smaller bid-ask spreads and a smaller price impact. Simulated results confirm that price

elasticity has substantial effects on bid-ask spreads and price impact. Higher price elasticity

results in narrower bid-ask spreads and a smaller price impact.

After examining the individual effect of each channel, we try to provide a unified

explanation of the previously documented and seemingly contradictory empirical findings in the

corporate bond market by combining the effects of all three channels (inventory capacity, total

capital cost, and price elasticity) simultaneously. To do so, we compare the simulation results

under two sets of scenarios that correspond to the pre-regulation period and the post-regulation

period. More specifically, in the post-regulation period, dealers’ inventory capacity is lower

(than in the pre-regulation period), total capital cost is higher, and competition from non-bank

dealers is greater. The simulation results show that our simple model can successfully match the

basic observed liquidity patterns documented in the prior literature (e.g., Anderson and Stulz,

2017; Bao et al., 2018; Bessembinder et al., 2018; Dick-Nielsen and Rossi, 2018; Trebbi and

Xiao, 2018): that is, average inventory decreases considerably, price impact increases

significantly, and the bid-ask spread narrows slightly in the post-regulation period.

Furthermore, our model can also match some more complex empirical irregularities. For

example, there is empirical evidence that price impact becomes higher in periods of market stress

after the crisis (Anderson and Stulz, 2017; Bao et al., 2018). The simulation results show that our

simple model can also match this pattern (i.e., price impact indeed becomes larger when the

dealers’ inventory positions move closer to the limit of inventory capacity). This is because the

market maker reduces both bid and ask prices more sharply in our model to lower the inventory

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position when its inventory position is close to the limit.5 Another major concern regarding post-

crisis liquidity in the corporate bond market is that large trades are difficult to execute (Anderson

and Stulz, 2017). Our model-simulated results again show that, in the post-regulation scenario,

larger trades indeed have a higher probability of being rejected by the market maker as a result of

the tightening limit on inventory capacity and the higher capital cost.

Finally, our model also generates two new predictions that have not been examined by the

existing literature. First, the simulated order rejection rate increases for bonds with higher risk

weights and larger trade sizes in the post-regulation period. Thus, the first new empirical

prediction from our model is that riskier bonds with higher weights in risk-weighted assets

(RWAs) according to the Basel III capital requirements will experience decreases in average

trade sizes. In other words, we expect to see a higher percentage of small trades among these

riskier bonds in the post-regulation period. Second, the simulated price impact is larger for

riskier bonds after the Basel III capital requirements are considered. This is because riskier bonds

take up a larger inventory capacity in the post-regulation period, which leads to a larger price

impact. Therefore, the second new empirical prediction from our model is that the ratio of the

price impact of riskier bonds to that of less risky bonds increases significantly.

To empirically test these new predictions, we use bond transaction data in TRACE for the

period from January 2004 to December 2014. Following prior studies, we divide the sample

period into four periods: pre-crisis, crisis, post-crisis, and regulatory periods. For each period, we

calculate the distribution of trade size for bonds with different risk weights using either the

number of bond transactions or trading volume. Consistent with our first prediction, we find that

the portion of large trades declines considerably in the regulatory period relative to the pre-crisis

5 The market maker lowers the ask price to reduce the intensity of incoming sell orders and lowers the bid price to

increase the intensity of incoming buy orders.

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level, and the decline is particularly significant for risky bonds. To test the second prediction, we

calculate price impact using the Amihud (2002) measure for bonds with different risk weights

over the four periods. The result shows that the ratio of the average Amihud measure for bonds

with higher risk weights to that for bonds with the lowest risk weight increases significantly in

the regulatory period, which is consistent with the second prediction.

The remainder of the paper is organized as follows. Section 2 reviews various strands of

research related to our study and highlights our contributions. Section 3 describes the setting,

calibration, and simulation of our model with regulation-related features. Section 4 presents a

unified explanation of prior empirical findings regarding bond liquidity. Section 5 conducts

empirical tests of two new model-generated predictions of the effects of capital requirements on

liquidity. Section 6 concludes the paper.

2 Related literature

Our paper contributes to several strands of literature. First, the paper contributes to the recent

literature on the effects of the post-crisis regulatory changes on market making in the corporate

bond market. An ongoing debate is whether the Dodd-Frank Act strengthens liquidity or not.

Bessembinder et al. (2018) find that the post-crisis regulations lower the participation and capital

commitment of bank-affiliated dealers in the corporate bond market. Bao et al. (2018) examine

the liquidity of the corporate bond following credit rating downgrades and find that the Volcker

Rule increases the illiquidity of stressed bonds. Anderson and Stulz (2017) find that liquidity is

lower after extreme VIX increases. Dick-Nielsen and Rossi (2018) find that the cost of

immediacy increased significantly for corporate bonds after the crisis by using index exclusions

as a natural experiment. Randall (2017) shows that liquidity is worse when inventory costs rise,

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particularly for lower-rated bonds. Wang, Wu, and Zhong (2018) find that the co-movement in

illiquidity increased significantly after the post-crisis regulations took effect. These findings are

consistent with the predictions in Duffie (2012) that the post-crisis regulations would reduce the

quality and capacity of market making by banks. Trebbi and Xiao (2018), on the other hand, use

time-series methods and find no structural breaks of reduced market liquidity for the regulatory

period. Our paper contributes to this literature in several ways. First, our model provides a

unified explanation of the previously documented empirical findings regarding bond liquidity.

More importantly, we highlight various effects of Basel III capital requirements on bond

liquidity. Second, our model offers two new testable predictions that have not been examined

before. Finally, we empirically test the predictions using bond data in TRACE and find evidence

consistent with our model predictions.

Second, our paper is related to the literature on inventory models. Market makers provide

the service of immediacy by holding assets in their own inventory. Amihud and Mendelson

(1980) develop an inventory model in which buy and sell orders from investors are depicted by

price-dependent Poisson processes. They derive the optimal pricing policy and show that bid-ask

spreads is a declining function of inventory position. Stoll (1978) suggests that there exists an

optimal inventory level for market makers. Ho and Stoll (1981) derive the optimal bid and ask

prices assuming both the stochastic nature of orders and the time variation of stock prices. The

inventory model has been extended by incorporating asymmetric information. O’Hara and

Oldfield (1986) study the effect of risk aversion on bid-ask spreads. Madhavan and Smidt (1993)

develop a model that combines inventory and information effects. We extend the Amihud and

Mendelson (1980) model by incorporating features related to the recent financial regulation

reforms. We demonstrate that the extended Amihud and Mendelson (1980) model is able to

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study important policy issues, such as how the post-crisis regulations affect dealers’ inventory

and market liquidity.

Third, our paper is also related to the strand of literature on the risk-bearing capacity of

market makers. Trading in OTC markets is facilitated by market makers who are subject to

financial constraints. When market makers’ capacity is reduced, asset prices tend to be more

expensive and the market becomes less liquid (e.g., Stoikov and Saglam, 2009; Comerton-Forde

et al., 2010; Kitwiwattanachai and Pearson, 2014; Gârleanu, Pedersen, and Poteshman, 2009;

Junge and Trolle, 2015; Cao et al., 2016; Siriwardane, 2017; Friewald and Nagler, 2018). Our

paper provides theoretical analyses for these empirical studies. We show that when the inventory

capacity of the market maker is reduced, the average inventory position of dealers becomes

smaller and the price impact becomes larger. More importantly, large trades become more

difficult to execute, leading to overall market illiquidity especially in periods of distress.

Finally, our paper is related to the large literature on the liquidity of corporate bonds in

general. For example, Schultz (2001) provides early evidence on bond transaction costs using

institutional trading data. Some studies develop measures of bond liquidity based on transaction

data (e.g., Mahanti et al., 2008; Bao, Pan, and Wang, 2011). Other studies have linked bond

liquidity to pricing (e.g., Chen, Lesmond, and Wei, 2007; Lin, Wang, and Wu, 2011; Feldhütter,

2012; Bao and Pan, 2013), information transparency (e.g., Edwards, Harris, and Piwowar, 2007),

regulations (e.g., Ellul, Jotikasthira, and Lundblad, 2011), financial crisis (e.g., Dick-Nielsen,

Feldhütter, and Lando, 2012; Friewald, Jankowitsch, and Subrahmanyam, 2012), information

efficiency (e.g., Ronen and Zhou, 2013), and related markets (e.g., Das, Kalimipalli, and Nayak,

2014).

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3 The model

The model in our study is based on the inventory model in Amihud and Mendelson (1980), with

added features relevant to the post-crisis regulations.

3.1 Base assumptions

We start from a set of base assumptions that are the same as those in Amihud and Mendelson

(1980):

(A) All bond trades are intermediated through a single market maker. No direct trades

between investors are permitted.

(B) The market maker is a price setter. She sets an ask price, 𝑃𝑎, at which she accepts a

buy order with one unit and a bid price, 𝑃𝑏, at which she accepts a sell order with one

unit. 𝑃𝑎 and 𝑃𝑏 are functions of the market maker’s inventory position.

(C) The arrivals of buy and sell orders are described by two independent Poisson

processes, with arrival rates 𝐷(𝑃𝑎) and 𝑆(𝑃𝑏).

(D) The permissible inventory levels are {−𝐿, −𝐿 + 1, −𝐿 + 2, … , 𝐿 − 2, 𝐿 − 1, 𝐿}, where

L is an integer.

(E) The market maker maximizes her expected profit per unit time.

3.2 New features relevant to the post-crisis regulations

We assume that the market maker is an affiliate of a bank and thus is subject to the post-crisis

regulations. The post-crisis regulations impose the following constraints on the market maker:

(F) As stated in assumption (D), there is a limit on the maximum capacity 𝐿 of the

inventory. 𝐿 is determined by a number of factors, including the market maker’s

maximum capital, and the required Basel capital ratio. For a given level of capital, the

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higher the required Basel capital ratio, the smaller the effective maximum capacity 𝐿.

Since a bank’s capital is determined by many factors that are beyond the scope of our

study, we examine the effects of inventory capacity on market-making activities by

directly varying 𝐿 in our model.6

(G) Capital held for the inventory incurs a cost 𝑐, where 𝑐 = 𝑅𝑒 ⋅ 𝑘, 𝑅𝑒 is the required

return on equity and 𝑘 is the capital held for a certain level of inventory 𝐼. Capital 𝑘 is

proportional to inventory 𝐼 : that is, 𝑘 = 𝛼 ⋅ 𝐼 , where 𝛼 is the Basel capital ratio.

Hence, the capital cost 𝑐 is given by 𝑐 = 𝐶 ⋅ 𝐼, where 𝐶 ≡ 𝛼 ⋅ 𝑅𝑒. The inventory 𝐼 can

be expressed as

𝐼 ≡ ∑ 𝑅𝑊𝐴𝑖

𝑛

𝑖=1= ∑ 𝑅𝑊𝑖 ⋅ 𝐴𝑖

𝑛

𝑖=1,

where 𝑅𝑊𝐴𝑖 is the risk-weighted asset for bond 𝑖, 𝑅𝑊𝑖 is the risk weight of bond 𝑖, 𝑛

is the number of bonds, and 𝐴𝑖 is the cumulative amount of bond 𝑖 in the inventory.

𝐴𝑖 can be expressed in terms of orders:

𝐴𝑖 = ∑ 𝑏𝑗 ⋅ 𝑆𝑖𝑗

𝑚𝑖

𝑗=1

,

where 𝑆𝑖𝑗 is the size of trade 𝑗 for bond 𝑖, 𝑏𝑗 is equal to 1 if an incoming order is a sell

order and to -1 otherwise, and 𝑚𝑖 is the number of orders for bond 𝑖. The capital cost

𝑐 increases with the Basel capital ratio 𝛼.

(H) Following Amihud and Mendelson (1980), we assume that the demand and supply of

bonds are linear functions of bid and ask prices, 𝐷(𝑃𝑎) = 𝛾𝐷 − 𝛽𝐷 ⋅ 𝑃𝑎 and 𝑆(𝑃𝑏) =

𝛾𝑆 + 𝛽𝑆 ⋅ 𝑃𝑏. 𝛽𝐷 and 𝛽𝑆 represent the price elasticities of the buy and sell order arrival

rates, respectively. We assume that 𝛽𝐷 and 𝛽𝑆 take the same value 𝐸.

6 Varying 𝐿 is equivalent to varying the required Basel capital ratio while keeping capital 𝐾 constant.

11

The problem faced by the market maker is to maximize the expected profit per unit time (net

of the capital cost) by setting bid and ask prices as a function of inventory position, subject to

limits on inventory capacity. Since the expected profit per unit time is equal to the product of the

bid-ask spread and the arrival rates of orders, there is an optimal choice of bid and ask prices.

The objective function to be maximized by the market maker is

𝐺 = ∑ 𝜙𝑗(𝑞𝑗 − 𝑐𝑗),

𝐿

𝑗=−𝐿

(1)

where 𝜙𝑗 is the stationary probability of finding the process in state 𝑗, 𝑐𝑗 is the capital cost in

state 𝑗, and 𝑞𝑗 is the earning rate in state 𝑗 which is given by

𝑞𝑗 = 𝐷(𝑃𝑎(𝑗)) ⋅ 𝑃𝑎(𝑗) − 𝑆(𝑃𝑏(𝑗)) ⋅ 𝑃𝑏(𝑗). (2)

Equation (1) implicitly depends on the required Basel capital ratio 𝛼 because 𝐿 depends on 𝛼.

3.3 Calibrating the model

To fairly compare our simulated results to the empirical findings in the literature, we calibrate

the model parameters as closely as possible to empirical liquidity measures and trading activities

observed in TRACE.

Appendix A reports the values of the parameters used in the baseline model. We divide the

parameters into two groups. The first group consists of those parameters that are related to the

post-crisis regulations, including {𝐿, 𝐶, 𝐸} . We set inventory capacity 𝐿 in the pre-regulation

period to $10 million, which is about 10 times the average size of institutional trades in TRACE.7

Before the onset of the 2008 financial crisis, most dealer banks had only completed the

implementation of Basel I and had not completed the implementation of Basel II. It is commonly

7 The peak value of aggregate dealers’ inventory is about $300 billion in 2007. There are about 100,000 bonds in

TRACE. On average, dealers’ inventory per bond is about $3 million.

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believed that the capital requirement was not binding for trading corporate bonds before the

crisis. Therefore, the capital cost 𝐶 is set to 0% in the pre-regulation period. In the post-

regulation period, the capital cost 𝐶 is set to 1%, which is approximately equal to the product of

the Basel minimum capital ratio 𝛼 (8%) and required return on banks’ equity 𝑅𝑒 (12.8%).8 The

choice of 8% is conservative since Basel III may require a total capital requirement of up to 13%

if additional capital buffers are included.9 Setting 𝛼 to a higher value will strengthen our results.

Mizrach (2015) shows that the estimated bid-ask spread was in the range of 50 to 70 basis points

(bps) before the 2008 financial crisis. Hence, we set the price elasticity of order arrival rates 𝐸 to

150, which implies model bid-ask spreads around 60 bps.

The second group includes those parameters that determine arrival rates of buy and sell

orders and trade size. The average trading frequency of bonds in TRACE is about 200 times per

year. Therefore, we require 𝐷(𝑃𝑎) and 𝑆(𝑃𝑏) to be half of the average bond trading frequency in

TRACE (i.e., 100) by setting 𝛾𝐷 to 15,100 and 𝛾𝑆 to -14,900 when 𝑃𝑎 and 𝑃𝑏 are equal to the

bond face value. We choose the face value of bonds to be $100. The average trade size for

institutional trades in TRACE is about $1 million.10 Therefore, we set the unit of trade size 𝑆 to

be $1 million and consider the range of trade size to be $1 million to $5 million.

3.4 Monte Carlo simulations

Amihud and Mendelson (1980) provide a closed-form solution in the special case of linear

demand and supply functions. In our model, we include the features relevant to the post-crisis

regulations, which makes it difficult to find a closed-form solution for Equation (1). Therefore,

8 Using the data in the 2007 annual report of the Federal Financial Institutions Examination Council, the average

return on equity of national banks is about 12.8%. 9 Basel III also requires a mandatory “capital conservation buffer” of 2.5% of RWAs and a discretionary

countercyclical capital buffer of 0%~2.5% of RWAs. Therefore, Basel III may require a total capital requirement of

up to 10.5%~13% (8%+2.5%+0~2.5%) if these additional buffers are included. 10 Institutional trades are those with size > $100,000.

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we employ MC simulations which allow us to numerically solve for optimal bid and ask prices in

Equation (1). Furthermore, the MC simulations allow us to conveniently study trading activities

under a variety of market scenarios. In this subsection, we describe the basic settings of our MC

simulations. The technical details of MC simulations are provided in Appendix B.

In our model, the bid and ask prices are functions of inventory positions. We maximize the

objective function in Equation (1) by varying a vector of bid prices {𝑏−𝑚 , 𝑏−𝑚+1, 𝑏−𝑚+2, … 𝑏𝑚}

and a vector of ask prices {𝑎−𝑚 , 𝑎−𝑚+1, 𝑎−𝑚+2, … 𝑎𝑚} that are defined on a set of discretized

inventory positions evenly spaced between −𝐿 and 𝐿 , where 𝑚 is the number of discrete

inventory positions. We set 𝑚 equal to 5 in a trade-off between computational convenience and

accuracy.11 We use linear interpolation to calculate bid and ask prices for inventory positions

falling between the discretized grids. The optimal bid and ask prices maximize the dealer’s profit

in Equation (1). We are interested in the average bid-ask spread between the optimal bid and ask

prices:

𝐴𝑣𝑔 𝑏𝑖𝑑 − 𝑎𝑠𝑘 𝑠𝑝𝑟𝑒𝑎𝑑 =1

2𝑚 − 1∑ (𝑎𝑖 − 𝑏𝑖 )

𝑚−1

𝑖=−𝑚+1

, (3)

where we exclude the two end points in calculating the average bid-ask spread. We define the

price impact of a trade on the midpoint of bid and ask prices that moves the inventory level from

𝐿𝑖 to 𝐿𝑗 as follows:

𝑃𝑟𝑖𝑐𝑒 𝑖𝑚𝑝𝑎𝑐𝑡 = |

0.5(𝑎𝑖 + 𝑏𝑖) − 0.5(𝑎𝑗 + 𝑏𝑗)

𝐿𝑖 − 𝐿𝑗

|. (4)

The above price impact measures the price change as a result of a one-unit change in inventory

position. For a given set of model parameters, we report price impact averaged over all orders in

the MC simulations.

11 A larger number of 𝑚 increases the accuracy of the price functions but requires more computing time.

14

Without loss of generality, the size of each buy or sell order is a multiple of one unit of $1

million. The arrival times of buy or sell orders are simulated as a Poisson process, with intensity

specified by the demand and supply functions. For each MC simulation, we simulate arrival of

buy and sell orders from time 0 to time 1.12 An order is complete if the market maker’s inventory

is within the maximum capacity after the trade. An order with size 𝑆 is rejected if the market

maker’s inventory exceeds the maximum capacity after the trade (i.e., 𝑆 + 𝐼 > 𝐿 or −𝑆 + 𝐼 <

−𝐿). The purpose of this feature is to emulate the empirical observed pattern that block trades are

difficult to execute through dealers after the crisis. Trading volume can be easily obtained in the

MC simulations by aggregating the size of completed orders as follows:

𝑇𝑟𝑎𝑑𝑖𝑛𝑔 𝑣𝑜𝑙𝑢𝑚𝑒 =1

𝑁∑ ∑ 𝑆𝑖𝑡𝑗

𝑀𝑖

𝑗=1

,

𝑁

𝑖=1

(5)

where 𝑀𝑖 is the number of orders in the path 𝑖, 𝑁 is the number of MC paths, and 𝑆𝑖𝑡𝑗 is the size

of an order at time 𝑡𝑗 in the path 𝑖. We define average inventory position in the MC simulations

as follows:

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐼𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦 =1

𝑁∑ ∑ |𝐼𝑖𝑡𝑗

| ⋅ (𝑡𝑗+1 − 𝑡𝑗)

𝑀𝑖

𝑗=1

𝑁

𝑖=1

, (6)

where 𝐼𝑖𝑡𝑗 is the inventory level at time 𝑡𝑗 in the path 𝑖. To obtain a good convergence result, we

use 640,000 paths in the MC simulations.

4 A unified explanation of post-crisis regulations on liquidity

In this section, we first analyze the individual effects of the regulation-related parameters (i.e.,

inventory capacity, total capital cost, and price elasticity) on liquidity and market-making

12 The unit time in our simulation can be viewed as 1 year.

15

activities. Then, we examine the combined effects of the regulation-related parameters to provide

a unified explanation of the seemingly puzzling prior empirical findings.

4.1 Individual effects

4.1.1 Effects of inventory capacity

As a result of the heightened capital requirements in Basel III, market makers can hold less

trading assets for a given amount of capital. In this subsection, we investigate how inventory

capacity affects liquidity and market-making activities.

We vary the market maker’s maximum inventory capacity 𝐿 from $5 million to $10 million.

For each inventory capacity, we run the MC simulations to obtain optimal bid and ask prices by

maximizing the profit of the market maker per unit time in Equation (1). The size for both buy

and sell orders is fixed to be $1 million. As shown in Amihud and Mendelson (1980), the

optimal bid and ask prices for the market maker are a decreasing function of inventory position.

This is because when inventory position is close to the maximum limit, the market maker

decreases ask and bid prices to lower the arrival rate of sell orders and increase the arrival rate of

buy orders.

<Figure 2>

Panel A of Figure 2 shows the simulated optimal ask price as a function of inventory

position for different inventory capacities. Consistent with the prediction in Amihud and

Mendelson (1980), the ask price decreases with inventory position. More importantly, the slope

of the ask price is steeper when inventory capacity 𝐿 is smaller. Panel B of Figure 2 shows a

similar decreasing pattern for the optimal bid prices. As shown in Panel C, the bid-ask spreads

exhibit “U” shapes. The “U” shape is flatter when inventory capacity is larger. These results

16

suggest that a smaller inventory capacity leads to the market maker having steeper pricing

functions.

Having obtained the optimal bid and ask prices, we examine the impact of inventory

capacity on the liquidity of completed orders. The simulated liquidity measures are summarized

in Panel A of Table 1. Overall, inventory capacity affects all aspects of liquidity. In column (1),

trading volume decreases slightly when the inventory capacity shrinks. This is because the

market maker can accommodate fewer trades with a smaller inventory capacity. It is more

interesting to analyze the market maker’s inventory position since the most noticeable

observation in the corporate bond market after the crisis is the large decline in market makers’

inventory. In column (2), the average inventory position decreases significantly from $3.50

million to $1.98 million when inventory capacity shrinks from $10 million to $5 million. This is

consistent with the view that the post-crisis regulations reduce market makers’ inventory

capacity, which in turn reduces the inventory position of market makers. In column (3), the

dependence of the average bid-ask spread on inventory capacity appears to be marginal. When

the inventory capacity shrinks from $10 million to $5 million, the bid-ask spread only increases

slightly from $0.68 to $0.69. This is consistent with the prior empirical finding that the price-

based liquidity measures on corporate bonds after the 2008 financial crisis are actually not worse

than those before the crisis (e.g., Anderson and Stulz, 2017; Bessembinder et al., 2018; Trebbi

and Xiao, 2018). In column (4), the price impact increases significantly from 0.0103 to 0.0319

when the inventory capacity shrinks from $10 million to $5 million.

<Table 1>

17

4.1.2 Effects of total capital cost

In recent years, banks have recovered from the big loss in the 2008 financial crisis and held

adequate capital. Banks could allocate more capital for their market-making business. Therefore,

inventory capacity may not be the only reason explaining the deteriorating liquidity in the

corporate bond market. In this subsection, we investigate the impact of the capital cost of

inventory on market making.

<Figure 3>

The capital cost of inventory reduces the profit of market making, and thus market makers

are unwilling to hold large inventory positions. To examine how the capital cost affects market

making, we change the coefficient of the capital cost 𝐶 in assumption (G) in the model. We vary

𝐶 from 0% to 1% and run MC simulations to solve for optimal bid and ask prices. Panels A and

B of Figure 3 show the optimal ask and bid prices as a function of inventory position for

different levels of capital cost. Higher capital cost results in steeper pricing functions, indicating

that the market maker is unwilling to hold a large inventory when the capital cost is high. Panel

C of Figure 3 shows that the effect of capital cost on bid-ask spreads is not substantial.

Panel B of Table 1 presents simulated liquidity measures for different levels of capital cost.

Columns (1) and (3) indicate that the effects of capital cost on volume and average bid-ask

spread are marginal. However, the impact of capital cost on average inventory position is sizable.

Column (2) shows that average inventory position decreases from $3.5 million to $1.82 million

when the capital cost increases from 0% to 1%. This is expected because the market maker tends

to avoid a large inventory to reduce the capital cost. As shown in column (4), higher capital cost

also leads to a large increase in price impact as a result of the steeper pricing functions.

18

4.1.3 Effects of price elasticity

In the corporate bond market, market makers are not monopolistic price-setters. Competition

from non-bank liquidity providers or electronic trading platforms prevents the market maker

from charging monopolistic prices. The electronic trading of corporate bonds has been

increasingly adopted in recent years. In April 2012, BlackRock announced plans to launch a

crossing system called the Aladdin Trading Network. In June 2012, Goldman Sachs introduced

GSessions, another crossing system, to market. Several other systems are in operation. In the

United States, electronic trading volume as a percentage of all trading volume in U.S. IG

corporate bonds grew from 10% in 2011 to 14% in 2012.13 MarketAxess Holdings’s market

share of overall U.S. IG corporate bond trading hit 16.6% in June 2013.

The potential impact of electronic trading on market making is to reduce the bargaining

power of market makers (e.g., Hendershott and Madhavan, 2015). Thus, the price elasticity of

order arrival intensity will increase. As a result of the competition from electronic trading

platforms and possibly from non-bank dealers, bank-affiliated dealers may be forced to reduce

the bid-ask spreads and price impact. This helps to explain why price-based liquidity measures

indicate a better liquidity after the crisis than before the crisis.

In our model, we examine this competition effect on market making by changing the price

elasticity 𝐸 in the demand and supply functions. Greater competition leads to a higher price

elasticity, which implies that the market maker has less power to charge monopolistic prices.

<Figure 4>

We vary price elasticity 𝐸 from 150 to 187.5. This magnitude of price elasticity means that

the number of trades per year increases by 150 in response to a one-dollar decrease in price.

Panels A and B of Figure 4 show that price elasticity has significant effects on optimal ask and

13 “Corporate Bond E-Trading: Same Game, New Playing Field”, McKinsey & Company, 2013.

19

bid prices. High price elasticity reduces ask prices and increases bid prices. In addition, higher

price elasticity leads to flatter ask and bid pricing functions. Panel C of Figure 4 shows that the

bid-ask spread is lower when price elasticity is higher.

Panel C of Table 1 presents simulated liquidity measures for different levels of price

elasticity. Columns (1) and (2) suggest that price elasticity has no significant effect on volume

and average inventory position. However, price elasticity does have a considerable impact on the

bid-ask spread. As shown in column (3), when price elasticity increases from 150 to 187.5, the

bid-ask spread drops significantly from $0.68 to $0.54, indicating that when investors have

alternative ways to trade bonds, the market maker makes less profit per transaction. Column (4)

shows that price elasticity has a moderate effect on price impact.

<Table 3>

4.2 Unified explanation of previous empirical findings

In the prior subsections, we examined the individual effect of inventory capacity, capital cost,

and price elasticity on liquidity. In this subsection, we provide a unified explanation of previous

empirical findings in the corporate bond market by examining the combined effects of these

model parameters (L, C, and E) on liquidity under different scenarios that are relevant to the

post-crisis regulations.

4.2.1 Reduction in dealer inventory and bid-ask spread

We consider several scenarios in our model to examine liquidity measures in the pre-regulation

and post-regulation periods. Before the 2008 financial crisis, capital requirements were not

enforced for dealers in the corporate bond market. Furthermore, there was no strict risk

management. Dealers had a larger inventory capacity and faced less competition from other

liquidity providers. Therefore, for the pre-crisis period, we choose an inventory capacity of $10

20

million per bond, a capital cost of 0%, and a price elasticity of 150. In the post-regulation period,

dealers have a smaller inventory capacity, a higher capital requirement, and more competition

from other liquidity providers. To reduce uncertainty in estimating model parameters for the

post-regulation scenarios, we consider a set of scenarios by choosing an inventory capacity of $5

million or $6 million (i.e., a 40%~50% decrease in inventory capacity), a cost of capital of 1%,

and a price elasticity of 165 or 172.5 (i.e., a 10%~15% increase in price elasticity).14 These

choices of parameters try to capture the potential impact of the post-crisis regulations on dealers’

inventory. (See Appendix A for more details about the choice of parameters).

Table 2 shows the comparison of simulated liquidity measures between the pre-regulation

and post-regulation scenarios. In general, we find that liquidity deteriorates in the post-regulation

scenarios relative to the pre-regulation scenario. Trading volume decreases in the post-regulation

scenarios (column (1)). Consist with the observation that dealers’ aggregate inventory of

corporate bonds declines significantly after the crisis, column (2) shows that the average

inventory position is almost reduced by half in the post-regulation scenarios as a result of a

smaller inventory capacity and a higher cost of capital. Consistent with the finding in

Bessembinder et al. (2018) that the trade execution costs of corporate bonds are reduced after the

2008 financial crisis, column (3) shows that the average bid-ask spread is even smaller in the

post-regulation scenarios as a result of higher price elasticity (Trebbi and Xiao, 2018). The price

impact in the post-regulation scenarios is about two times larger than that in the pre-regulation

scenario, as shown in column (4).

In summary, these results shed light on the puzzling liquidity patterns which show that after

the 2008 financial crisis, dealers’ inventory is significantly reduced and the price impact during

times of stress is larger, whereas the bid-ask spread is even narrower.

14 In setting these parameters, we consider the aggregate effect of increased capital requirements in Basel III.

21

<Table 2>

4.2.2 Post-crisis liquidity during times of stress

In this subsection, we examine the impact of post-crisis regulations on market making in a

stressed trading environment. As documented in Anderson and Stulz (2017) and Bao et al.

(2018), the price impact is significantly larger after the crisis than before the crisis following

large bond sales. Motivated by this empirical evidence, we examine the effect of inventory

capacity on trading activities when the market maker holds a large positive inventory position

(i.e., 𝐼 close to 𝐿).

In our model, the price impact comes from the decreasing slope of the pricing function. As

the inventory increases, the absolute value of the slope also decreases to discourage sell orders

and encourage buy orders. Thus, the price impact will increase as the position moves to the upper

limit of the inventory. We report the average price impact at each inventory position in Table 3.

Each row represents the price impacts at different positions for a given inventory capacity. In

general, we observe that when the inventory position increases close to the inventory capacity,

the price impact increases significantly. In the pre-regulation scenario (Panel A), the price impact

is quite small until the inventory position is large (I=$8 million, $9 million). In the post-

regulation scenarios (Panel B), however, the price impact becomes large when the inventory

position increases just above $4 million. For example, in the second row, the price impact jumps

from 0.038 to 0.142 when the inventory position increases from $3 million to $4 million. This

result is consistent with the findings in Anderson and Stulz (2017) and Bao et al. (2018) that the

price impact is larger after the crisis, especially in times of stress.

<Table 3>

22

4.2.3 Execution of large trades

Another major concern regarding the post-crisis liquidity in the corporate bond market is that

large trades are difficult to execute (Anderson and Stulz, 2017). We investigate this issue in our

MC simulations by including orders of different sizes. We consider additional order sizes

ranging from $2 million to $5 million, and each order size accounts for 5% of the total number of

orders. Large orders are more likely to be rejected by the market maker because of the limit on

the inventory capacity. We count rejected orders as a percentage of total orders for different

order sizes. Table 4 presents the simulated results. It is clear that larger trades have considerably

higher probabilities of being rejected by the market maker (columns (3)-(5)). For example, when

the inventory capacity is $5 million, more than 30% of orders with a size of $5 million are

rejected by the market maker in the post-regulation scenarios and only about 11% of orders with

a size of $5 million are rejected in the pre-regulation scenarios. Column (1) shows that orders

with a size of $1 million have very low rejection rates in both the pre- and post-regulation

scenarios.

In summary, another main effect of reducing inventory capacity on market making is that it

is difficult for the market maker to intermediate large trades, and the average inventory position

of the market maker also becomes smaller. These results are consistent with the empirical

finding in Bessembinder et al. (2018) that bank-affiliated dealers reduce their capital

commitment after the 2008 financial crisis.

<Table 4>

23

5 New testable predictions and empirical results

Our analyses in the previous section provide a unified explanation for the previous empirical

findings in the corporate bond market. In this section, we first derive two model predictions of

the effects of the post-crisis regulations that have not examined before and then empirically test

these predictions using a large set of corporate bond data in TRACE.

5.1 Basel III capital requirements

Under Basel III, the minimum capital adequacy ratio that banks must maintain is 8% of RWAs.

The risk weight (RW) is divided into five risk levels: 20% for bonds with the highest investment

grades (IGs) (AAA to AA-), 50% for bonds with the second highest IGs (A+ to A-), 75% for

bonds with the third highest IGs (BBB+ to BBB-), 100% for high-yield bonds (BB+ to BB-), and

150% for bonds rated below BB-.15 Basel III capital rules imply that the capital charge increases

quickly as credit rating decreases. Bonds with higher risk weights consume more of the

inventory space of market makers. Therefore, we expect that Basel III has important implications

for bond liquidity.

5.2 Implications of Basel III capital requirements for bond liquidity

Our model allows us to examine the effects of the Basel III capital requirements on bond

liquidity. We first consider the effect of the Basel III capital requirements on trade size for bonds

with different risk weights. As noted in Section 5.1, riskier bonds are assigned larger weights

when calculating RWAs and thus take up more of the inventory space of dealers after the

regulations took effect. We consider risk weights in the range of 20% to 150% in the simulations.

The assumption (G) in Section 3.2 implies that a bond trade takes up an inventory space

15 For details, see “High-level summary of Basel III reforms”, Bank for International Settlements, 2017. In contrast,

Basel I applies a constant risk weight of 100% to all bonds with different credit ratings.

24

proportional to its risk weight. For example, a $1 million bond trade with a risk weight of 100%

takes up $1 million of inventory space, and a trade of the same size with a risk weight of 20%

takes up only $0.2 million of inventory space. We simulate the order rejection percentage for

bonds with different trade sizes and risk weights and present the results in Table 5. Columns (1)

to (5) clearly show that the simulated rejection percentage increases for bonds with higher risk

weights. Furthermore, the rejection percentage is significantly higher for larger size trades.

Therefore, the percentage of small trades for riskier bonds will increase after the regulations took

effect. We summarize these results in the following prediction:

Prediction 1: The percentage of small trades becomes larger for bonds with higher risk

weights according to the Basel III capital requirements.

<Table 5>

Second, we simulate the effect of the Basel III capital requirements on the price impact of

bonds with different risk weights. As shown in Panel A of Table 6, riskier bonds have a larger

price impact in the post-regulation scenarios.16 For instance, in the first post-regulation scenario

(the second row), the price impact for the bond with a risk weight of 150% is 0.0576, whereas

the price impact for the bond with a risk weight of 20% is 0.00611. We calculate the ratios of the

price impact of riskier bonds to those of safer bonds and present the results in Panel B of Table 6.

Overall, the calculated ratios in the post-regulation scenarios (columns (2)-(5)) are several times

those in the pre-regulation scenario (column (1)). For example, the ratio (RW=75%/RW=20%) is

1 in the pre-regulation scenario and increases to above 4 in the post-regulation scenarios. This

result leads to the following testable prediction:

Prediction 2: The ratio of the price impact of riskier bonds to that of less risky bonds is

16 In the pre-regulation scenario, we assume that a constant risk weight of 100% is applied to all bonds, which is

consistent with the Basel I capital requirements.

25

higher after the Basel III capital requirements took effect.

<Table 6>

5.3 Data and empirical results

In this section, we describe our data and empirically test the two predictions using transaction

data on corporate bonds in TRACE.

5.3.1 Data and sample description

We obtain corporate bond data from TRACE and rating history files from Mergent’s Fixed

Income Securities Database (FISD) for the period from January 1, 2004 to March 31, 2014.

We remove transactions with known errors in TRACE using the method in Dick-Nielsen

(2009). To exclude transactions with a retail size, we require a trading volume greater than 1000.

We also require transaction prices between 50 and 150 to exclude observations with extreme

prices. We merge TRACE transactions with a FISD bond rating by CUSIP. After applying these

filters, we obtain a sample of 61,790,231 transactions with 89,193 unique bonds.

Table 7 presents the descriptive statistics of our sample. As shown by the number of

transactions, most of the transactions in TRACE are concentrated in investment grade and high-

yield bonds. A-rated and BBB-rated bonds have the largest trading volume. Average trading size

ranges from 141,708 to 479,962.

<Table 7>

Following Bessembinder et al. (2018), we divide the sample period into four periods: pre-

crisis, crisis, post-crisis, and regulatory. The pre-crisis period is from January 1, 2006 to June 30,

2007; the crisis period is from July 1, 2007 to April 30, 2009; the post-crisis period is from May

1, 2009 to June 30, 2012; and the regulatory period is from July 1, 2012 to March 31, 2014.

26

Since the Volcker Rule became effective on April 1, 2014, we choose the end date of the

regulatory period to be March 31, 2014 to isolate the impact of Basel III capital requirements.

5.3.2 Distribution of trade size

We begin our investigation with the distribution of trade size. As noted in sections 4.2.2 and

5.2, our model predicts that larger trades have higher rejection rates by the market maker in the

post-regulation scenarios. More importantly, our model also predicts that large and risky trades

are more likely to be rejected. To test these predictions,

We first calculate the distribution of trade size for all bonds over the four periods. We

classify trade size in par value into three categories: small (less than $100,000), median (between

$100,000 and $1,000,000), and large (greater than $1,000,000). Panel A of Table 8 presents the

number of trades in each size category as a percentage of the total number of trades over the four

periods. The percentage of the large trade category declines from 16.16% in the pre-crisis period

to 11.16% during the crisis and to 10.48% in the regulatory period. The percentage of the small

(median) trade group increases from 71.71% (12.12%) in the pre-crisis period to 74.85%

(14.67%) in the regulatory period. Consistent with our simulated results in Table 4 which show

that large trades are more likely to be rejected by the market maker in the post-regulation

scenarios, these results suggest that it is more difficult to execute block trades after the

regulations took effect.

Panel B of Table 8 presents the trading volume in each size category in par value as a

percentage of the total trading volume over the four periods. Consistent with the results in Panel

A, Panel B shows that the percentage of the large trade category declines from 86.17% in the

pre-crisis period to 77.34% in the regulatory period, whereas the percentage of the small (median)

27

trade group increases from 4.12% (9.71%) in the pre-crisis period to 6.30% (16.40%) in the

regulatory period.

<Table 8>

We next summarize the number of trades in each size category as a percentage of the total

number of trades by different risk weights. The results are reported in Table 9. The most striking

result, presented in Panel A, is that for bonds with the lowest risk weight (20%), the percentage

of the large trade category is 14.64% in the regulatory period, higher than that (8.50%) in the

pre-crisis period. In contrast to the result in Panel A, panels B to E show that the percentage of

the large trade category for bonds with higher risk weights decreases significantly in the

regulatory period compared with the pre-crisis period. For example, in Panel C, the percentage of

the large trade category for bonds with a risk weight of 75% decreases from 25.36% to 9.37%.

These results suggest that the post-crisis regulations have a differential effect on trade sizes and

large trades concentrated in highly rated bonds.

<Table 9>

Table 10 reports trading volume in each size category in par value as a percentage of the

total trading volume by different risk weights. Overall, the results in Table 10 are consistent with

those in Table 9. Panel A shows that the percentage of the large trade category (82.73%) in the

regulatory period is roughly equal to that (83.56%) in the pre-crisis period. As can be seen in

panels B to E, the percentage of the large trade category declines significantly in the regulatory

period relative to the pre-crisis period.

<Table 10>

28

5.3.3 Price impact of higher-rated bonds and lower-rated bonds

We examine whether the ratio of the price impact of lower-rated bonds to that of higher-rated

bonds increases in the regulatory period. For each bond in one of the four periods, we calculate

the price impact using the method in Amihud (2002) as follows:

𝐴𝑚𝑖ℎ𝑢𝑑 =1

𝑁𝑡∑

|𝑝𝑘 − 𝑝𝑘−1

𝑝𝑘−1|

𝑆𝑖𝑧𝑒𝑘

𝑁𝑡

𝑘=1

, (7)

where 𝑁𝑡 is the number of trades in the period 𝑡, 𝑝𝑘 is the price of the 𝑘th trade, and 𝑆𝑖𝑧𝑒𝑘 is the

size of the 𝑘th trade (in millions of U.S. dollars). Then, we aggregate the price impact measures

across bonds for the five rating categories and over the four periods. Following Bessembinder,

Kahle, Maxwell, and Xu (2009) and Bao et al. (2018), we exclude trades with $100,000 or less in

par amount to circumvent the noise from small trades in calculating the Amihud (2002) price

impact.17

The aggregated price impact measures are presented in Panel A of Table 11. We also report

the ratios of the price impact of bonds with high risk weights to the price impact of bonds with

the lowest risk weight in Panel B of Table 11. Overall, the price impact for all levels of risk

weight (R1 to R5) increases significantly during the crisis period and drops to a level even lower

than its pre-crisis level after the crisis. For instance, the price impact for bonds with a risk weight

of 20% (R1) is 0.048 in the pre-crisis period, increases to 0.096 in the crisis period, and

decreases to 0.014 in the regulatory period. This result is consistent with the findings in

Bessembinder et al. (2018) and Anderson and Stulz (2017). More importantly, Panel B of Table

11 shows that the ratios of price impact increase significantly in the regulatory period compared

to the pre-crisis period. For example, the ratio of the price impact with a risk weight of 50% to

17 Our results still hold using all trades, including those with $100,000 or less in par amount.

29

that with a risk weight of 20% (R2/R1) is 0.779 in the pre-crisis period and increases to 2.411 in

the regulatory period. Even in the crisis and post-crisis periods, this ratio is only close to 1.

Consistent with our model prediction in Section 5.2, these results suggest that the capital

requirements in Basel III have a differential effect on liquidity for bonds with different credit

ratings. Bonds with lower ratings experience a large increase in price impact after the post-crisis

regulations took effect.

<Table 11>

6 Conclusion

On the basis of a simple inventory model that extends the model in Amihud and Mendelson

(1980) with features related to post-crisis regulations, this paper explains the following puzzling

liquidity patterns in the corporate bond market after the 2008 financial crisis: (1) improved price-

based liquidity measures, such as the bid-ask spread; (2) a large reduction in dealers’ inventory

of corporate bonds and a reduction in dealers’ capital commitment; (3) increased price impact in

periods of market stress; and (4) increased difficulty in the execution of large trades.

Our theoretical analysis of the effects of the post-crisis regulations on bond liquidity

suggests that the capital requirements in Basel III have important implications for market making

by banks. Together, reduced inventory capacity, higher cost of capital, and more elastic prices

are likely to cause a large reduction in dealers’ inventory in the corporate bond market.

Furthermore, block size trades are difficult for market makers to intermediate, and price impact

is significantly higher in periods of market stress. Our model generates the link between the

Basel III capital requirements and the liquidity of bonds with different risk weights.

30

Using the corporate bond transaction data in TRACE, we empirically examine the effects of

the capital requirements in Basel III on bond liquidity. We find that the percentage of large

trades declines after the post-crisis regulations took effect. More importantly, the percentage of

large trades declines more for riskier bonds. The capital requirements also have a differential

effect on liquidity across bonds with different levels of risk. Specifically, the ratio of the price

impact of bonds with higher risk weights to that of bonds with the lowest risk weight increases

significantly in the regulatory period. These empirical findings are consistent with the

predictions of our inventory model.

The purpose of the post-crisis regulations is to ensure the stability of the financial system.

However, an unintended consequence is that the more stringent capital requirements effectively

redistribute the liquidity for bonds with different credit ratings (i.e., higher liquidity for higher-

rated bonds and lower liquidity for lower-rated bonds). Therefore, while the capital requirements

make banks safer, they leave the high-yield sector more vulnerable to potential liquidity shocks.

31

Appendix A: Model parameters

This table presents the calibrated parameter values of our inventory model.

Description Notation Value Explanations

Regulation-related parameters

Inventory capacity 𝐿 $5m~10m In the pre-crisis period, 𝐿 is set to $10m per bond; in

the regulatory period, 𝐿 is set to $5m~10m per bond.

Capital cost 𝐶 0%~1.0% 𝐶 ≡ 𝛼 ⋅ 𝑅𝑒

Basel capital ratio 𝛼 0%~8%

In the pre-crisis period, 𝛼 is set to 0; in the regulatory

period, 𝛼 is set to 8% according to bank capital

requirement defined in the Basel Accord.

Return on equity 𝑅𝑒 12.8% 𝑅𝑒 is fixed and estimated using data in Federal

Financial Institutions Examination Council Annual

Report 2007.

Price elasticity of

buy or sell orders 𝐸 150~187.5

In the pre-crisis period, 𝐸 is set to 150, which is calibrated to match the pre-crisis bond bid-ask spreads

in TRACE; in the regulatory period, 𝐸 is set to

150~187.5.

Risk weight of

bond RW 1/5~1

In the post-regulation period, we divide the risk

weights of bonds into five levels (20%, 40%, 60%,

80%, and 100%).

Other calibrated parameters

Parameter in the

arrival rate of buy

orders 𝛾𝐷 15,100

𝛾𝐷 is calibrated so that 𝐷(𝑃𝑎) = 𝛾𝐷 − 𝛽𝐷 ⋅ 𝑃𝑎 =100,

which is approximately half of the average number of

trades per liquid bond per year in TRACE, where 𝑃𝑎 is

set to 𝐹 and 𝛽𝐷 is set to 𝐸.

Parameter in the

arrival rate of sell

orders 𝛾𝑆 -14,900

𝛾𝑆 is calibrated so that 𝑆(𝑃𝑏) = 𝛾𝑆 + 𝛽𝑆 ⋅ 𝑃𝑏 =100,

which is approximately a half of the average number of

trades per liquid bond per year in TRACE, where 𝑃𝑎 is

set to 𝐹 and 𝛽𝐷 is set to 𝐸.

Trade size S $1m~5m The average trade size for institutional trades in TRACE is about $1m. Thus, the trade size is assumed

to be $1m~5m.

Bond face value 𝐹 $100 A common choice of bond face values.

32

Appendix B: Monte Carlo simulations

Each MC simulation consists of two main steps. The first step is to generate the arrival time 𝑡 of

buy or sell orders. The arrival time 𝑡 follows a Poisson process with an arrival rate 𝜆, where 𝜆 is

the arrival rate of buy orders 𝐷(𝑃𝑎) or sell orders 𝑆(𝑃𝑏). It is well known that 𝑡 is exponentially

distributed. The probability density function (pdf) of an exponential distribution is

𝑓(𝑡; 𝜆) = {𝜆𝑒−𝜆𝑡 𝑡 ≥ 0,0 𝑡 < 0.

The exponential distribution is generated using the Mersenne Twister generator developed in

Matsumoto and Nishimura (1998). This generator is the most widely used general-purpose

pseudorandom number generator (PRNG). It was the first PRNG to provide fast generation of

high-quality pseudorandom integers. This study employs the most commonly used version,

which is based on the Mersenne prime 219937−1.

The second step is to maximize the market maker’s profit in Equation (1). The maximization

procedure is done using a function “max_using_approximate_derivatives” in Dlib C++ Library.18

This function performs unconstrained minimization of a nonlinear function and numerically

approximates the gradient. To reduce MC noise in each step of maximization, the same random

seed is used.

To achieve high performance, the MC simulation code is written in C++. The source code is

compiled using GCC 5.4.0. It takes about 8.8 seconds to simulate 640,000 paths on a server with

Intel Xeon Gold 5118 CPU. The maximization procedure takes about 1000 steps, and the total

computational time is around 8,800 seconds.

18 The source code and documents of Dlib are available at http://dlib.net.

33

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36

Table 1: Individual effects of inventory capacity, capital cost and price elasticity on

liquidity

Panel A reports simulated liquidity measures for different levels of inventory capacity. Inventory

capacity, defined in Section 3.1, ranges from $5 million to $10 million. Price elasticity 𝐸 is 150,

and the cost of capital 𝐶 is 0. Panel B reports simulated liquidity measures for different capital

costs. Capital cost, defined in Section 3.2, ranges from 0% to 1%. Price elasticity 𝐸 is 150, and

inventory capacity 𝐿 is 10. Panel C reports simulated liquidity measures for different price

elasticities. Elasticity, defined in Section 3.1, ranges from 150 to 187.5. Inventory capacity 𝐿 is

10, and the cost of capital 𝐶 is 0. Volume is the average number of orders accepted by the market

maker per unit time, defined in Equation (5). Inventory is the absolute size of the inventory (long

or short) averaged over time, defined in Equation (6). Bid-ask spread is the average bid-ask

spread for the optimal inventory, defined in Equation (3). Price impact is the average absolute

difference in the midpoint of bid and ask prices between two inventory levels, defined in

Equation (4). 640,000 paths are used for the MC simulation and trade size is fixed at 1 in all

panels.

Simulated liquidity measures

(1) (2) (3) (4) Volume ($m) Inventory ($m) Bid-ask spread ($) Price impact

Panel A: Inventory capacity (L)

5 95.48 1.98 0.69 0.0319

6 96.48 2.28 0.68 0.0239

7 97.66 2.57 0.68 0.0189

8 98.15 2.81 0.67 0.0156

9 97.71 3.16 0.68 0.0126

10 97.66 3.50 0.68 0.0103

Panel B: Capital cost (C)

0.0% 97.66 3.50 0.68 0.0103

0.2% 97.31 2.81 0.68 0.0141

0.4% 97.46 2.57 0.68 0.0168

0.6% 97.39 2.31 0.68 0.0197

0.8% 97.13 2.16 0.68 0.0228

1.0% 96.24 1.82 0.69 0.0328

Panel C: Elasticity (E)

150.0 97.66 3.50 0.68 0.0103

157.5 98.85 3.44 0.64 0.0103

165.0 98.37 3.28 0.61 0.0102

172.5 98.44 3.49 0.58 0.0093

180.0 98.14 3.47 0.56 0.0088

187.5 98.15 3.50 0.54 0.0088

37

Table 2: Combined effects of the post-crisis regulations on liquidity measures

This table reports the combined effects of inventory capacity, total capital cost, and price

elasticity on liquidity and trading activities. Inventory capacity is the maximum long or short

position the market maker can take. Volume is the average number of orders accepted by the

market maker per unit time, defined in Equation (5). Average inventory is the absolute size of the

inventory (long or short) averaged over time, defined in Equation (6). Avg bid-ask spread is the

average bid-ask spread for the optimal inventory, defined in Equation (3). Avg price impact is the

average absolute difference in the midpoint of bid and ask prices between two inventory levels,

defined in Equation (4). Panel A shows simulated liquidity measures in the pre-regulation

scenario, and Panel B shows simulated liquidity measures for the post-regulation scenarios. The

model parameters are L (inventory capacity), C (capital cost), and E (elasticity). 640,000 paths

are used for the MC simulation.

Simulated liquidity measures

(1) (2) (3) (4)

Scenarios Volume ($m) Inventory ($m) Bid-ask spread ($) Price impact

Panel A: Pre-regulation

L=10, C=0%, E=150 97.66 3.50 0.68 0.0103

Panel B: Post-regulation

L=5, C=1%, E=165 94.70 1.67 0.64 0.0346

L=5, C=1%, E=172.5 95.12 1.70 0.61 0.0324

L=6, C=1%, E=165 94.65 1.81 0.63 0.0282

L=6, C=1%, E=172.5 96.96 1.77 0.60 0.0293

38

Table 3: Combined effects of the post-crisis regulations on price impact in periods of stress

This table reports the simulated price impact for different inventory capacities. For each

inventory capacity, we report the price impact for each possible inventory position. Inventory

position is reported in each column (in million USD), and inventory capacity is reported in each

row. Price impact is defined in Equation (4). 640,000 paths are used for the MC simulation.

Simulated price impact at different inventory levels

Scenarios I=1 I=2 I=3 I=4 I=5 I=6 I=7 I=8 I=9 I=10

Panel A: Pre-regulation

L=10, C=0%, E=150 0.005 0.005 0.009 0.009 0.008 0.008 0.015 0.015 0.061 0.061

Panel B: Post-regulation

L=5, C=1%, E=165 0.034 0.024 0.029 0.038 0.142 L=5, C=1%, E=172.5 0.030 0.025 0.024 0.036 0.134 L=6, C=1%, E=165 0.025 0.037 0.016 0.000 0.067 0.091 L=6, C=1%, E=172.5 0.014 0.046 0.023 0.002 0.070 0.051

39

Table 4: Combined effects of the post-crisis regulations on the percentage of rejected

orders for trades of different sizes

This table reports the percentages of incoming orders that are rejected by the market maker in the

pre- and post-regulation scenarios. Trade size ranges from $1 million to $5 million. 640,000

paths are used for the MC simulation.

Simulated percentage of rejected orders

(1) (2) (3) (4) (5)

Scenarios Size=$1m Size=$2m Size=$3m Size=$4m Size=$5m

Pre-regulation L=10, C=0%, E=150 0.03% 1.15% 3.88% 7.35% 11.51%

Post-regulation L=5, C=1%, E=165 0.06% 3.47% 9.94% 19.36% 31.16%

L=5, C=1%, E=172.5 0.06% 3.61% 10.37% 19.72% 31.57%

L=6, C=1%, E=165 0.04% 2.02% 6.44% 12.87% 21.38%

L=6, C=1%, E=172.5 0.03% 1.63% 5.60% 11.82% 20.59%

40

Table 5: Simulated average rejection rate in the post-regulation scenarios with different

risk weights

This table reports the percentage of rejected incoming orders with different risk weights (in rows)

and different trade sizes (in columns). The rejection rate is averaged over the four post-regulation

scenarios. Trade size ranges from $1 million to $5 million, and risk weight ranges from 20% to

150%. 640,000 paths are used for the MC simulation.

Simulated average percentage of rejected orders

(1) (2) (3) (4) (5)

Size=$1m Size=$2m Size=$3m Size=$4m Size=$5m

RW=20% 0.00% 0.00% 0.01% 0.03% 0.07%

RW=50% 0.01% 0.24% 1.07% 2.52% 4.75%

RW=75% 0.33% 2.08% 5.53% 10.67% 17.29%

RW=100% 0.05% 2.68% 8.09% 15.94% 26.17%

RW=150% 1.02% 8.43% 21.07% 38.56% 58.13%

41

Table 6: Simulated ratios of the price impact of bonds with different risk weights

This table reports the ratios of the simulated price impact of bonds with different risk weights in

the pre- and post-regulation scenarios. Price impact is the average absolute difference in the

midpoint of bid and ask prices between two inventory levels, defined in Equation (4). 640,000

paths are used for the MC simulation. Panel A shows simulated average price impact for bonds

with different risk weights. Panel B shows the ratios of price impact for bonds with different risk

weights.

Panel A: Price impact for assets with different levels of risk Simulated price impact

(1) (2) (3) (4) (5)

Scenarios RW=20% RW=50% RW=75% RW=100% RW=150%

Pre-regulation

L=10, C=0%, E=150 (Pre1) 0.0104 0.0104 0.0104 0.0104 0.0104

Post-regulation

L=5, C=1%, E=165 (Post1) 0.00611 0.0174 0.0263 0.0396 0.0576

L=5, C=1%, E=172.5 (Post2) 0.00563 0.0162 0.0245 0.0371 0.0538

L=6, C=1%, E=165 (Post3) 0.00543 0.0143 0.0225 0.0310 0.0488

L=6, C=1%, E=172.5 (Post4) 0.00554 0.0150 0.0237 0.0327 0.0511

Panel B: Ratio of price impacts Pre-regulation Post-regulation

(1) (2) (3) (4) (5)

Scenarios Pre1 Post1 Post2 Post3 Post4

RW=50% / RW=20% 1.0 2.85 2.88 2.63 2.71

RW=75% / RW=20% 1.0 4.30 4.35 4.14 4.28

RW=100% / RW=20% 1.0 6.49 6.58 5.71 5.90

RW=150% / RW=20% 1.0 9.42 9.54 8.99 9.23

42

Table 7: Descriptive statistics of bond transactions in TRACE

This table presents the descriptive statistics for our bond sample across different credit ratings.

The sample period is from January 1, 2006 to March 31, 2014. Trading size is expressed in USD.

Bond transaction data are obtained from TRACE.

Rating No. of bonds No. of transactions Avg. size

AAA 3,304 1,916,279 340,486

AA+ 2,289 869,088 479,962

AA 6,194 3,545,340 253,051

AA- 9,624 3,771,934 336,766

A+ 8,882 5,358,511 308,996

A 10,862 7,663,887 303,703

A- 7,869 6,302,427 316,427

BBB+ 7,404 5,532,054 370,631

BBB 7,090 6,584,862 388,640

BBB- 4,276 5,674,536 310,581

BB+ 3,243 2,531,881 207,662

BB 2,853 1,619,359 215,846

BB- 2,564 1,914,598 227,375

B+ 2,951 1,914,128 229,714

B 2,445 1,426,995 249,045

B- 2,781 2,255,929 223,806

CCC+ 1,770 1,410,262 270,380

CCC 719 637,968 277,070

CCC- 372 292,736 329,859

CC 1,102 480,306 181,025

C 599 87,151 141,708

43

Table 8: Distribution of trade size in TRACE

This table presents the trade size distribution of all corporate bonds in TRACE for different

periods. The distribution is based on the number of trades. The pre-crisis period is from January

1, 2006 to June 30, 2007; the crisis period is from July 1, 2007 to April 30, 2009; the post-crisis

period is from May 1, 2009 to June 30, 2012; and the regulatory period is from July 1, 2012 to

March 31, 2014. Panel A shows the distribution based on the number of trades. Panel B shows

the distribution based on trading volume in par value.

Panel A: By number of trades

Trade size Pre-crisis Crisis Post-crisis Regulatory

<= $100K 71.71% 78.50% 78.34% 74.85%

> $100K & <= $1M 12.12% 10.34% 11.97% 14.67%

> $1M 16.16% 11.16% 9.69% 10.48%

Panel B: By trading volume


<= 100K 4.12% 6.22% 6.93% 6.30%

> 100K & <= 1M 9.71% 11.33% 14.54% 16.40%

> 1M 86.17% 82.45% 78.53% 77.30%

44

Table 9: Distribution of trade size by risk weights based on the number of trades

This table presents the trade size distribution of corporate bonds with different risk weights over

different periods. The distribution is based on the number of trades in TRACE. The pre-crisis

period is from January 1, 2006 to June 30, 2007; the crisis period is from July 1, 2007 to April 30,

2009; the post-crisis period is from May 1, 2009 to June 30, 2012; and the regulatory period is

from July 1, 2012 to March 31, 2014. Panel A shows the distribution of trades with a credit

rating of AAA to AA-. Panel B shows the distribution of trades with a credit rating of A+ to A-.

Panel C shows the distribution of trades with a credit rating of BBB+ to BBB-. Panel D shows

the distribution of trades with a credit rating of BB+ to BB-. Panel E shows the distribution of

trades with a credit rating of B+ to C.

Panel A: Trades with a rating of AAA to AA- (RW=20%)


<= 100K 80.95% 83.73% 78.59% 66.47%

> 100K & <= 1M 10.55% 9.49% 12.26% 18.89%

> 1M 8.50% 6.77% 9.16% 14.64%

Panel B: Trades with a rating of A+ to A- (RW=50%)


<= 100K 76.22% 82.37% 80.28% 75.46%

> 100K & <= 1M 11.65% 9.84% 11.89% 14.63%

> 1M 12.13% 7.79% 7.82% 9.90%

Panel C: Trades with a rating of BBB+ to BBB- (RW=75%)


<= 100K 60.33% 77.96% 78.86% 76.17%

> 100K & <= 1M 14.31% 10.45% 12.11% 14.45%

> 1M 25.36% 11.59% 9.04% 9.37%

Panel D: Trades with a rating of BB+ to BB- (RW=100%)


<= 100K 76.04% 66.34% 76.16% 74.42%

> 100K & <= 1M 9.78% 12.22% 12.15% 14.97%

> 1M 14.19% 21.44% 11.69% 10.62%

Panel E: Trades with a rating of B+ to C (RW=150%)


<= 100K 55.26% 59.67% 73.89% 73.63%

> 100K & <= 1M 15.79% 13.00% 11.42% 13.44%

> 1M 28.94% 27.33% 14.69% 12.93%

45

Table 10: Distribution of trade size by risk weights based on trading volume

This table presents the trade size distribution of corporate bonds with different risk weights over

different periods. The distribution is based on trading volume in par value in TRACE. The pre-

crisis period is from January 1, 2006 to June 30, 2007; the crisis period is from July 1, 2007 to

April 30, 2009; the post-crisis period is from May 1, 2009 to June 30, 2012; and the regulatory

period is from July 1, 2012 to March 31, 2014. Panel A shows the distribution of trades with a

credit rating of AAA to AA-. Panel B shows the distribution of trades with a credit rating of A+

to A-. Panel C shows the distribution of trades with a credit rating of BBB+ to BBB-. Panel D

shows the distribution of trades with a credit rating of BB+ to BB-. Panel E shows the

distribution of trades with a credit rating of B+ to C.

Panel A: Trades with a rating of AAA to AA- (RW=20%)


<= 100K 6.15% 8.27% 5.96% 3.66%

> 100K & <= 1M 10.29% 11.85% 12.46% 13.61%

> 1M 83.56% 79.88% 81.58% 82.73%

Panel B: Trades with a rating of A+ to A- (RW=50%)


<= 100K 4.10% 7.08% 7.12% 5.38%

> 100K & <= 1M 8.70% 11.32% 14.15% 14.64%

> 1M 87.20% 81.60% 78.73% 79.98%

Panel C: Trades with a rating of BBB+ to BBB- (RW=75%)


<= 100K 1.62% 4.50% 6.30% 6.31%

> 100K & <= 1M 5.75% 8.90% 13.26% 14.94%

> 1M 92.63% 86.60% 80.44% 78.75%

Panel D: Trades with a rating of BB+ to BB- (RW=100%)


<= 100K 8.65% 4.91% 9.05% 10.49%

> 100K & <= 1M 15.63% 14.55% 19.89% 26.95%

> 1M 75.72% 80.54% 71.07% 62.56%

Panel E: Trades with a rating of B+ to C (RW=150%)


<= 100K 4.07% 4.40% 8.46% 9.11%

> 100K & <= 1M 15.98% 14.38% 19.62% 24.52%

> 1M 79.95% 81.22% 71.92% 66.37%

46

Table 11: Price impact ratios for bonds with different risk weights

This table reports the price impact measures for corporate bonds with different risk weights over

different periods. Following Amihud (2002), we calculate the price impact measure defined in

Equation (7). Bond prices and trading volumes are obtained from TRACE. The pre-crisis period

is from January 1, 2006 to June 30, 2007; the crisis period is from July 1, 2007 to April 30, 2009;

the post-crisis period is from May 1, 2009 to June 30, 2012; and the regulatory period is from

July 1, 2012 to March 31, 2014. Following Bessembinder et al. (2009) and Bao et al. (2018), we

exclude trades with $100,000 or less in par amount in calculating the Amihud (2002) measure.

Panel A reports the price impact measures for bonds with different risk weights (from 20% to

150%) over the four periods. Panel B reports ratios of price impact of bonds with higher risk

weights to price impact of bonds with lower risk weights. RW=50% / RW=20% is the ratio of the

price impact for bonds with a risk weight of 50% to that for bonds with a risk weight of 20%.

Other ratios are similarly defined.

(1) (2) (3) (4) Pre-crisis Crisis Post-crisis Regulatory

Panel A: Price impact for different ratings

RW=20% (AAA to AA-) 0.048 0.096 0.044 0.014

RW=50% (A+ to A-) 0.038 0.106 0.053 0.034

RW=75% (BBB+ to BBB-) 0.017 0.082 0.043 0.029

RW=100% (BB+ to BB-) 0.044 0.075 0.049 0.028

RW=150% (B+ to C) 0.033 0.111 0.081 0.038

Panel B: Price impact ratios

RW=50% / RW=20% 0.779 1.102 1.199 2.411

RW=75% / RW=20% 0.351 0.848 0.972 2.057

RW=100% / RW=20% 0.913 0.781 1.100 1.941

RW=150% / RW=20% 0.684 1.155 1.819 2.697

47

Figure 1: Aggregate dealer inventory

This figure shows primary dealers’ aggregate net position of corporate securities from 2001 to

2015. The data are obtained from the Federal Reserve Bank of New York. The net position is in

billion USD.

0

50

100

150

200

250

300

2001 2003 2005 2007 2009 2011 2013 2015

48

Figure 2: Dependence of optimal bid and ask prices on inventory size

This figure shows the optimal bid and ask prices as a function of inventory for different

inventory sizes. Panel A shows the optimal ask prices, Panel B shows the optimal bid prices, and

Panel C shows the optimal bid-ask spreads. The lines in the plots represent different inventory

capacities. The horizontal axis shows the inventory level. In panels A, B, and C, the vertical axis

shows the ask price, the bid price, and the bid-ask spread, respectively.

Panel A: Optimal ask prices

Panel B: Optimal bid prices

100.1

100.2

100.3

100.4

100.5

100.6

100.7

-10 -5 0 5 10

Ask

pri

ce

Inventory position

5678910

99.3

99.4

99.5

99.6

99.7

99.8

99.9

-10 -5 0 5 10

Bid

pri

ce

Inventory position

5678910

49

Panel C: Optimal bid-ask spreads

0.6

0.7

0.8

0.9

1

-10 -5 0 5 10

Bid

-ask

sp

read

Inventory position

5678910

50

Figure 3: Dependence of optimal bid and ask prices on the capital cost

This figure shows the bid and ask prices as a function of inventory for different levels of capital

cost. The size of inventory is fixed at 5. Panel A shows the optimal ask prices, Panel B shows the

optimal bid prices, and Panel C shows the optimal bid-ask spreads. The lines in the plot represent

different capital costs. The horizontal axis shows the inventory level. In panels, A, B, and C, the

vertical axis shows the ask price, the bid price, and the bid-ask spread, respectively.

Panel A: Optimal ask prices

Panel B: Optimal bid prices

100.0

100.1

100.2

100.3

100.4

100.5

100.6

100.7

-10 -5 0 5 10

Ask

pri

ce

Inventory position

0.00%0.20%0.40%0.60%0.80%1.00%

99.3

99.4

99.5

99.6

99.7

99.8

99.9

100

-10 -5 0 5 10

Bid

pri

ce

Inventory position

0.00%0.20%0.40%0.60%0.80%1.00%

51

Panel C: Optimal bid-ask spreads

0.6

0.7

0.8

0.9

1

-10 -5 0 5 10

Bid

-ask

sp

read

Inventory position

0.00%0.20%0.40%0.60%0.80%1.00%

52

Figure 4: Dependence of optimal bid and ask prices on price elasticity

This figure shows the bid and ask prices as a function of inventory for different price elasticities.

The size of inventory is fixed at 5. Panel A shows the optimal ask prices, Panel B shows the

optimal bid prices, and Panel C shows the optimal bid-ask spreads. The lines in the plot represent

different price elasticities. The horizontal axis shows the inventory level. In panels A, B, and C,

the vertical axis shows the ask price, the bid price, and the bid-ask spread, respectively.

Panel A: optimal ask prices

Panel B: optimal bid prices

100.1

100.2

100.3

100.4

100.5

100.6

100.7

-10 -5 0 5 10

Ask

pri

ce

Inventory position

150.0157.5165.0172.5180.0187.5

99.3

99.4

99.5

99.6

99.7

99.8

99.9

-10 -5 0 5 10

Bid

pri

ce

Inventory position

150.0157.5165.0172.5180.0187.5

53

Panel C: optimal bid-ask spreads

0.5

0.6

0.7

0.8

0.9

1

-10 -5 0 5 10

Bid

-ask

sp

read

Inventory position

150.0157.5165.0172.5180.0187.5

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Post-Crisis Regulations, Market Making, and Liquidity in the … · 2019. 9. 27. · Post-Crisis Regulations, Market Making, and Liquidity in the Corporate Bond *Market Xinjie Wang