Coarse Pricing in QE Auctions*
Yusuke Tsujimoto †September 13, 2023

Abstract
This paper documents and studies the practice of coarse pricing by highly sophisticatedinstitutions in an important market. Although the Fed explicitly sets the tick size of 1/256th
in its QE-driven reverse auctions to purchase Treasury securities, the offer prices of primary
dealers—the only direct participants—exhibit strong clustering on coarser grids. This coarse
pricing has decreased over time but surged temporarily during March 2020. It grows when
the offer-to-cover ratio is low, indicating a competitive constraint. Importantly, primary dealers with larger market shares price more finely, and laggard dealers employ coarse pricing
especially when precise pricing is difficult. With regard to the Fed’s purchase costs, coarsely
priced offers have higher prices, conditional on winning the auction. This difference mightreflect lower auction-winning chances of coarsely priced offers. The results are consistent with
the notion that dealers’ coarse pricing results from the information cost associated with pricing precision, as proposed by Grossman et al. (1997), but not with dealer collusion. Topmost
dealers uniquely advance market efficiency in these multi-trillion-dollar operations of the Fed.
JEL classification codes: E58, G14, G4, D43.
Keywords: Price clustering, Quantitative easing, Treasury bond, Information processing.

* I am indebted to my supervisory committee, Mark Huson, Vikas Mehrotra, and Randall Morck for their comments
and guidance. This paper has benefited from helpful feedback received from Adelina Barbalau, Sebastian Fossati,
Hsuan Fu, Takeo Hoshi, Naoto Isaka, Hyeik Kim, Sofia Priazhkina, Bruce Riegal, Edward Ritter, Masahiro Watanabeand seminar participants at the University of Alberta and Waseda University, and participants at the Canadian Economics Association Conference 2023. This research was conducted while I was at the University of Alberta. All errors
are my own.
† School of Commerce, Waseda University, 1-6-1 Nishiwaseda, Shinjuku, Tokyo 169-8050, Japan, email: ytsujimoto@waseda.jp.

1

Introduction

In March 2009, the Fed launched a large-scale purchase program of long-term Treasury securities, now
marked as the beginning of its quantitative easing (QE) purchases of government bonds. After more thana decade of the Fed’s QE, there is currently abundant and growing literature on its macroeconomic effects.
Yet, we still know little about the market where these massive purchases take place. This is rather puzzling
given the Fed’s enormous purchase sizes and the role of the U.S. Treasury market as the “single most important financial market in the world” (Group of Thirty, 2021). To fill this gap, this paper closely examines
offer-level data from the Fed’s reverse auctions (hereafter called QE auctions) for Treasury security purchases,1 and document that primary dealers (PDs) of the Federal Reserve Bank of New York (FRBNY)—the
only direct participants in QE auctions—exhibit coarse pricing behavior.
Specifically, in QE auctions the Fed sets the uniform tick size of 1/256th (i.e., 0.390625 cents) per $100
par value for all target Treasury notes and bonds, although quotes of those securities are seldom posted at
this level of fineness in secondary trading venues. Figure 1 demonstrates the existence of coarse pricing
among (successful) offers in QE auctions, and perhaps more surprisingly, a striking difference between two
PDs: Morgan Stanley (with the second largest market share in my sample of QE auctions) and Credit Suisse(with the seventh largest market share). Whereas Morgan Stanley’s price endings are almost uniformly
distributed, Credit Suisse’s distribution displays a strong clustering of price endings on 0, 4/256, 8/256,. . . ,252/256 (i.e., 1/64ths).2 Coarse pricing has been documented in various financial (and non-financial) markets,3 but it is particularly surprising in this setting. First, Treasury securities are one of the world’s mostliquid and heavily researched asset classes. Second, PDs (and institutions indirectly participating in QE
auctions) are highly sophisticated investors with expertise in fixed-income valuation. Third, in QE auctions
there is no incentive to sacrifice pricing precision for execution priority; offers are treated equally as long as
1 As discussed in Section 2, some of the reverse auctions in my sample period are better framed as traditional market
operations rather than QE. However, for convenience, I use the term QE auctions to refer to all the Fed’s reverse auctionsof Treasury coupons.
2 Note that Credit Suisse’s smaller sample size can lead to greater variability in realized proportions of price endings,
but not clustering on a specific subset of price endings.
3 They include stock exchanges (Harris, 1991; Ikenberry and Weston, 2008; Bhattacharya et al., 2012), gold markets(Ball et al., 1985), future markets (Ap Gwilym et al., 1998a,b; Kuo et al., 2015), and particularly relevant to this research
are dealer markets such as NASDAQ (Christie et al., 1994; Christie and Schultz, 1994, 1999) and municipal bond markets(Li, 2007; Griffin et al., 2023). Nikiforov and Pilotte (2017a,b, 2019) also document price-end clustering of Treasury
coupon securities in the secondary market.

1they are submitted by the QE auction closure.
This paper studies the causes and consequences of this previously undocumented practice in QE auctions. Market microstructure literature suggests two possible reasons. First, dealers’ coarse pricing mightwork as a mechanism to coordinate among themselves, thereby extracting higher profits. A well-discussed
example is NASDAQ market makers’ avoidance of odd-eighth quotes, first documented by Christie andSchultz (1994). This practice was controversial especially because virtually no market makers deviated fromthe practice for certain stocks (Christie and Schultz, 1994). While disagreement exists about what caused thisapparently anti-competitive practice,4 it is a fact that this practice mechanically limited the minimum possible bid-ask spread to two eighths for the affected stocks. Nevertheless, this collusion view is inconsistent
with the coarse pricing patterns observed in QE auctions. As suggested in Figure 1, top dealers, who bydefinition account for the bulk of transactions, engage in coarse pricing much less frequently, regardless of
the Treasury security type.
An alternative view for dealers’ coarse pricing is the “competitive theory of clustering” of Grossman
et al. (1997), which was also proposed to explain the NASDAQ market maker behavior. According to thisview, dealers, even when competing in the financial market, engage in coarse pricing due to information
processing costs associated with increased pricing precision. Grossman et al. (1997, p. 25) state, “Finer unitsof trade allow for more accurate pricing. But this is a mixed blessing. It takes time and effort to obtain
more precise valuations of assets,” and as a result, “[t]he precise degree of coarseness chosen will dependon the balance between the benefits and costs of a finer grid.” In the QE auction context, pricing precision’s
information costs can lead PDs to price on grids coarser than the Fed’s 1/256ths grid, especially because
secondary-market transactions are predominantly based on coarser grids (mainly 1/64ths or 1/128ths).
My results are consistent with the theory of Grossman et al. (1997). First, there exists a positive andstrong association between the pricing fineness of (accepted) offers and the PD’s market share in QE auctions. This result is expected if it is not costless for dealer banks to develop and employ sophisticatedpricing technology tailored for this special QE market. (Note that the costs include not only capital investments but also, and perhaps more importantly, the attraction, retention, and deployment of human capital;
4 Christie and Schultz (1994) interpret it as indicating implicit collusion, yet others point out institutional features

limiting dealer competition (Demsetz, 1997; Kandel and Marx, 1997).

2

highly-skilled traders are a critical asset in high finance). Because the return from pricing technology increases with the investment size (Arrow, 1987; Peress, 2004), the cost-benefit tradeoff indicates a positive
association between the market share and sophisticated pricing.
The estimated between-PD difference is economically sizable. This paper proposes a method to quantify PD-level pricing fineness in a manner that is directly interpretable and comparable. It infers the proportion at which a particular PD used each of the following four possible pricing grids—1/32nds, 1/64ths,
1/128ths, and 1/256ths. Because offered Treasury security types vary by PD and time, this method controls
for this heterogeneity, estimating the proportions in case the PD prices a ‘typical’ security in QE auctions:off-the-run Treasury note maturing in 5–10 years. Take Figure 1 again. The estimated proportion of using
the finest 1/256ths pricing grid is 94.9% for Morgan Stanley and 40.5% for Credit Suisse. More generally, according to my baseline specification, a one percentage point increase in QE auction market share translates
into a 2.4 percentage point increase in the use of the finest 1/256ths grid.
Second, if the cost of increasing pricing precision drives dealers’ coarse pricing, we should observe
greater coarse pricing when it is more difficult—and therefore more costly—to precisely price the security(Ball et al., 1985; Grossman et al., 1997). I find a number of results in line with this prediction. First, coarse
pricing is more pronounced for Treasury securities whose valuation is supposedly more difficult: thosewith longer remaining maturities (i.e., greater interest rate risk) and with high volatility. The second andperhaps more direct evidence is that offers for securities that are less finely priced in the secondary market
are also priced less finely in QE auctions.5 When a specific Treasury security’s secondary-market prices
are fine, it should be easier for PDs to price the security precisely in QE auctions—either because the finesecondary-market prices directly help them confidently price the security on a less coarse grid, or because
fine prices in the secondary market indicate the intrinsic ease of precisely pricing the security. Moreover, I
document that this effect is particularly large for non-top dealers.
Over time, PDs price more finely in QE auctions. This competition between PDs in upgrading pricing
precision can be seen as direct micro-evidence of investor evolution in the adaptive markets view of Lo(2019). Notably, the Treasury security trading landscape has undergone remarkable changes during the
5 Specifically, I collect daily closing ask prices of Treasury securities on the trading day preceding each QE auction.

3

analysis period. Joint Staff Report (2015) and Brainard (2018) indicate that in the 2010s dealers increasingly
adopted fintech, such as systems for automated Treasury security trading, yet at different speeds.
The COVID-19 crisis period is, however, an exception to the trend toward greater pricing fineness. In
massive Treasury purchase auctions of March 2020, pricing fineness temporarily plunged. This is a period
in which the Treasury security market exhibited rare malfunctioning; in contrast to typical crisis episodes,
the prices of long-term Treasury securities—arguably the world’s safe haven—decreased in the wake of
the COVID-19 crisis.6 Yet, multivariate analysis indicates that the massive purchase sizes themselves, andthe resulting low offer-to-cover ratios, contributed to the rise of coarse prices during this period. The inverse relationship between the offer-to-cover ratio and the prevalence of coarse pricing indicates the role ofcompetition in curbing this phenomenon.
The last part of this paper investigates coarse pricing from the viewpoint of the Fed’s QE operation
costs. Admittedly, the lack of losing offer data limits the thoroughness of this analysis. My empiricalapproach is to compare the levels of offer prices among accepted offers for the same Treasury security inthe same QE auction. The fixed-effects regressions show that conditional on winning, coarsely priced offers
are more highly priced. Note that this does not necessarily mean that coarsely priced offers are generally
more highly priced. To win a QE auction an offer needs to have a sufficiently low price. Consequently,
higher prices, conditional on winning, can be observed as a result of a reduced probability of winning theauction. In any case, this exercise indicates that coarse pricing has cost implications for the Fed.
Two data limitations should be acknowledged. The first is that the offer-level data disclosed in accordance with the Dodd-Frank Act includes all accepted offers but not losing offers. This data curtailment
precludes recovering some key structural parameters, such as dealers’ marginal valuations (Boneva et al.,2020). In the context of this paper, this limitation’s most direct consequence is that I can observe the pricing
fineness of only winning offers, although the main interest lies in understanding how PDs price in general.
This data curtailment can introduce a bias, as offers with different price-end fineness might have different
auction-winning probabilities. The previous paragraph’s finding suggests that coarsely priced offers havelower auction-winning probabilities. If this is the case, the extent of coarse pricing observed among winning
6 The driver was investors dashing for cash, selling even safe Treasury securities to dealers at a massive scale (Duffie,

2020; He et al., 2022; Schrimpf et al., 2020; Vissing-Jorgensen, 2021).

4

offers should be regarded as a lower bound for dealers’ actual use of coarse pricing, i.e., the extent of coarse
pricing unconditional on the QE auction outcome.
Second, although QE auctions permit PDs to submit not only their own offers but also their clients’offers, the data do not flag client offers.7 Consequently, the between-dealer variation might reflect different
proportions of client offers. For example, the positive association between PD market share and pricing
fineness can arise if client offers constitute a larger proportion for larger-market-share PDs and if client
offers tend to be more finely priced. It appears highly unlikely that different client proportions solely
explain the huge between-PD gap in pricing fineness.8 Nevertheless, this issue should be kept in mind
when interpreting the results concerning between-dealer differences.
This paper is policy-relevant. First, this paper sheds new light on the competition and microstructureof this important yet understudied market. In implementing QE-driven purchases, the Fed monitors “theperformance of operations” and “the extent and concentration of dealer participation in operations” (Potter,
2013, p. 4). More broadly, there is an ongoing discussion about the optimal size—and “diversity”—of the
Fed counterparties.9 Based on the current primary dealer system, the Fed’s market operations are solelyintermediated by large Treasury security dealers designated as primary dealers. On one hand, Potter (2015)
says that the system requires “established, regulated market participants” and “must be of an appropriate
size to provide adequate execution capacity and competitive pricing.” On the other hand, he also notes that“[s]taff time and resources are required to monitor and manage” relationships with counterparties.10 This
paper’s results highlight the special role of topmost dealers in facilitating and enhancing price competition
and market efficiency.
Second, the Fed’s massive purchase scale means that coarse pricing can lead to substantial cost implications. In my sample period, the total amount of QE purchases of Treasury coupons is $3.92 trillion (withthe average per auction being $4.12 billion). Then, one tick size change multiplied by the total purchase
7 This is also the case of the proprietary U.K. QE auction data analyzed by Boneva et al. (2020).
8 At the same time, I do not exclude the possibility that PDs’ precise pricing capabilities are influenced by client

offers. For example, finely priced offers from clients, or just receiving more client offers, might be helpful for the PD toincrease the precision of its own offers.
9 The “diversity” debate has emerged due to the increasing presence of non-dealers (most notably the so-called“principal trading firms”) in the secondary market.
10 The Fed’s primary dealer pilot program (2013–2014), which temporarily allowed four relatively small dealers toparticipate in the Fed’s operations, was one of the attempts to search for a better balance.

5

size amounts to $15.3 billion. The tick size is also economically significant in comparison to the Fed’s transaction costs in QE auctions. Song and Zhu (2018) estimate that the weighted average cost (relative to thebest ask price at auction closing) is merely 0.71 cents per $100 par value. This is roughly equivalent to two
minimum ticks (0.78 cents per $100 par value).
This paper contributes to three strands of literature: First, this paper contributes to the literature on
dealers’ coarse pricing. In addition to the NASDAQ market maker behavior mentioned above, Li (2007) and
Griffin et al. (2023) document that municipal bond dealers often price on a coarse grid (in eighths). Griffin
et al. (2023) further show that dealers enjoy higher markups from such transactions. Note that the suspectedmechanism here is the conflict of interest between well-informed sellers (dealers) and less-informed buyers(including retail customers) under decentralized bilateral transactions.11 These market characteristics arequite different from QE auctions, where many PDs compete to sell more transparent Treasury securities. I
show that in QE auctions dealers’ coarse pricing is consistent with the theory of Grossman et al. (1997), in
which they face a tradeoff between the pricing precision’s cost and competitive advantage.
Second, this paper contributes to the emerging literature on the implementation mechanisms of QE.
Song and Zhu (2018) study the Fed’s preference in QE auctions and PDs’ bidding behavior using the samedata as this paper.12 They first confirm that as the Fed’s public disclosures suggest, the Fed prefers “cheap”
Treasury securities, i.e., those whose secondary market prices are low relative to those implied by the yieldcurve model. Song and Zhu (2018) then show that PDs extract higher profits when offering such securities.
Boneva et al. (2020) analyze U.K. QE auction data containing both winning and losing offers. They structurally estimate U.K. primary dealers’ marginal valuations of offered gilts, and show that these valuations
are related to the interest rate risk and the regulatory capital requirements the dealers face. Laséen (2023)
also uses comprehensive offer-level data in Sweden to study primary dealer behavior in those auctions.13
This paper complements these papers in understanding price competition in QE auctions, taking cues fromprice-end patterns.
11 The municipal bond market is unique among over-the-counter markets in that the participation of retail investors

is high (see Bessembinder et al. (2020)).
12 PDs’ bidding behavior in Treasury issuance auctions is studied by Hortaçsu and Kastl (2012), Hortaçsu et al. (2018),
and Allen et al. (2020).
13 More broadly, the implementation cost of QE is also studied by D’Amico and King (2013) in the U.S., Breedon (2018)
in the U.K., and Schlepper et al. (2020) in the Eurosystem. In addition, An and Song (2022) study the Fed’s purchaseprices in its agency MBS operations.

6Third, this paper adds to the literature on the limited sophistication of sophisticated investors. Previous papers study institutional investors’ herding (Wermers, 1999; Nofsinger and Sias, 1999; Griffin et al.,2003) and heuristic decisions (Akepanidtaworn et al., 2021; Wang, 2020) in stock trading.14 This paper documents a manifestation of information processing constraints in the context in which it is arguably least
expected—pricing of Treasury securities, the world’s most heavily researched asset class, by the New York
Fed’s primary dealers, one of the most sophisticated and influential investor groups in the global economy(He et al., 2017; Goldberg, 2020).15 Notably, pricing fineness of the market’s most sophisticated investors isparticularly interesting because it can even speak to the limit of market efficiency (Grossman and Stiglitz,
1980; Mondria et al., 2022).

2

Institutional background2.1

Primary dealers

PDs are major dealers in U.S. Treasury securities designated by the FRBYNY as (sole) counterparties of the
Fed’s market operations. Most notably, PDs are allowed to submit competitive bids in Treasury issuance
auctions (and profit from selling the purchased securities in the secondary market). A primary dealer status
can also help the dealer attract large customers such as foreign central banks (Rennison and McLannahan,2016). Their main requirements are as follows. First, PDs are expected to bid a certain amount in every
Treasury issuance auction. Second, for other Fed operations (including QE operations), each PD is expectedto participate “at levels commensurate with its size and presence in the market.”16 The last requirement
is assisting the Fed to formulate monetary policy. During my sample period, the number of PDs changed
14 Wang (2020) shows that institutional investors, like retail investors but to a lesser extent, exhibit the “round-numberbias”—i.e., the tendency to submit orders with rounded price endings such as “.0”—in stock trading, and that this bias islower for larger institutions. Note that his research question is fundamentally different from mine. Price-end clustering
measured this way likely reflects traders’ cognitive reference points (also see Bhattacharya et al. (2012), whose method
Wang (2020) follows).
15 Fleming et al. (2005) and Goldreich (2015) also study bounded rationality of PDs. They document evidence of suboptimal bids in Treasury bill issuance auctions (until an auction rule change in 2004). One notable difference between
their datasets and mine is that theirs are aggregated auction-level data, precluding PD-level analysis.
16 The Fed’s “Administration of Relationships with Primary Dealers” statement published on March 24, 2016 (https://www.newyorkfed.org/markets/pridealers_policies.html)

7

from 18 in 2010 to 24 in 2020.17 Table 1 lists PDs in my sample period.

2.2

Phases of the Fed’s QE

Table 2 lists major events among the Fed’s QE purchases of Treasury securities. As shown in the last column,
this paper divides the sample period into five sub-periods: QE2 (August 17, 2010–September 20, 2020), MEP(September 21, 2011–December 11, 2012), QE3 (December 12, 2012–October 29, 2014), QE pause (October 30,
2014–March 11, 2020), and QE4 (March 12–June 29, 2020). Appendix A provides a summary of the historyof the Fed’s QE. Figure 2 shows the time series of the Fed’s purchases of Treasury coupon securities.
The following clarifications will prove useful. First, the first sub-period (QE2) centers around, but
is not limited to, the so-called “QE2” round, which was implemented from November 3, 2010 throughJune 22, 2011. Before and after the round, the Fed reinvested the proceeds of maturing securities into
Treasury securities to maintain its balance sheet size. Those purchases are included in the QE2 sub-period.
Second, the MEP and QE3 sub-periods exactly match the periods of the Maturity Extension Program (MEP)
and the so-called “QE3,” respectively. Third, the MEP was different from the other QE rounds in that
the Fed financed the purchases of long-term Treasury securities by selling short-term securities (thereby
not changing the aggregate bank reserves). Fourth, although the QE pause sub-period starts in October
2014, most of the purchases took place in August 2019 or thereafter. During this period, the Fed resumed
reinvesting in Treasury securities to maintain a sufficient size of bank reserves. Lastly, the QE4 sub-period
covers QE purchases in response to the COVID-19 crisis (up to the sample conclusion of June 2020). Notethat the primary purpose of the post-pandemic purchases was to tame the disruptions in the market (seeAppendix A for more discussions).

2.3

Structure of QE auctions

The Desk of the FRBNY administers QE auctions, as in the case of other Fed market operations. Auctionlevel summary statistics are provided in Table 3. Details of the QE auction protocol vary from phase tophase, yet a typical timeline is as follows. First, the QE auction schedule for the next monthly cycle is
17 The current and historical lists of PDs are available at https://www.newyorkfed.org/markets/primaryde

alers.

8announced around the previous monthly cycle’s end. The schedule lists the dates of auctions, together with
their target maturity ranges (e.g., Treasury coupons with 4.5 to 7 years remaining to maturity) and expectedpurchase sizes.18 Securities excluded from the auction, such as those scarce in the secondary market, areannounced at the auction start. PDs are therefore well informed of which securities are included in theauction.
Second, on the auction date, PDs can submit up to nine offers per security. Each offer consists of priceand quantity. Unlike Treasury issuance auctions, QE auctions are price-discriminatory; the offer price is theprice at which the PD sells the security to the Fed if the offer is accepted. While PDs are the only direct
participants, they can also place their clients’ offers on their behalf. Both the minimum offer size and theprice increment are set at $1 million. The tick size is 1/256th per $100 par value for all Treasury notes andbonds.
The Fed’s QE auctions are multi-object auctions, as each auction accepts multiple Treasury securities
within the target maturity range.19 Therefore, the FRBNY adopts the following approach to rank offers
for different securities. It first calculates the benchmark price for each auction-target security by applying
its proprietary yield-curve model to secondary market prices (Sack, 2011). Then, offer prices normalized
by their benchmark prices are directly comparable regardless of the differences in the offered securities.
The FRBNY purchases from the offers with the lowest prices relative to the model-implied price until its
desired total purchase amount is reached. Note that this protocol means that the Fed prefers securities
whose market prices are below the prices implied by the Fed’s model, i.e., “cheap” securities (Song andZhu, 2018).
In my sample period, the median auction time is 45 minutes, and the auction close time ranges from9:50 a.m. to 3:05 p.m., with 90.9% of them ending in the morning. It is exceedingly rare for an auction to
end after 2 p.m. (1.6%).20 The settlement is typically the next day, meaning that PDs can cover their short
18 In early QE phases, the FRBNY disclosed the expected purchase ranges, while only the maximum purchase amounts

are announced in later phases.
19 In the average QE auction, 30.7 unique Treasury securities meet the basic eligibility criteria of QE auctions, with 3.8of them being explicitly excluded for other reasons. Among the remaining 26.9 CUSIPs, 17.0 are purchased (Panel B of
Table 3).
20 More specifically, until 2019, the vast majority (95.9%) of QE auctions ended at 11:00 a.m. In the post-COVID-19
period, the Fed tended to hold multiple QE auctions (for different target maturities) at different times on the same day,resulting in more diverse auction close time.

9

positions anytime in the afternoon of the auction date.
Lastly, the FRBNY releases the auction result in three steps. First, immediately after each auction, the
FRBNY releases quantity-related information, such as the total offered and accepted amounts and the purchased amount per security. Participating PDs are notified of their auction outcomes at this time. Second,
the price-related information is disclosed at the end of each monthly auction cycle. The information is aggregated to the (purchased) security level. The disclosed items are the total offered and purchased amounts,
the weighted average and highest accepted prices, and the proportion of accepted offers among the highest
price offers. Lastly, about two years after the auction, the FRBNY releases disaggregated data on accepted
offers, which is the primary dataset used in this study.

3

Data

The FRBNY publicly discloses all accepted offers of QE auctions after the enactment of the Dodd-Frank WallStreet Reform and Consumer Protection Act on July 21, 2010.21 The offer-level data includes the Treasury
CUSIP, offer price, offered amount, and the identity of the PD who submitted the offer. I retrieved all
accepted offers for nominal Treasury notes and bonds.22 My sample period spans from August 17, 2010
(the date of the first QE auction post-Dodd-Frank Act) to June 29, 2020 (the last auction date in 2020Q2).
Treasury security information was obtained from TreasuryDirect.23
Table 4 reports the descriptive statistics of my sample offers. Security types dramatically vary overtime. The vast majority of the purchased securities are off the run, with no on-the-run securities being
purchased after the QE3 period. Also, the composition of remaining maturities varies strikingly over time.
While 20–30 years account for 50.8% in the MEP period, 0–5 years is the majority (53.0%) in the QE4 period.
These target security composition changes highlight the importance of controlling for security types in theempirical analysis. The bottom rows of Table 4 concern the market shares of PDs who submitted offers.
The market share is based on trade amounts and calculated for each sub-period. Following Song and Zhu
21 https://www.newyorkfed.org/markets/omo_transaction_data.html
22 Therefore, my sample does not include Treasury bills or TIPS.
23 https://www.treasurydirect.gov/instit/annceresult/annceresult_query.htm

10(2018), I group PDs into top five PDs and non-top five PDs.24

4

Result

4.1

Setup

The market convention for quoting Treasury coupon securities is per $100 par value, with the decimal partbeing a multiple of 1/32 or a multiple of a fraction of 1/32 (that is, 1/64, 1/128, or 1/256).25 In QE auctions,the tick size is set to one eight of 1/32, i.e., 1/256. Throughout the paper, a Treasury security prices’ decimal
part is denoted as d/256. That is, d ∈ D = {0, 1, 2, . . . , 255} and d = 1 means the price ending is 1/256.
To analyze price-end clustering, which indicates dealers’ use of coarser pricing grids, I divide D into
four mutually exclusive subsets:

D = {0, 1, 2, . . . , 255} = X32 ∪ X64 ∪ X128 ∪ X256 , where







 X32 = {0, 8, 16, . . . , 248}







 X64 = {4, 12, 20, . . . , 252}

(1)





X128 = {2, 6, 10, . . . , 254}









 X = {1, 3, 5, . . . , 255}
256

In words, each subset represents the possible coarsest pricing grid that the PD could have used to arrive at
the particular d. For example, if d ∈ X64 , the possible coarsest pricing grid used by the dealer is 1/64ths.
This is because the dealer can arrive at d ∈ X64 based on the pricing grid of 1/64ths, 1/128ths, or 1/256ths,
but not on the 1/32nds grid. Note that only in the case of d ∈ X256 , the pricing grid used by the PD can be
identified with certainty (to be 1/256ths).
24 The remaining results are robust to the use of alternative cut-off points, such as top four or top seven. Section 4.7
conducts detailed PD-level analysis.
25 For example, given a $1,000 face value, a price quote of $101.09375 (= $101 + 3/32) indicates that the bond’s priceis $1,010.9375.

11

4.2

Testing price-end clustering on coarser grids in QE auctions

I follow Kuo et al. (2015) and Bhattacharya et al. (2018) to statistically test the existence of price-end clustering on coarser grids. Specifically, for each price ending d ∈ D = {0, 1, 2, . . . , 255}, I calculate the percentage
of offers with this price ending among all offers. To facilitate interpretation, this percentage is then subtracted by the expected percentage under the uniform priced-end distribution, which is 1/256 × 100 =0.390625%. This outcome variable is regressed on the following variables:

Percentd − 0.390625 = α + β 1 D32 + β 2 D64 + β 3 D128 + ϵi,j ,(2)where D32 , D64 , and D128 take the value of one if the price ending d belongs to X32 , X64 , and X128 , respectively, and zero otherwise.
The regression results are reported in Panel A of Table 5. Column 1, which pools all sample (winning)offers, shows strong price-end clustering on coarser grids. According to the constant, the percentage ofa d in X256 = {1, 3, 5, . . . , 255} being selected is 0.177% (as it is 0.213 percentage points less than what is
expected under the uniform distribution). This is due to clustering on grids coarser than 1/256ths; the three
price-end type dummy variables are positive and statistically significant, meaning that price endings in D32 ,
D64 , and D128 are more likely to be selected. Based on Column 1, the probability of d in X32 being chosen
is 0.177% + 0.839% = 1.016%. Likewise, the probabilities of a price ending in X64 and X128 are 0.719%
and 0.340%. Importantly, the differences between D32 and D64 and between D64 and D128 are statisticallysignificant. This result implies that PDs tend to use all of the coarse pricing grids, namely 1/32nds, 1/64ths,
and 1/128ths.26

4.3

Estimating the proportions in which different pricing grids are used

The price-end clustering observed among accepted offers reflects dealers’ use of coarse pricing grids. To
directly gauge this behavior of dealers, I propose a method to infer the likelihoods that PDs employed
the four possible pricing grids, based on the coefficients of Specification 2 (i.e., the observed price-end
26 For example, if PDs used only the pricing grids of 1/128ths and 1/256ths, the coefficients of D and D should32
64

not be greater than that of D128 .

12

clustering). It is important to note that, given that my sample includes only accepted offers, the following
inference is based on an implicit assumption that the price-end type is independent of the auction-winningprobability.27 If PDs always use the finest 1/256ths pricing grid (grid-256ths), then Pr[d ∈ X32 ] = P[d ∈X64 ] = 12.5%, Pr[d ∈ X128 ] = 25%, and Pr[d ∈ X256 ] = 50%. I denote these conditional probabilities
grid-256ths

as ϕ32grid-256ths, ϕ64

grid-256ths, ϕ128grid-256ths, and ϕ256

, respectively. Table 6 summarizes these conditional

probabilities for all the pricing grids.
Now, suppose that dealers use the 1/32nds pricing grid (grid-32nds) with probability λ32 , and the pricing grids of 1/64ths, 1/128ths, and 1/256ths with probabilities λ64 , λ128 , and λ256 , respectively. Then, theunconditional probabilities of price-end types are:grid-32nds+ λ64 × ϕ32

grid-32nds+ λ64 × ϕ64

ϕ32 = λ32 × ϕ32
ϕ64 = λ32 × ϕ64

grid-64ths+ λ128 × ϕ32

grid-64ths+ λ128 × ϕ64

grid-32nds+ λ64 × ϕ128

grid-32nds+ λ64 × ϕ256

ϕ128 = λ32 × ϕ128
ϕ256 = λ32 × ϕ256

grid-128ths+ λ256 × ϕ32

grid-128ths+ λ256 × ϕ64

grid-64ths+ λ128 × ϕ128

grid-64ths+ λ128 × ϕ256

grid-256ths

grid-256ths

grid-128ths+ λ256 × ϕ128

grid-128ths+ λ256 × ϕ256,,

grid-256ths

grid-256ths,.

The log-likelihood function can be defined as follows:ln L(λ) = Fraction32 × ln ϕ32 + Fraction64 × ln ϕ64 + Fraction128 × ln ϕ128 + Fraction256 × ln ϕ256 ,(3)

where Fraction32 is the observed fraction of d ∈ X32 , and Fraction64 , Fraction128 , and Fraction256 are similarly defined.28 I maximize this function to obtain the estimates of λ = {λ32 , λ64 , λ128 , λ256 }, with a constraint of each element of λ being between 0 and 1.
Panel B of Table 5 reports the predicted proportions in which the four pricing grids are used. The finest
1/256ths grid is predicted to be employed 45.4% of the time, and the coarsest 1/32nds grid 9.5% of the time.
The rest is accounted for by the two pricing grids between these two extremes.
27 As discussed later, I find some evidence suggesting that coarsely priced offers have a lower chance of winning a

QE auction. If this is true, coarsely priced offers are underrepresented within the sample of accepted offers, leading toa downward bias in the predicted proportions of using coarse pricing grids. This potential sampling bias, arising fromthe data limitation, should be kept in mind.
28 These ratios can be computed from the coefficients of Specification 2.

13

4.4

Price-end clustering by time and by PD

Table 5 also shows that price-end clustering in QE auctions greatly varies by time and the PD submitting
the offer. In Columns 2–4, I repeat Specification 2 separately for each sub-period. There are two notablefindings. First, PDs tend to price more finely over time. While the predicted proportion of the 1/256ths
grid use is only 14.5% in the QE2 period, it accounts for more than 50% in the subsequent periods. Second,
the predicted proportion of the coarsest 1/32nds grid use jumped to a much higher value than the earlier
periods (21.2%) in the QE4 period. More specifically, Figure 3 illustrates that this surge in the coarsest
pricing took place during the massive purchases of March 2020. The extent to which changing auction
characteristics can explain these time-series patterns is examined in the next section.
The remaining columns of Table 5 show a stark difference in pricing fineness between top and nontop dealers. The last column of Panel A establishes the statistical significance, with the coefficients of allof D32 , D64 , and D128 being significantly lower for top dealers. Panel B also indicates that the difference is
economically sizable; for instance, the predicted proportion of the 1/256ths grid use is 66.2% for top dealersand 26.3% for non-top dealers.

4.5

Determinants of coarse pricing in QE auctions

To analyze heterogeneity in coarse pricing, I regress the fineness of offer prices on security-, offer-, andauction-characteristics. The key idea is that the price-end type, X32 , X64 , X128 , or X256 , indicates the likelihood with which coarser pricing grids are used, with X32 being the highest and X256 the least. (To be
precise, the probability of coarse pricing is zero if d is in X256 .) I therefore perform the ordered logit analysis with the dependent variable being Price-end fineness, which takes the value of one if the offer’s priceending is in X32 , two if it is in X64 , three if it is in X128 , and four if it is in X256 . This analysis aims to uncoverwhich factors increase (or decrease) the likelihood of precise pricing. Table 7 lists the definitions of variables
used in this offer-level analysis. The descriptive statistics are provided in Table 8.
One key explanatory variable is Cheapness, which is included to analyze the effect of the Fed’s algorithm
in ranking offers. As explained in Section 2.3, the Fed’s protocol prefers securities that are deemed undervalued based on the Fed’s yield curve model. Although the model is not publicly disclosed, it is a standard

14

cubic spline model (Sack, 2011), and Song and Zhu (2018) find that their yield curve estimation result—andthe ensuing cheapness measure—can explain the Fed’s purchase behavior. I thus estimate a cubic spline
model following Song and Zhu (2018), and define Cheapness as the percentage difference between the yieldcurve-implied price and the actual secondary-market price. The details of the yield curve estimation aresummarized in Internet Appendix A.
Table 9 shows the estimation results. Panel A reports the coefficients and the marginal effects on theprobability of Price-end fineness = 4, that is, D256 = 1.29 The coefficient of Top five is positive and significant
in all models. According to Model 1, the probability of the price ending being the finest type (i.e., D256 = 1)
is 14.9 percentage points higher for top dealers’ offers. This is sizable given that only 22.7% of sample offers
have this finest price-end type (Table 8). Models 2 and 3 show that the marginal effect remains stable evenwhen additional control variables are added.
In Model 1, Cheapness is negative and significant, meaning that prices are coarser when the Treasury
security is deemed undervalued and therefore preferred by the Fed. Because strategic dealers can extract
higher profits when delivering cheaper securities in QE auctions (Song and Zhu, 2018), those dealers might
be using coarse pricing as a device to set a higher price for cheaper securities. However, the evidence is notconsistent with this view.
First, Panel B further investigates this association by splitting the sample by Top five and reporting themarginal effect of Cheapness on all possible values of Price-end fineness. It shows that the marginal effects
of Cheapness are generally more pronounced for non-top dealers. Strikingly, the marginal effect on theprobability of Price-end fineness = 1 is 0.357 for those dealers, meaning that a one standard deviation change
of Cheapness translates into an 8.50 percentage point increase in this probability. The stronger association
for non-top dealers does not align well with the narrative that the effect of Cheapness is a result of dealers’deliberate coarse pricing for cheap securities. Instead, a more plausible explanation is that when the security
is deemed undervalued by the Fed, it is more likely that even coarsely priced—or less sophisticatedly
priced—offers can win the QE auction. (In Section 4.8, I show that coarsely priced offers are more highly
priced among accepted offers.) This is because offers for cheap securities have an intrinsically higher chance
29 The ordered logit model can derive marginal effects for each possible value of the outcome variable.

15

of winning a QE auction based on the Fed’s algorithm to rank offers.
Second, the effect of Cheapness is sensitive to the inclusion of controls, with the coefficient even changing
its sign. In particular, Cheapness is no longer negatively associated with Price-end fineness after controlling
for three basic security types—on-the-run status, maturity, and Treasury bond dummy (Model 2 in Panel
A of Table 9). On one hand, this result relates to the fact that Cheapness is highly positively correlated withremaining maturity.30 On the other hand, this result is inconsistent with the view that dealers use coarse
pricing deliberately and specifically for cheap securities. Rather, this result indicates that certain security
types (most notably maturity) are associated with both pricing fineness and Cheapness.
Model 3 provides three additional insights. First, Voaltility, the standard deviation of the offered security in the five-day period leading up to the auction date, is negatively associated with pricing fineness.
This finding is consistent with the hypothesis of Ball et al. (1985) and Grossman et al. (1997) that valuation
uncertainty, and the resulting greater cost in precise pricing, drives coarse pricing.
Second, the result of Offer-to-cover suggests the role of competition in restricting coarse pricing. According to Model 3, one standard deviation increase in Offer-to-cover translates into a 1.99 percentage point
increase in the probability of D256 = 1. Note that this is the effect after controlling for period dummies, andmoreover, this result is not driven by massive post-COVID-19 purchases; the coefficient remains significant
at the 1% level even if the QE4 period is dropped.
Third, pricing fineness is greater in later phases than in the first QE2 phase (the model’s baseline period),
and this is more evident after controlling for security- and auction-characteristics. In Model 1, the coefficientof QE4 is not significant with a low marginal effect on the probability of D256 = 1 (a 3.8 percentage point
increase). Yet, the variable is significant at the 1% level in Model 3, which controls for security- and auctiontype variables. Most notably, Volatility is high and Offer-to-cover is low in the QE4 period, and these factors
contributed to deteriorated pricing fineness in the post-pandemic QE operations.
30 The correlation between Cheapness and remaining maturity in years is 0.754.

16

4.6

Pricing fineness in the secondary market for Treasury securities

I now turn my attention to the secondary market for Treasury securities with the purpose of better understanding the QE auction market’s coarse pricing.

4.6.1

Trading venues and tick sizes

Since the launches of eSpeed and BrokerTec in 1999 (the latter of which started service in 2000), trades ofon-the-run Treasuries have almost entirely moved to fully electronic systems (Mizrach and Neely, 2006).
These markets now accommodate non-dealers, with the most significant non-dealer participants being theso-called principal trading firms (PDFs) (Harkrader and Puglia, 2020). However, once a security goes offthe-run, the trading volume plunges, and most of the trading migrates to more traditional voice-assisted
systems (Barclay et al., 2006). Off-the-run trading is also characterized by high market fragmentation, as
opposed to the on-the-run trading arena, where two platforms, BrokerTec and Dealerweb, dominate others(McPartland, 2018).
Tick sizes vary by transaction venue and maturity. The main electronic venues of on-the-run securities,
such as BrokerTec and eSpeed (then acquired by Dealerweb), set the tick sizes of 1/128th for 2-, 3-, and 5year notes and 1/64th for 7- and 10-year notes and 30-year bonds in the period investigated in this section.31
Conversely, in the case of off-the-run trading on voice-assisted markets, it is customary, albeit not a rule, thatbrokers display tick sizes coarser than 1/256th (such as 1/64th and 1/128th), except for securities nearing
maturity.

4.6.2

Pricing fineness in the secondary market for Treasury securities

To understand pricing fineness in the secondary market, I repeat the price-end clustering regression withsecondary-market price data. For each Treasury security purchased in a QE auction, I obtained the security’s daily closing ask price on the preceding trading day from Bloomberg.32 The result of repeating
31 These venues lowered the tick size of 2-year notes to 1/256th on November 19, 2018 (Fleming et al., 2022). Note,however, that the Fed purchased on-the-run securities only until August 2014.
32 I do not use the CRSP Treasury data for studying price endings due to its apparent anomaly. For unknown reasons,the proportion of price endings in D128 doubled on March 7, 2012; while the average proportion was 23% for thepreceding ten trading days, it jumped to 48% in the next ten trading days, and this was not a temporary phenomenon
but a permanent shift. Bloomberg data does not exhibit such dramatic and puzzling price-end patterns.

17

Specification 2 with the secondary-market price data is reported in Table 10. According to Column 1 ofPanel B, the predicted proportion of the 1/256ths grid use is 13.4%. The sub-period analysis reveals that
this value remained virtually zero for the first three sub-periods (QE2, MEP, and QE3). The 1/64ths and1/128ths grids account for a large fraction, and, there is little evidence of the coarsest 1/32nds grid use.33
Table 11 then estimates the ordered logit model of the determinants of pricing fineness of the secondarymarket price data. Importantly, maturity exhibits a strong nagative association with pricing fineness, as
was the case for QE auction offers. (On-the-run and Bond have the same signs, but they are not statisticallysignificant in this secondary-market data analysis.) For example, relative to the baseline maturity of five toten years, maturity of less than five years is associated with a 9.7 (18.6) percentage point higher probability
Secondaryof D256

Secondary= 1 (D128= 1). To sum up, Table 11 suggests a link in pricing fineness between thesecondary market and QE auctions.

4.6.3

Relating pricing fineness in the secondary market to that in QE auctions

To directly test the association, Table 12 regresses QE auction offers’ Price-end fineness on the price-end type
Secondary

dummies of the security’s secondary market price; D256

is a dummy variable taking the value of one

if the price ending of the Treasury security’s closing ask on the trading day preceding the QE auction is in
Secondary

X256 , and I similarly define D32

Secondary, D64

Secondary, and D128

Secondarymodel with the baseline category being D32. Specifically, I estimate the ordered logit= 1. The coefficients show whether finer prices in thesecondary market indicate finer offer prices in QE auctions (i.e., a higher Price-end fineness).
Secondary

The positive coefficients of D128

Secondaryand D256

in Column 1 confirm the association. They remain

statistically significant even when security-type controls (On-the-run, Maturity0-5 , Maturity10-20 , Maturity20-30 ,and Bond) are added (Column 4). To demonstrate the economic significance, Figure 4 plots the predictedprobabilities according to the models of Columns 2 and 3 of Table 12. In particular, the predicted probabilities for the two extreme price-end types, X256 and X32 , greatly vary by the secondary-market price-end
Secondarytype for non-top dealers (Panel B of Figure 4). When D32= 1, the predicted probability of X256 is
Secondarymerely 11.5%. However, it jumps to 29.2% (i.e., a 153.9% increase) when D256

= 1. On the other hand,

33 Nikiforov and Pilotte (2017a) document a similar price-end distribution using tick-level data on Treasury notes.

18

the predicted probability of X32 drops from 42.8% to 19.1%.
These results strongly substantiate the costly nature of pricing precision as a driver of coarse pricing
in QE auctions. There are two possible channels through which finer prices in the secondary market areassociated with a lower cost of precise pricing for PDs. First, having fine secondary-market prices as inputs
to the model makes it easier for dealers to confidently price securities on a fine grid. Second, securities withfine prices in the secondary market (e.g., short-maturity securities) are expected to be intrinsically easierto precisely price. That non-top dealers’ pricing fineness is highly sensitive to the fineness of secondary
market prices suggests that pricing easiness relaxes the limit of precise pricing that lagging PDs face.

4.7

Dealer-level analysis

4.7.1

PD-level pricing finenessThis section looks more closely at PD-level differences in coarse pricing. Pricing fineness varies by not only
PD but also offered security type (Table 9). I thus quantify pricing fineness of each PD, after controllingfor basic security characteristics. More specifically, I employ a variant of Specification 2, in which theright-hand side of the model includes dummy variables for three basic security characteristics—on-therun status, remaining maturity, and Treasury note vs. bond—and PD × sub-period fixed effects. Appendix
B explains more details about this method.34 This approach allows for predicting the probabilities of priceendings in X32 , X64 , X128 , and X256 for each PD in each sub-period in the case of offering ‘typical’ security
in QE auctions, namely, off-the-run Treasury note maturing in 5–10 years. These predicted probabilities arethen fed into the method of Section 4.3 to estimate the proportions in which different price grids were used.
The summary statistics of the estimated PD-level proportions can be found in Table A.2 of Appendix B.

4.7.2

Pricing fineness and market share

PD-level analysis also points to a strong association between pricing fineness and market share. Figure 5
plots dealer market share and the predicted proportion of using the finest 1/256ths grid. Panel A of Table
34 To include those dummy variables and fixed effects, the outcome variable, Percent − 0.390625, is calculated foreach security type s, PD j, and sub-period t. Because this means that the ratios are calculated from different numbers ofoffers, the regression weights observations by the number of underlying offers.

19

13 shows that market share is significantly and positively associated with the use of the finest pricing gridfor all the sub-periods other than QE2. The association becomes somewhat weaker but still remains, even
if I use the predicted proportion of using the 1/128ths or 1/256ths grid (Columns 5–8).
In Column 1 of Panel B, which pools the four sub-periods, Market share is statistically and economicallysignificant. The coefficient indicates that a one percentage point increase in Market share translates into a 2.4
percentage point increase in the finest 1/256ths grid use. Notably, this result is not driven by a few of thelargest PDs, as it is robust to the exclusions of Goldman Sachs (Column 2) and Morgan Stanley (Column 3).
In Column 4, I also remove small PDs (with less than 2% market share in the period). The result remains
highly similar. Panel C shows that the coefficients of Market share remain similar even if the regressions are
performed on the first differences.
Having established a strong positive association between pricing fineness and QE auction market share,
I also ask whether pricing fineness is related to more primitive dealer characteristics such as balance sheet
size, location, and experience. Internet Appendix B summarizes the data collection and the analysis result.
Panel A of Table IA.4 shows that the use of the finest 1/256ths grid is positively associated with the balance
sheet size and its use is lower for foreign PDs (i.e., PDs that are a subsidiary of a foreign-based financialgroup). However, these results are statistically significant only at the 10% level and become insignificant
if the dependent variable is replaced with the probability of using the 128ths or 256ths grid (Panel B).
Therefore, these variables’ explanatory power is weak at best. In contrast, the market share remains highly
significant even if PD-level controls are added. One interpretation is that the key driver of the betweenPD variation in the tendency of precise pricing in QE auctions is their presence in this specific market,
rather than their overall size or experience. At the same time, I note substantial measurement issues of mydealer-level variables; dealer size is measured infrequently and noisily,35 and experience is only roughly
measured.
35 For most PDs, a separate balance sheet item for Treasury securities does not exist, and I simply analyze the balance

sheet size.

20

4.8

Coarse pricing and the level of offer prices

Does coarse pricing relate to the level of offer prices, which ultimately determines the Fed’s QE purchasecosts? I examine this question by looking at differences in the level of prices within accepted offers for agiven Treasury security in the QE auction. Specifically, I define Price diff as the percentage difference (inbasis points) between the offer price and the lowest accepted offer price for the same security in the same
QE auction, and run the following regression:Price di f f = β 1 D32 + β 2 D64 + β 3 D128 + γ + ϵi,j ,(4)where D32 , D64 , and D128 take the value of one if the (accepted) offer’s price ending d belongs to X32 , X64 ,and X128 , respectively, and zero otherwise, and γ is CUSIP × QE-auction fixed effects. I measure Price diffonly when multiple accepted offers exist for the security in the QE auction.36 Absorbing between-security
variation, γ ensures that D32 , D64 , and D128 compare Price diff —the offer price normalized by the lowest
accepted offer for the security—for accepted offers with different price-end types within accepted offers forthe same security in the same QE auction.
Panel A of Table 14 shows that the mean Price diff, which is winsorized at the 2.5% and 97.5%, is 3.6
basis points. This is fairly small, being less than the average bid-ask spread for my sample QE auction
securities, 4.3 basis points (Table 8). This result is consistent with the claim of Song and Zhu (2018) that theQE auctions’ operation costs are fairly moderate.
The regression results are documented in Panel B of Table 14. Column 1 is the result of estimating Specification 4, and it shows that (accepted) offers with coarse price endings have significantly higher prices
compared to those with price endings in X256 . The differences are also economically significant. The coefficient of D32 (0.282) indicates that, relative to the mean value (3.61), (accepted) offers with the coarsest priceendings have, on average, a 7.81% higher Price diff, which is the percentage difference between the offer’s
price and the minimum accepted offer price for the security in the QE auction.
36 This restriction only slightly decreases the number of sample (accepted) offers: from 90,022 to 86,611. Conditionalon having multiple accepted offers, the average (median) number of accepted offers for a Treasury security in a QE
auction is 12.4 (10).

21Therefore, coarsely priced offers on average have higher prices, conditional on winning. Note that thisresult does not necessarily mean that coarsely priced offers, unconditional on the QE auction outcome,are on average more highly-priced. For an offer to win a QE auction, its price needs to be sufficiently low
relative to competing offers. Therefore, coarsely priced offers can have a higher average price conditional
on winning simply because they have a lower chance of winning a QE auction in the neighborhood of thelowest acceptable price.
The remaining columns in Panel B of Table 14 show that who submits offers also matters. While theoffer amount (Ln(offer amount)) has only an insignificant effect, Top five, the dummy for offers submitted by
a top five dealer, is negative and significant at the 5% level in Columns 3 and 4.37 Importantly, the inclusionof Top five materially lowers the effects of coarse pricing; the coefficients of D32 and D64 in Column 3 areslightly less than half of those in Column 1.
To summarize, Table 14 demonstrates that coarse pricing does indeed matter from the viewpoint of the
Fed’s QE operation costs. At the same time, prices are lower for accepted offers submitted by top dealers,
and the association between pricing fineness and the offer price level is nearly halved when the top fivedealer dummy is added to the right-hand side of the model. Therefore, the association partially reflects thefact that coarsely priced offers are more likely to come from non-top five dealers, whose accepted prices
tend to be more highly priced.

5

Discussion of client offers

The positive association between market share and pricing fineness is consistent with the notion that largermarket-share PDs are more willing to seek precise pricing; the information processing costs of pricing
precision can lead to increasing returns from pricing technology (Arrow, 1987; Peress, 2004). This section
discusses how the existence of client offers can affect this interpretation.
In QE auctions non-PDs (such as hedge funds and money managers) are allowed to submit offers indirectly through PDs (Sack, 2011). Unfortunately, these client offers cannot be discerned in my data. Con37 Note that in the presence of CUSIP × QE-auction fixed effects, any Treasury security-characteristic variables will
be subsumed.

22

sequently, one concern is that the positive association between pricing fineness and market share mightreflect different proportions of client offers and different extent of coarse pricing between PDs and clients.
Consider a (seemingly more plausible) situation in which top PDs route more client offers due to their more
extensive customer network. In such a case, in order for client offers alone to produce the positive association between PD market share and pricing fineness, clients must have a tendency to price more finely
than PDs. There is one institutional reason to suspect this. In these primary dealer-intermediated markets,clients are aware that PDs can revise their own offers after observing client offers, possibly deliberately
undercutting them (Hortaçsu and Kastl, 2012). As such, coarse pricing can be particularly costly for clients.
The observed between-PD variation, however, still rejects client offers as a sole explanation. Consider
the MEP period as an example (Panel B of Figure 5). In this period, the estimated proportion of using the1/256ths pricing grid reached 100% for Morgan Stanley, the market share leader. Of course, this should be
the case only if both the PD (Morgan Stanley) and its clients always used the finest pricing grid. This result,however, is inconsistent with the assumption that clients tend to price more finely than PDs. Therefore,
there must be at least some between-PD differences in the tendency of coarse pricing.38

6Conclusion

This paper sheds new light on price competition in a massive, important, yet still understudied market:
QE reverse auctions of Treasury securities. To my knowledge, this paper is the first to document dealers’
practice of submitting coarsely priced offers in this market. As such, it complements the work of Song and
Zhu (2018), who conduct auction theory-based analysis of dealer behavior in the Fed’s QE auctions. On
one hand, coarse pricing is on a downward trend. On the other hand, coarse pricing surged during theFed’s massive pandemic-driven purchases in March 2020. I also show that conditional on winning a QE auction, coarsely priced offers are more highly priced. Therefore, this practice has relevance for policymakers
designing and monitoring this market.
The cross-sectional analysis of pricing fineness reveals that it varies by the ease of precisely pricing the
38 Notably, this logic does not exclude the possibility that PDs learn from client offers; observing a larger number ofclient offers might help top PDs to price more finely than others.

23security. Offer prices are more finely priced when valuation uncertainty is limited (e.g., low volatility) andwhen the security is finely priced in the secondary market prior to the auction. This paper also documents
strong association between PD market share and pricing fineness—offers submitted by larger-market-share
PDs are more finely priced. The relationship is economically significant; my benchmark specification indicates a one percentage point increase in market share translates into a 2.39 percentage point increase in theestimated proportion of using the finest 1/256ths grid. Collectively, my results are consistent with Grossman et al.’s (1997) theory that information costs of increasing pricing precision lead to coarse pricing ofdealers. In contrast, I do not find evidence that coarse pricing works as a coordinating mechanism for PDs
to maintain high spreads. Yet, the results do imply that competition plays a role in constraining coarse
pricing.
This paper thus illustrates the special importance of the topmost dealers in the Fed’s counterparty
framework—they can uniquely facilitate price competition and market efficiency in Fed operations. Also,from a theoretical standpoint, this paper presents rare micro-level evidence that information processing
costs can lead to a trade-off in pricing precision even among highly sophisticated investors.

24

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29

Figures
Figure 1: Price-end clustering of QE auction offers: Morgan Stanley vs. Credit Suisse
Panel A: Morgan Stanley
N. of (winning) offers = 16,245
1.8%
1.6%
1.4%
1.2%
1%
.8%

Probability of each d under the uniform distribution = 0.39%

.6%
.4%
.2%
0%
0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

136

144

152

160

168

176

184

192

200

208

216

224

232

240

248 255

80

88

96

104

112

120

128

136

144

152

160

168

176

184

192

200

208

216

224

232

240

248 255

d: Decimal part of the offer price as a multiple of 1/256

Panel B: Credit Suisse
N. of (winning) offers = 2,672
1.8%
1.6%
1.4%
1.2%
1%
.8%
.6%
.4%
.2%
0%
0

8

16

24

32

40

48

56

64

72

d: Decimal part of the offer price as a multiple of 1/256d ∈ {0,2,4,...,254}d ∈ {1,3,5,...,255}

This figure shows the distributions of the decimal part of the offer price as a multiple of 1/256 (d). The sampleis (winning) offers in QE auctions from August 17, 2010 to June 29, 2020. The data is obtained from the FRBNY.

30

Figure 2: QE operations of Treasury notes and bonds
Panel A: N. of QE acutions
80
70

QE4: 3/13~6/30/2020
QE2: 8/17/2010~(Sample period start)
MEP: 9/21/2011~

60

QE3:
12/12/2012~10/29/2014
Between QE3 and QE4

50
40
30
20
10
0

2010

2011

2012

2013

2014

2015

2016

2017

2018

2019

2020

Panel B: Purchased amount ($ billions)
750
700
650
600
550
500
450
400
350
300
250
200
150
100
50
0

2010

2011

2012

2013

2014

2015

2016

2017

2018

2019

2020

This figure shows the time series of the Fed’s QE purchases of Treasury notes and bonds from August 17,
2010 to June 29, 2020. The data source is the Treasury securities operation results disclosed by the FRBNY at
https://www.newyorkfed.org/markets/desk-operations/treasury-securities.

31

Figure 3: Price-end types of (successful) offers in QE auctions
Coarsest: d is in {0/256, 8/256, 16/256,..., 248/256}
Finest:d is in {1/256, 3/256, 5/256,..., 255/256}

60%

50%
March 2020

40%
April 202030%
April 2020

20%

March 2020

10%

0%

2010

2011

2012

2013

2014

2015

2016

2017

2018

2019

2020

In this figure, successful offers in QE auctions are aggregated monthly, and each displayed symbol’s size corresponds to the amount purchased in the month. For each month, the price-end types of (successful) offers arecomputed. The sample period is from August 17, 2010 to June 29, 2020.

32

Figure 4: Predicted probabilities of price-end types of QE auction offers based on Models 2 and 3of Table 12Panel A: Top five = 1 (Model 2)

Panel B: Top five = 0 (Model 3)

100%

100%

90%

90%

11.5

32.1

11.2
21.2
29.2

31.6

80%

80%

42.0

18.7

18.4

48.1

70%

70%

60%

60%

26.0

23.9

23.9

27.0

50%

26.9

28.6

50%
24.1

40%

23.2
19.1

40%

26.3

19.3

30%

d is in...

X256
X128
X64
X32

23.1

30%
16.2
14.3

20%
24.8

10%

25.2
17.7

42.8

20%

43.6
26.6

10%

19.1

14.4

0%

0%
Sec.

D

Sec.

Sec.

Sec.

D 64=1 D 128=1 D 256=1
32=1
Secondary-market price-end type

D

Sec.
32

Sec.

Sec.

Sec.

=1 D 64=1 D 128=1 D 256=1
Secondary-market price-end typeThis figure shows the predicted probabilities of price-end types of QE auction offers based on Models 2 and 3of Table 12. Model 2 uses the sample of Top f ive = 1, and Model 3 that of Top f ive = 0. The predicted probaSecondary

bilities are calculated for each of the four possible types of the secondary-market price ending: D32= 1,Secondary
Secondary
Secondary
Secondary
D64= 1, D128= 1, and D256
= 1. D32
takes the value of one if the decimal part of theTreasury security’s closing ask quote on the trading day preceding the QE auction is in X32 , and zero otherwise.
Secondary
Secondary
Secondary
D64= 1, D128= 1, and D256
= 1 are similarly defined.

33

Estimated prop. of the finest grid use

Estimated prop. of the finest grid use

Figure 5: Market share and the predicted proportion of using the finest 1/256ths pricing grid
Panel A: QE2

Panel B: MEP

100%

100%

90%

90%

80%

80%

70%

70%

60%

Goldman Sachs

40%

40%

UBS
Credit Suisse

20%

HSBC
RBS/NatWest
NomuraJefferies
Barclays
J.P. Morgan
DeutscheBank
BofA/Merrill
RBC
MizuhoDaiwa
BNP Paribas

10%
0%
0%

5%

10%

Morgan Stanley

MizuhoNomuraJefferies

20%

J.P. Morgan
RBC
BofA/Merrill
SocieteGenerale
HSBC
Daiwa
BNP
Paribas

10%

20%

25%

30%

35%

0%

Citigroup

90%

Societe Generale

60%

25%

30%

35%

40%
J.P. MorganBankof NovaBank
ScotiaDeutscheJefferies
Mizuho
UBS

30%
20%

SocieteGenerale
Daiwa
BNP Paribas

25%

30%

35%

Barclays
Bank of Nova Scotia
RBS/NatWest

10%

0%

Goldman Sachs

Citigroup
Morgan Stanley
BofA/Merrill
Fargo Pierpont
RBC WellsAmherst
MizuhoCredit Suisse
Deutsche
Bank
Daiwa
BMO
Jefferies
NomuraTDJ.P. Morgan
CantorFitzgerald
BNP
Paribas

50%

40%

10%

20%

HSBC

70%

Barclays
NomuraCantor Fitzgerald
RBC
HSBC
Credit Suisse
RBS/NatWestBofA/Merrill

20%

15%

80%

Goldman Sachs

30%

10%

Panel D: QE4

BMO

80%

50%

5%

100%

Morgan Stanley

60%

UBS

30%

Panel D: QE3 (including QE pause)

70%

Barclays
Credit Suisse
Goldman Sachs
RBS/NatWest

0%
15%

100%
90%

Cantor Fitzgerald

50%

Cantor Fitzgerald

30%

Deutsche Bank

60%

Citigroup

50%

Morgan Stanley

Citigroup

0%
0%

5%

10%

15%

20%

Market share

25%

30%

35%

0%

5%

10%

15%

20%

Market share

The x-axis measures the (traded amount-based) market share in QE auctions in the sub-period. The y-axis
represents the estimated proportion of using the finest 1/256ths pricing grid. A PD is included in a panel only
if it has at least 100 winning offers in the sub-period and if it was designated as a PD at the beginning of the
sub-sample period. Each panel includes the fitted line.

34

Tables
Table 1: Primary dealer list
Name

Parent location

Goldman Sachs
Morgan Stanley
Citigroup
Barclays
BofA/Merrill
HSBC
Credit Suisse
RBS/NatWest
Deutsche Bank
J.P. Morgan
BMO
NomuraBNP Paribas
Daiwa
Jefferies
Bank of Nova Scotia
RBC
Amherst Pierpont
MizuhoTD
Wells Fargo
UBS
Societe Generale
Cantor Fitzgerald
MF Global

Domestic
Domestic
Domestic
Foreign
Domestic
Foreign
Foreign
Foreign
Foreign
Domestic
Foreign
Foreign
Foreign
Foreign
Domestic
Foreign
Foreign
Domestic
Foreign
Foreign
Domestic
Foreign
Foreign
Domestic
Domestic

PD designation date(post-8/17/2010)

10/4/2011–

10/4/2011–
5/6/2019–
2/11/2014–
4/18/2016–
2/2/2011–
2/2/2011–10/31/2011

Pilot program primary dealers (July 2013–July 2014)
CabreraG.X. Clarke
Loop
Mischler

Amount sold ($ billions)N. of offers accepted595.787
503.859
256.365
253.379
253.098
207.759
190.467
187.855
186.736167.096126.949
119.014114.22998.55590.03489.31488.63885.40567.25666.32156.64856.20736.13517.1742.847

8,612
16,245
4,103
6,915
5,725
2,930
2,672
2,820
4,112
2,836
2,896
2,782
11,117
1,577
1,903
1,711
1,610
1,572
1,637
424
883
1,607
1,531
1,338
110

0.477
0.315
0.044
0.023

162
154
23
15

Primary dealers during my sample period (2010Q3–2020Q2) are listed. Note that four dealers participated in the primary dealer pilot program in July 2013–July 2014. See Internet Appendix B.1 for data sources of the parent location andperiod as a primary dealer. The amount sold and the number of winning offers are based on my sample QE auctionsof Treasury notes and bonds.

35

Table 2: Timeline of major events in the Fed’s QE purchases of Treasury securities
Date
March 18, 2009

Event
Announcement of “QE1” of Treasury securities: The Fed would purchase up to$300 billion of Treasury coupon securities in the next six months.
Purchases of Treasury coupon securities began.
QE1 ended.
The FOMC announced the plan to reinvest proceeds of maturing Treasury securities, agency debt, and agency MBS in Treasury coupon securities.

PeriodThe Fed started disclosing detailed operation result data (roughly two years after
the operation date).
Announcement of “QE2”: The Fed would purchase $600 billion of Treasury
coupon securities by June 2011.
The FOMC announced the end of QE2 and the plan to reinvest proceeds of maturing debt in Treasury coupon securities.

QE2

Announcement of the Maturity Extension Program: The Fed would purchase $400
billion of long-term Treasury securities (maturing in 6 to 30 years) based on proceeds from selling short-term ones (maturing in less than 6 years) by June 2012.
The termination date of the Maturity Extension Program was extended to December 2012. The Fed would continue the purchases of long-term Treasuries (andsales of short-term Treasuries) at the same pace.

MEP

December 12, 2012

Announcement of “QE3” of Treasury securities: The Fed would continue the purchases of long-term Treasury securities, but unlike in the previous Maturity Extension Program, it would not match the purchase amounts with the proceeds fromselling short-term securities.

QE3

October 29, 2014

QE3 ended. However, the Fed would continue reinvesting in Treasury coupon
securities to maintain its balance sheet size at $4.5 trillion.
The FOMC announced the intention of initiating the balance sheet normalization
program (reducing reinvestment in Treasury securities) this year, if the economic
condition allows.
The FOMC announced the start of the balance sheet normalization program in
October 2017.
The balance sheet normalization program was concluded. The Fed would reinvest
up to $20 billion per month in Treasury securities.

QE
pause

Beginning of QE4” round: The Fed would purchase Treasury securities of various maturities to address highly unusual disruptions in Treasury financing markets associated with the coronavirus outbreak.” The massive purchases started on
March 13, 2020.
Announcement of QE4: The Fed would purchase at least $500 billion of Treasury
securities “over coming months.”

QE4

March 25, 2009
October 29, 2009
August 10, 2010

August 17, 2010
November 3, 2010
June 22, 2011

September 21, 2011

June 20, 2012

June 14, 2017

September 20, 2017
July 31, 2019

March 12, 2020

March 15, 2020

Sources: Announcements and events listed on the website of the Federal Reserve.

36

Table 3: Descriptive statistics of QE auctions
Panel A: N. of QE auctions per auction date
Mean number of Distribution of the number of QE auctions
PeriodQE auctions
1
2
3 4 5
6
Total
QE2
1.019
159
3
0 0 0
0
162
MEP
1.049
176
9
0 0 0
0
185
QE3
1.012
343
4
0 0 0
0
347
QE pause
1.017
58
1
0 0 0
0
59
QE4
2.493
39
9
4 4 5 12
73
Total
1.153
775 26 4 4 5 12826
Panel B: N. of eligible, included, and purchased securities per QE auction
Eligible securitis
Included securities
Purchased securities
PeriodN
Mean
S.d.
Median Mean
S.d.
Median Mean
S.d.
Median
QE2
165
27.4
6.08
28.0
25.2
5.28
26.0
13.2
5.20
13.0
MEP
194
19.1
4.11
19.0
16.5
4.54
17.0
13.6
4.02
14.0
QE3
351
22.2
3.07
22.0
18.3
2.76
19.0
12.9
4.64
13.0
QE pause
60
38.0
7.84
40.0
33.8
8.56
36.5
11.4
6.41
10.0
QE4
182
60.0
30.41
52.0
53.9
29.12
49.0
33.8
19.68
29.0
Total
952
30.7
20.42
24.0
26.9
19.23
20.0
17.0
12.61
14.0
Panel C: Submitted and purchased amounts per QE auction
Submitted amount ($ milllions) Accepted amount ($ millions)
PeriodN
Mean
S.d.
MedianMean
S.d.
MedianQE2
165 20,641
8,481
20,9495,182 2,414
6,260
MEP
194
9,223
4,914
6,486
3,193 1,406
2,512
QE3
351
8,004
4,250
5,870
2,264 1,268
1,575
QE pause
60
9,549
4,172
9,530
1,768
520
1,801
QE4
182 18,797 12,725
14,2708,477 5,967
6,000
Total
952 12,603
9,227
10,426
4,116 3,797
3,000

Offer-to-cover ratio
Mean
4.996
2.838
3.775
5.549
2.344
3.634

S.d.
4.212
0.599
1.373
2.279
0.853
2.301

Median
3.899
2.774
3.521
5.189
2.104
3.159

Panel D: Winning offers and dealers per QE auction
Period

N

QE2
MEP
QE3
QE pause
QE4
Total

165
194
351
60
169
939

N. of winning offers
Mean
94.2
128.4
80.1
30.9
115.8
95.9

S.d.
55.3
58.5
42.9
23.8
66.0
58.3

Median
89
117
72
24
110
86

N. of winning dealers

Mean N. of winning offers
per winning dealer

Mean
15.2
16.2
15.6
11.0
19.3
16.0

Mean
5.96
7.71
5.03
2.65
5.66
5.71

S.d.
3.33
3.64
3.88
4.33
4.09
4.26

Median
16
17
16
11
20
17

S.d.
3.12
2.75
2.30
1.35
2.59
2.86

Median
5.61
7.38
4.64
2.16
5.13
5.18

This table presents desctiptive statistics of QE auctions of Treasury notes and bonds held from August 17, 2010
to June 29, 2020. The sample period is divided into five sub-periods based on QE phases: QE2 (8/17/2010–
9/19/2011), MEP (9/23/2011–12/10/2012), QE3 (12/13/2012–10/27/2014), QE pause (2/23/2016–3/3/2020),
and QE4 (3/13/2020–6/29/2020). Panel A reports the number of separate QE auctions (based on the Fed’s
operation ID) that the Fed conducted per QE auction date. Panels B–D report QE auction-level characteristics.
The sample size for Panel D is slightly smaller because there were 13 instances in which two separate QE auctions
targeting the same set of Treasury securities were held on the same date. Panel D treats those pairs of QE
auctions as a single observations, because the publicly disclosed offer-level data does not indicate which of thetwo auctions each offer belongs to in such cases.

37

Table 4: Descriptive statistics of (accepted) offers in QE auctions
N

QE2
15,549MEP
24,912QE328,126

QE pause
1,857

QE4
19,578

Total
90,022

Offer size ($ millions)
Mean
S.d.
Min
Median
Max

55
121.1
1
25
5,008

24.9
78.2
1
5
1,875

28.3
65.5
1
10
1,500

57.1
129.8
1
25
2,150

78.8
128.1
1
49
4,749

43.5
99.7
1
15
5,008

On-the-runs vs. off-the-runs (%)
Off-the-run
On-the-run

89.0
11.0

96.0
4.0

95.8
4.2

100.0
0.0

100.0
0.0

95.7
4.3

Remaining maturities (years; %)
0–5
5–10
10–20
20–30

34.4
50.8
6.6
8.2

0.0
37.3
7.0
55.8

10.7
37.9
4.7
46.7

46.3
30.7
5.4
17.5

53.0
22.7
2.1
22.2

21.8
36.5
5.1
36.6

Security types (%)
2Y notes
3Y notes
5Y notes
7Y notes
10Y notes
30Y bonds

1.0
10.2
18.3
22.6
23.8
24.0

0.0
0.0
0.0
15.3
18.3
66.4

0.0
0.0
6.3
15.7
21.6
56.5

7.5
8.5
18.1
22.5
19.3
24.0

8.6
11.8
18.8
17.5
18.0
25.2

2.2
4.5
9.6
17.3
20.3
46.2

Market share of the primary dealer (%)
Top five dealers
Non-top five dealers

33.6
66.4

57.0
43.0

56.1
43.9

31.1
68.9

37.4
62.6

47.9
52.1

This table reports the types of the sample offers for QE auctions of Treasury notes and bonds from August 17, 2010 toJune 29, 2020. Primary dealer market shares are measured based on the trade amount in QE auctions in the sub-period.

38

Table 5: Price-end clustering on coarser grids
Panel A: Regression resultsAll

Sub-period

PD market share(1)
0.839***(0.017)QE2(2)
1.155***(0.024)MEP(3)0.794***(0.020)QE3(4)0.584***(0.017)

QE pause(5)0.571***(0.052)QE4(6)1.037***(0.048)Top five(7)
0.493***(0.012)

Non-top five(8)
1.157***(0.023)(7) - (8)-0.665***(0.026)D64

0.542***(0.008)0.872***(0.020)0.536***(0.016)0.492***(0.014)0.378***(0.042)0.374***(0.013)0.332***(0.007)0.735***(0.011)-0.403***(0.013)D128

0.163***(0.004)0.322***(0.011)

0.093***(0.007)0.156***(0.008)0.196***(0.025)0.134***(0.007)0.117***(0.005)0.206***(0.005)-0.089***(0.007)

Constant-0.213***(0.002)-0.334***(0.002)-0.189***(0.003)-0.174***(0.003)-0.168***(0.012)-0.210***(0.004)-0.132***(0.003)-0.288***(0.002)0.156***(0.003)N
Adjusted R2

256
0.980

256
0.967

256
0.956

256
0.937

256
0.542

256
0.910

256
0.954

256
0.981

D32 − D64

0.297***[265.535]

0.282***[79.976]

0.259***[107.412]

0.093***[18.534]

0.194***[9.090]0.663***[181.815]

0.161***[154.367]

0.423***[264.990]

D64 − D128

0.378***[1977.001]

0.550***[578.012]

0.443***[684.574]

0.335***[467.250]

0.182***[15.948]0.240***[296.327]

0.215***[866.277]

0.528***[1894.439]

21.21
15.36
17.15
46.28

5.15
13.76
14.92
66.17

13.53
33.82
26.37
26.28

D32

Panel B: Estimated proportions in which different pricing grids are used (%)
grid-32nds
9.52
9.04
8.28
2.96
6.19
grid-64ths
24.22
35.2
28.36
21.47
11.63
grid-128ths
20.89
41.23
11.84
19.99
25.09
grid-256ths
45.38
14.53
51.53
55.58
57.08

C Panel A reports the results of estimating Specification 2. The dependent variable, Percentd − 0.390625, is the percentage ofoffers with price endings being d among all offers, minus 0.390625. The right-hand side variables are D32 , D64 , and D128 , whichtake the value of one if the price ending d belongs to X32 , X64 , and X128 , respectively, and zero otherwise. Column 1 uses all
sample (winning) offers. In Columns 2–6, the ratios are calculated separately for each sub-period: QE2, MEP, QE3, QE pause,and QE4. Columns 7 and 8 repeat the analysis for offers of top five PDs and non-top five PDs, respectively. PDs are ranked
based on trade amounts in QE auctions in each sub-period. Column 9 tests the differences in the coefficients between Columns
7 and 8. Heteroskedasticity-robust standard errors are reported in parentheses. At the bottom, differences between coefficientsare tested and their F statistics are reported in bracket parentheses. ***, **, and * indicate significance at 1%, 5%, and 10% levels.
Panel B reports the predicted proportions in which PDs used the 1/32nds, 1/64ths, 1/128ths, and 1/256ths pricing grids, basedon the method described in Section 4.3.

Table 6: Price-end type distributions conditional on a pricing grid used
Conditional on using grid-g = . . .
grid-32nds grid-64ths grid-128ths grid-256ths
grid-g

ϕ32= Pr[d ∈ X32 | grid-g]
grid-gϕ64= Pr[d ∈ X32 | grid-g]
grid-gϕ128 = Pr[d ∈ X128 | grid-g]
grid-gϕ256 = Pr[d ∈ X256 | grid-g]

1
0
0
0

0.5
0.5
0
0

0.25
0.25
0.5
0

0.125
0.125
0.25
0.5

This table summarizes the conditional probabilities of price ending d in X32 , X64 , X128 , and X256 , for each of the four pricing
grids, grid-32nds, grid-64ths, grid-128ths, grid-256ths.

39

Table 7: Definitions of the variables for the analysis of the determinants of pricing fineness
Variable
D32
D64
D128
D256
Top f ive
CheapnessOn-the-run
Maturity0-5
Maturity5-10
Maturity10-20
Maturity20-30
Bond
Bid-askVolatility
Ln(offer amount)
Ln(outstanding)Ln(total purchases)
Offer-to-cover

Description
A dummy variable that takes a value of one if the price ending is in X32 ={0/256, 8/256, 16/256, . . . , 248/256}
A dummy variable that takes a value of one if the price ending is in X64 ={4/256, 12/256, 20/256, . . . , 252/256}
A dummy variable that takes a value of one if the price ending is in X128 ={2/256, 6/256, 10/256, . . . , 254/256}
A dummy variable that takes a value of one if the price ending is in X256 ={1/256, 3/256, 5/256, . . . , 255/256}
A dummy variable that takes the value of one if the PD is a top five dealer based on the
trade amount in QE auctions in the sub-period
Yield-curve-implied price minus market mid-price, divided by market mid-price (basedon the previous trading day’s close price; in percent; winsorized at the 2.5% and 97.5%)
A dummy variable that takes a value of one if the security is an on-the-run security
A dummy variable that takes a value of one if the remaining maturity is in (0,5) years
A dummy variable that takes a value of one if the remaining maturity is in [5,10) years
A dummy variable that takes a value of one if the remaining maturity is in [10,20) years
A dummy variable that takes a value of one if the remaining maturity is in [20,30) years
A dummy variable that takes a value of one if the security is a Treasury bond
Bid-ask spread, divided by the mid quote (based on the previous trading day’s
end-of-the-day bid and ask quotes; in percent; winsorized at the 2.5% and 97.5% levels)
Standard deviation of the security’s returns in the previous five trading days (percent;winsorized at the 1% and 99% levels)
Natural logarithm of the amount of the offer
Natural logarithm of the publicly-held outstanding par value of the offered CUSIP
Natural logarithm of the total purchase amount of the QE auction
Offer-to-cover ratio of the QE auction (winsorized at the 2.5% and 97.5% levels)

Data sources
FRBNY
FRBNY
FRBNY
FRBNY
FRBNY
CRSP
TreasuryDirect
TreasuryDirect
TreasuryDirect
TreasuryDirect
TreasuryDirect
TreasuryDirect
CRSP
CRSP
FRBNY
CRSP
FRBNY
FRBNY

This table lists the definitions and data sources of the variables used in the subsequent cross-sectional analysis of price-end
fineness of QE auction offers.

Table 8: Descriptive statistics of the variables for the analysis of the determinants of pricing fineness
D32
D64
D128
D256
Top f ive
CheapnessOn-the-run
Maturity0-5
Maturity5-10
Maturity10-20
Maturity20-30
Bond
Bid-askVolatility
Ln(offer amount)
Ln(outstanding)Ln(total purchases)
Offer-to-cover

Mean

S.d.

Min.

Median

Max.

N

0.325
0.230
0.218
0.227
0.479
0.180
0.043
0.218
0.365
0.051
0.366
0.462
0.043
0.552
23.15324.01322.023
2.968

0.468
0.421
0.413
0.419
0.500
0.238
0.204
0.413
0.481
0.220
0.482
0.499
0.015
0.455
1.782
0.620
0.797
0.993

0.000
0.000
0.000
0.000
0.000
-0.251
0.000
0.000
0.000
0.000
0.000
0.000
0.012
0.027
20.72321.40919.052
1.459

0.000
0.000
0.000
0.000
0.000
0.101
0.000
0.000
0.000
0.000
0.000
0.000
0.044
0.434
23.431
24.12222.032
2.773

1.000
1.000
1.000
1.000
1.000
0.723
1.000
1.000
1.000
1.000
1.000
1.000
0.074
2.079
29.24225.04123.942
5.778

90,02290,02290,02290,02290,022
89,96490,02290,02290,02290,02290,02290,022
89,96489,28190,022
85,97490,02290,022This table reports the descriptive statistics of the variables used in the subsequent cross-sectional analysis of price-end finenessof QE auction offers. See Table 7 for variable definitions. Cheapness, Bid-ask, Volatility, and O f f er-to-cover are winsorized atthe 2.5% and 97.5% levels.

40

Table 9: Ordered logit regression of the determinants of price-end fineness of QE auction offers
Panel A: Ordered logit regression of Price-end fineness
(1)
(2)

(3)

Coef.
0.895***(0.217)

Marg. eff.
0.149***(0.039)Coef.
0.929***(0.216)

Marg. eff.
0.151***(0.038)Coef.
0.966***(0.228)Marg. eff.
0.156***(0.038)-1.072***(0.226)-0.179***(0.029)0.429***(0.142)0.069***(0.021)0.252*(0.138)0.041*(0.022)On-the-run-0.385***(0.122)-0.062***(0.024)-0.532***(0.118)-0.086***(0.024)

Maturity0-5

0.623***(0.089)0.117***(0.020)0.410***(0.076)0.072***(0.016)

Maturity10-20-0.419***(0.113)-0.063***(0.018)-0.405***(0.152)-0.059***(0.019)Maturity20-30-0.456***(0.096)-0.068***(0.016)-0.139(0.134)-0.022(0.021)

Bond-0.443**(0.219)-0.072**(0.030)-0.462**(0.201)-0.075***(0.027)

Bid-ask

0.484(0.657)0.078(0.105)Volatility-0.484***(0.104)-0.078***(0.017)Ln(offer amount)-0.066(0.060)-0.011(0.010)Ln(outstanding)-0.006(0.039)-0.001(0.006)Ln(total purchases)-0.012(0.058)-0.002(0.009)Offer-to-cover

0.125***(0.027)0.020***(0.004)Top f ive

Cheapness

MEP

0.476
(0.309)0.070(0.052)0.919***(0.261)0.136***(0.050)0.900***(0.227)0.129***(0.043)QE30.812***(0.214)0.132***(0.038)1.011***(0.197)0.153***(0.035)0.930***(0.186)0.134***(0.033)

QE pause

0.898***(0.207)0.149***(0.037)0.871***(0.203)0.127***(0.033)0.749***(0.206)0.103***(0.032)QE4

0.270(0.193)0.038(0.028)0.238(0.172)0.029(0.022)

0.613***(0.212)0.081***(0.031)

N
Pseudo R2

89,9640.036

89,964
0.053(Continued)

41

85,8660.060

Table 9: Continued
Panel B: The association of Cheapness and Price-end fineness for Top five = 1 and Top five = 0
Marginal effect of Cheapness on the probability of. . .
Coefficient

Price-end fineness = 1(⇐⇒ D32 = 1)

Price-end fineness = 2(⇐⇒ D64 = 1)

Price-end fineness = 3(⇐⇒ D128 = 1)

Price-end fineness = 4(⇐⇒ D256 = 1)

-0.633***(0.210)0.112**(0.044)0.038***(0.011)-0.013(0.016)-0.136***(0.040)0.357***(0.045)-0.026(0.018)-0.157***(0.032)-0.173***(0.032)

Test of the difference in the marginal effects
χ2 statistics
17.33***

6.11**

22.92***

0.94

Sample: Top five = 1
Cheapness

Period dummies
N
Pseudo R2
Sample: Top five = 0
Cheapness

Period dummies
N
Pseudo R2✓
43,0770.017-1.540***(0.207)✓46,8870.017

This table estimates the ordered logit model in which the dependent variable is Price-end fineness, which takes the value of one
if D32 = 1, two if D64 = 1, three if D128 = 1, and four if D256 = 1. Panel A uses all sample (accepted) offers in QE auctions andreports the coefficients and marginal effects on the probability of Price-end fineness = 4 (i.e., D256 = 1). For the definitions of thevariables, see Table 7. In Panel B, the sample is split by Top five. In addition to the coefficients, the table reports the marginal
effects on the probabilities of Price-end fineness = 1, 2, 3, and 4. Standard errors are three-way clustered by CUSIP, auction date,and PD. They are reported in parentheses. Standard errors for marginal effects are obtained by using the delta method. At thebottom, the differences in the marginal effects between Top f ive = 1 and Top f ive = 0 are tested. To preferm this test, I pool thetwo sub-samples and run the ordered logit model in which the independent variables are Top five, Cheapness, the dummies forthe sub-periods, the interaction term of Top five and Cheapness, and those of Top five and the sub-period dummies. ***, **, and *indicate significance at 1%, 5%, and 10% levels.

42

Table 10: Price-end clustering on coarser grids of secondary market prices
Panel A: Regression resultsAll

Sub-period

(1)

QE2
(2)

MEP
(3)

QE3
(4)

QE pause
(5)

QE4
(6)

1.159***(0.017)1.426***(0.045)1.465***(0.030)1.481***(0.031)0.848***(0.067)0.708***(0.025)1.173***(0.015)1.304***(0.043)1.513***(0.034)1.500***(0.038)0.748***(0.059)0.768***(0.025)D128

0.187***(0.005)0.186***(0.013)0.074***(0.007)0.072***(0.006)0.350***(0.030)0.309***(0.010)Constant-0.338***(0.002)-0.388***(0.001)-0.391(.)-0.391(.)-0.287***(0.012)-0.262***(0.004)

256
0.990

256
0.947

256
0.978

256
0.976

256
0.674

256
0.927

-0.014[0.400]0.122*[3.807]-0.048[1.126]-0.019[0.146]0.100[1.315]-0.059*[2.950]0.986***[4052.203]

1.119***[621.373]

1.439***[1693.223]

1.427***[1397.315]

0.399***[38.994]

0.459***[310.069]

Secondary

D32

Secondary

D64

SecondaryN
Adjusted R2
Secondary− D64

Secondary− D128

D32

D64

Secondary

Secondary

Panel B: Estimated proportions in which different pricing grids are used (%)
grid-32nds
0.00
3.90
0.00
0.00
3.21
grid-64ths
62.66
71.60
90.56
90.75
25.51
grid-128ths
23.95
23.76
9.44
9.25
44.75
grid-256ths
13.38
0.74
0.00
0.00
26.53

0.00
27.46
39.55
32.99

Panel A tests the price-end clustering on coarser grids of my sample Treasury securities in the secondary market. For each
Treasury security purchased in a QE auction, I obtain that security’s closing ask price on the trading day preceding the QE
auction from Bloomberg. The regression specifications are identical to Specification 2, except that this analysis uses not offer
Secondaryprices in QE auctions but secondary market prices. The dependent variable, Percentd− 0.390625, is the percentage ofSecondary
Secondary
Secondaryprice endings being d, minus 0.390625. The right-hand side variables are D32, D64, and D128
, which take thevalue of one if the price ending d belongs to X32 , X64 , and X128 , respectively, and zero otherwise. Heteroskedasticity-robust

standard errors are reported in parentheses. At the bottom, differences between coefficients are tested and their F statistics arereported in bracket parentheses. ***, **, and * indicate significance at 1%, 5%, and 10% levels. Panel B reports the predictedproportions in which PDs used the 1/32nds, 1/64ths, 1/128ths, and 1/256ths pricing grids, based on the method described in
Section 4.3.

43

Table 11: Ordered logit regression of the determinants of price-end fineness of secondary market
pricesDependent variable: Price-end f inenessSecondary
Marginal effect on the probability of Price-end f inenessSecondary =...
Coefficient-0.179(0.136)

1Secondary(⇐⇒ D32= 1)
0.034(0.026)

2
Secondary(⇐⇒ D64= 1)-0.009(0.006)

3
Secondary(⇐⇒ D128= 1)-0.015(0.012)

4
Secondary(⇐⇒ D256= 1)-0.011(0.008)Maturity0-5

1.506***(0.117)-0.269***(0.019)-0.014(0.014)0.186***(0.012)

0.097***(0.014)

Maturity10-20-0.345**(0.145)0.081**(0.034)-0.042**(0.019)-0.030**(0.012)-0.009**(0.004)

Maturity20-30-0.435***(0.133)0.102***(0.031)-0.055***(0.017)-0.036***(0.012)-0.011***(0.004)Bond-0.131(0.135)0.025(0.026)-0.006(0.006)-0.011(0.012)-0.008(0.008)MEP

0.696***(0.118)-0.145***(0.024)0.059***(0.012)0.060***(0.009)0.026***(0.005)QE30.556***(0.118)-0.116***(0.024)0.050***(0.013)0.047***(0.009)0.019***(0.004)

QE pause

1.177***(0.177)-0.238***(0.035)0.075***(0.013)0.109***(0.017)0.054***(0.011)QE4

1.250***(0.133)-0.251***(0.027)0.075***(0.013)0.116***(0.012)0.059***(0.008)N
Pseudo R2

15,872
0.114

On-the-run

This table reports the result of estimating the ordered logit model of the price-end fineness of my sample Treasury securities
in the secondary market. For each Treasury security purchased in a QE auction, I obtain that security’s closing ask priceon the trading day preceding the QE auction from Bloomberg. Therefore, the outcome variable, Price-end f inenessSecondary ,Secondary
Secondary= 1, two if D64= 1,
Secondary
Secondary
three if D128
= 1, and four if D256
= 1. The remaining columns show the marginal effects on the probabilities ofPrice-end f inenessSecondary = 1, 2, 3, and 4. For the definitions of the explanatory variables, see Table 7. Standard errors areis the price-end fineness of the secondary market price and takes the value of one if D32

two-way clustered by CUSIP and auction date, and they are reported in parentheses. Standard errors for marginal effects are
obtained by using the delta method. ***, **, and * indicate significance at 1%, 5%, and 10% levels.

44

Table 12: Ordered logit regression of the price-end fineness of QE auction offers on the price-end
fineness of secondary market prices
Sample:Secondary

D64

Secondary

D128

Secondary

D256

Security-type controls
Period dummies
N
Pseudo R2

All(1)

Top f ive = 1(2)

Top f ive = 0
(3)

All
(4)

Top f ive = 1(5)

Top f ive = 0(6)-0.017(0.014)-0.023(0.015)-0.030(0.024)-0.014(0.012)-0.027(0.020)-0.018(0.022)0.594***(0.075)0.440***(0.096)0.734***(0.093)0.114***(0.042)0.065(0.062)0.132**(0.058)0.904***(0.137)0.695***(0.176)

1.164***(0.152)

0.336***(0.104)0.214(0.149)0.468***(0.117)

✓

✓

✓

✓
✓

✓
✓

✓
✓

90,0220.014

43,0960.018

46,9260.014

90,0220.031

43,0960.026

46,9260.045

This table reports the coefficients of the ordered logit regressin in which the dependent variable is Price-end fineness, whichtakes the value of one if the QE auction offer’s price ending is in X32 , two if it is in X64 , three if it is in X128 , and four if it is in
Secondary

X256 . D64

takes the value of one if the price ending of the Treasury security’s closing ask on the trading day preceding
Secondary

Secondary

the QE auction is in X64 , and zero otherwise. D128
and D256
are similarly defined. The secondary market priceend data come from Bloomberg. In Columns 4–6, the following scurity-type controls are included: On-the-run, Maturity0-5 ,Maturity10-20 , Maturity20-30 , and Bond. Standard errors are three-way clustered by CUSIP, auction date, and PD. ***, **, and *indicate significance at 1%, 5%, and 10% levels.

45

Table 13: Market share and the predicted proportion of using fine pricing grids
Panel A: Sub-period analysis
Dependent var.:
PeriodPricing grid256

Pricing grid128or256

QE2
(1)
1.006
(0.774)MEP(2)
2.395∗∗∗(0.293)QE3(3)3.384∗∗∗(0.859)QE4(4)2.970∗∗(1.069)

QE2
(5)
1.332
(0.975)MEP(6)2.071∗∗∗(0.519)QE3(7)2.158∗∗∗(0.504)

QE4
(8)
1.193
(0.993)

Constant

15.166∗∗(6.143)25.941∗∗∗(6.623)32.692∗∗∗(6.919)31.388∗∗∗(6.336)45.762∗∗∗(6.024)48.700∗∗∗(6.825)67.630∗∗∗(5.084)56.978∗∗∗(5.479)N
Adjusted R2

18
0.031

19
0.326

21
0.363

23
0.244

18
0.110

19
0.266

21
0.305

23
0.044

Market share

Panel B: Pooled analysis in levels
Dependent var.:

Pricing grid256

Market share(1)
2.393∗∗∗(0.313)

Excl.
GoldmanSachs(2)
2.431∗∗∗(0.386)

Period dummies
N
Adjusted R2

✓
81
0.383

✓
77
0.344

SampleAll

Pricing grid128or256

Excl.
MorganStanley(3)2.438∗∗∗(0.521)

Excl.
< 2% market
share
(4)
2.402∗∗∗(0.295)

✓
77
0.299

✓
55
0.486

(5)
1.836∗∗∗(0.405)

Excl.
GoldmanSachs(6)
1.703∗∗∗(0.435)

Excl.
MorganStanley(7)
2.264∗∗∗(0.490)Excl.
< 2% market
share
(8)
1.665∗∗∗(0.439)

✓
81
0.360

✓
77
0.317

✓
77
0.335

✓
55
0.444

All

Panel C: Pooled analysis in first differences
Dependent var.:∆Pricing grid256∆Market share(1)
2.431∗∗∗(0.457)

Excl.
GoldmanSachs(2)
2.413∗∗∗(0.523)

Period dummies
N
Adjusted R2

✓
63
0.278

✓
60
0.266

SampleAll∆Pricing grid128or256

Excl.
MorganStanley(3)2.297∗∗(0.893)

Excl.
< 2% market
share
(4)
2.574∗∗∗(0.330)

✓
60
0.156

✓
38
0.473

(5)
2.137∗∗∗(0.400)

Excl.
GoldmanSachs(6)
2.284∗∗∗(0.471)

Excl.
MorganStanley(7)2.260∗∗∗(0.736)Excl.
< 2% market
share
(8)
1.908∗∗∗(0.410)

✓
63
0.383

✓
60
0.375

✓
60
0.330

✓
38
0.439

All

This table reports the results of the OLS regressions in which the dependent variable is the predicted proportion of using thefinest 1/256ths pricing grid (Price grid256 ) or that of using the pricing grids of either 1/128ths or 1/256ths (Price grid128or256 ).
A PD is included in the sample if it has at least 100 winning offers in the sub-period and if it was designated as a PD at the
beginning of the sub-sample period. Market share is the trade amount-based market share of the PD in the sub-period. In PanelA, heteroskedasticity-robust standard errors are reported in parentheses. In Panels B and C, standard errors clustered for PD
are reported in parentheses. ***, **, and * indicate significance at 1%, 5%, and 10% levels.

46

Table 14: Coarse pricing and the level of prices among accepted offers in QE auctions
Panel A: Summary statistics of Price diff
N
Mean
S.d.
Min
Price diff
86,611
3.61
3.93
025%tile0.81

Median2.62

75%tile4.82

Panel B: Regressions of Price diff
(1)
D32
0.282*(0.139)(2)
0.278**(0.127)(3)
0.138(0.117)(4)
0.127(0.107)D64

0.247**(0.117)0.243**(0.108)0.119(0.110)0.108(0.100)D128

0.181**(0.073)0.178**(0.068)0.116*(0.062)0.109*(0.056)

Ln(offer amount)

0.035(0.052)Top five

CUSIP × QE auction FEs

0.050(0.045)-0.415**(0.177)-0.431**(0.165)

✓

✓

✓

✓

N
Adjusted R2

86,611
0.566

86,611
0.566

86,6110.569

86,611
0.569

D32 − D64

0.035
[0.484]0.066[0.978]0.035[0.481]0.066[1.062]0.020[0.195]0.002[0.001]

0.019[0.175]-0.001[0.000]D64 − D128

Max
18.50

Panel A shows the summary statistics of Price diff, which is the percentage difference (in basis points) between the offer priceand the minimum accepted price of offers for the same security in the same QE auction. Price diff is defined only when there
exist multiple winning offers for the security in the QE auction. This variable is winsorized at the 2.5% and 97.5%. Panel B
reports the OLS regression results with the dependent variable Price diff. All the models include CUSIP × QE-auction fixed
effects. D32 , D64 , and D128 take the value of one if the price ending d belongs to X32 , X64 , and X128 , respectively, and zerootherwise. Ln(Total assets) is the natural logarithm of the offer amount. Top five takes the value of one if the PD is one of the top
five dealers based on the trade amount in QE auctions in the sub-period, and zero otherwise. Standard errors are three-way
clustered by CUSIP, auction date, and PD. They are reported in parentheses. At the bottom, differences between coefficients aretested and their F statistics are reported in bracket parentheses. ***, **, and * indicate significance at 1%, 5%, and 10% levels.

47

Appendices
Appendix A

A brief history of the Fed’s QE purchases of Treasury securities

Table 2 lists major events of the Fed’s QE purchases of Treasury securities. The phases can be classified
as follows. First, on March 18, 2009, the Fed announced the commencement of purchasing up to $300
billion of Treasury coupon securities. The purchases of this so-called “QE1” round began on March 25 andended as planned six months later (on October 29).39 The Fed then announced on August 10, 2010 that it
would reinvest proceeds from maturing Treasury securities and other debt into Treasury coupon securities
to maintain its balance sheet size. The Fed resumed Treasury coupon security purchases following theannouncement.
Second, the “QE2” phase was announced on November 3, 2010. The announced purchase size of Treasury coupon securities was $600 billion. As planned, QE2 ended in June 2011, with the last purchase taking
place on June 22. Proceeds from maturing debt continued to be reinvested into Treasury securities.
Third, the Maturity Extension Program (MEP) was announced on September 21, 2011.40 While the
MEP still purchased long-term Treasury securities, the MEP differed from the preceding programs in thatit did not change the size of bank reserves—the purchases were funded by sales of shorter-term Treasury
securities (maturing in less than 3 years). The original plan was to purchase $400 billion of long-term
Treasury securities by June 2012. It was announced on June 20, 2012, however, that the Fed would continue
the purchases (and sales of shorter-term Treasuries) at the same pace until December 2012.
Fourth, the Fed announced the replacement of the MEP with “QE3” on December 12, 2012. Like QE1and QE2, QE3 did not entail sales of shorter-term Treasury securities. Unlike them, QE3 was open-ended
as it specified the (initial) monthly purchase amount, $45 billion, but neither the total purchase size nor thetermination date.41 QE3 was concluded on October 29, 2014.
Fifth, in the years following the QE3 conclusion, purchases of Treasury coupon securities were sporadic
and quite small. Moreover, purchases of long-term Treasuries were completely halted following the initia39 The purchases of agency MBS and agency debt continued until March 2010.
40 The MEP is also referred to as the “Operation Twist” after a similar program the Fed launched in 1961.
41 The announcement says, “If the outlook for the labor market does not improve substantially, the Committee willcontinue its purchases of Treasury and agency mortgage-backed securities” (https://www.federalreserve.gov/newsevents/pressreleases/monetary20121212a.htm).

48

tion of the balance sheet normalization program in October 2017. The balance sheet normalization program
was then concluded in August 2019, and the Fed resumed reinvesting proceeds of maturing securities inTreasury coupon securities (with the first purchase taking place on August 15, 2019). The reinvestment continued at the pace of roughly $15 billion per month until February 2020. The FOMC emphasized, however,that the purpose of Treasury security purchases during this period was to maintain a sufficient level of bankreserves, instead of restarting QE (Bernanke, 2022, pp. 248–252).
Lastly, the COVID-19 crisis led the Fed to launch massive operations in the Treasury market. On March
12, 2020, the Fed announced that, among other interventions, it would purchase Treasury securities ofvarious maturities from the next day. The massive new round of QE, which this paper dubs “QE4,” wasannounced on March 15. According to it, the Fed would purchase at least $500 billion of Treasury securities“over coming months.” While previous QE rounds had the main policy objective of stimulating the economy through lowering long-term rates, that of QE4 was different. The March 12 announcement clarified
that the purpose was “to address highly unusual disruptions in Treasury financing markets associated withthe coronavirus outbreak.”42

Appendix B

Dealer-level estimation of the proportions of pricing grids used

To quantify the extent of coarse pricing for each PD while controlling for offered security heterogeneity,
I employ a more elaborate version of Specification 2. The dependent variable, Percents,j,t,d − 0.390625,
is the percentage of offers with price endings being d among all offers for security type s submitted by
PD j during period t, minus 0.390625. More specifically, three basic Treasury security characteristics aretaken into account: on-the-run status, remaining maturity, and Treasury note vs. bond. Therefore, theratio is calculated for each possible combination of security type s, PD j, and period t. The full regression
42 https://www.newyorkfed.org/markets/opolicy/operating_policy_200312a

49

specification is:

Percents,j,t,d − 0.390625 = β 1 D32 + β 2 D64 + β 3 D128 + β 4 On-the-runs + β 5 Maturity0-5Y + β 6 Maturity10-20Y+ β 7 Maturity20-30Y + β 8 Bond + ( β 9 D32 + β 10 D64 β 11 D128 ) × On-the-run+( β 12 D32 + β 13 D64 + β 14 D128 ) × Maturity0-5Y+( β 15 D32 + β 16 D64 + β 17 D128 ) × Maturity10-20Y+( β 18 D32 + β 19 D64 + β 20 D128 ) × Maturity20-30Y+( β 21 D32 + β 22 D64 + β 23 D128 ) × Bond
+γZj,t + (δD32 + ζD64 + ηD128 ) × Zj,t + ϵi,d ,(5)where D32 , D64 , and D128 take the value of one if the cell’s price ending d (∈ D = {0, 1, 2, . . . , 255}) belongs
to X32 , X64 , and X128 , respectively, and zero otherwise; On-the-runs is the dummy variable for on-the-run
securities; Maturity0-5Y , Maturity10-20Y , and Maturity20-30Y are dummy variables for remaining maturities
being in (0, 5), [10, 20), and [20, 30) years, respectively; Bond is the dummy variable for Treasury bonds; Zj,tis the fixed effects for PD × period. (In this PD-level analysis I merge the QE-pause period into the QE3
period due to its small sample size.) Since Percents,j,t,d ’s are calculated from different numbers of offers, theregression weights the observations by the number of offers in the cell. The regression result is reported inTable A.1.
Based on Specification 5, the baseline security is an off-the-run Treasury note maturing in [5, 10) years,
which is the most common security type in my sample of QE auction offers. Moreover, by combining thecoefficients reported in Table A.1 with the coefficients of Zj,t and the interaction terms of Zj,t with D32 , D64 ,and D128 , I can predict the price-end type distributions at the PD × period level in the case of offering
the most common Treasury security type. These predicted PD × period-level values are then fed into the
maximum likelihood procedure detailed in Section 4.3 for estimating the proportions in which the four
pricing grids were used.
Table A.2 summarizes the estimated PD × period-level proportions in which the four pricing grids were50

used. There are two notable time-series patterns (which are in line with Table 5). First, PDs tend to pricemore finely over time. The median PD increases the use of the 1/256ths pricing grid from 15.6% in QE2 to29.6% in MEP and 48.5% in QE3. Second, the pricing fineness deteriorated in the QE4 period. The medianprobability of using the coarsest 1/32nds grid increased from 0% in QE3 to 18.2%.

Table A.1: OLS estimation of Specification 5
Coefficient
1.132***
1.038***
0.492***-0.062***0.054***-0.025**-0.027*-0.0290.185*
0.168
0.071*-0.440***0.192**
0.109**-0.165***
0.011
0.042
0.087***-0.001
0.033
0.415***
0.074-0.130***

D32
D64
D128
On-the-run
Maturity0-5Y
Maturity10-20Y
Maturity20-30Y
Bond
On-the-run × D32
On-the-run × D64
On-the-run × D128
Maturity0-5Y × D32
Maturity10-20Y × D32
Maturity20-30Y × D32
Maturity0-5Y × D64
Maturity10-20Y × D64
Maturity20-30Y × D64
Maturity0-5Y × D128
Maturity10-20Y × D128
Maturity20-30Y × D128
Bond × D32
Bond × D64
Bond × D128

Standard error(0.037)(0.023)(0.013)(0.020)(0.012)(0.012)(0.015)(0.018)(0.106)(0.106)(0.042)(0.073)(0.092)(0.053)(0.039)(0.077)(0.058)(0.033)(0.030)(0.034)(0.093)(0.055)(0.024)Zj,tZj,t × (δD32 + ζD64 + ηD128 )✓✓Number of observations. . .
before weighting
after weighting
Adjusted R2

161,024
23,045,632
0.324This table reports the OLS estimation of Specification 5. The dependent variable, Percents,j,t,d − 0.390625, is the percentage ofoffers with price endings being d among all offers for security type s submitted by PD j during period t, minus 0.390625. D32 ,
D64 , and D128 take the value of one if the price ending d belongs to X32 , X64 , and X128 , respectively, and zero otherwise. Onthe-run takes the value of one if the offered Treasury security is on-the-run. Maturity0-5Y , Maturity10-20Y , and Maturity20-30Y
are dummy variables for remaining maturities being in (0, 5), [10, 20), and [20, 30) years, respectively. Bond takes the value of
one if the offered security is a Treasury bond and zero if it is a Treasury note. Zj,t is the fixed effects for PD × period. Given
that the ratios are calculated from different numbers of offers, the OLS regression is weighted by the respective number ofoffers. Standard errors clustered by price ending (d) and PD (j) are reported in parentheses. ***, **, and * indicate significance
at 1%, 5%, and 10% levels.

51

Table A.2: Predicted proportions of using the 1/32nds, 1/64ths, 1/128ths, and 1/256ths pricing
grids for each PD
N

Mean

S.d.

Min

p25

Medianp75

Max

Panel A: Estimated proportion of using the 1/32nds grid
QE2
MEP
QE3
QE4

19
21
23
23

8.70
10.29
3.44
18.87

8.40
14.95
6.47
10.82

0.00
0.00
0.00
0.00

3.83
0.00
0.00
9.54

5.54
6.50
0.00
18.15

13.11
15.11
4.52
25.25

35.78
62.80
20.64
47.89

Panel B: Estimated proportion of using the 1/64ths grid
QE2
MEP
QE3
QE4

19
21
23
23

38.21
29.06
18.14
18.98

14.45
22.62
14.60
7.46

8.39
0.00
0.00
1.19

27.83
8.86
8.14
14.77

38.02
29.09
16.47
19.36

46.60
46.30
30.78
23.88

70.05
70.35
53.65
33.76

Panel C: Estimated proportion of using the 1/128ths grid
QE2
MEP
QE3
QE4

19
21
23
23

32.75
21.01
29.74
17.88

12.82
9.21
15.21
10.98

13.51
0.00
0.93
0.00

25.38
15.95
19.49
11.35

30.53
22.00
31.94
16.54

39.11
25.52
35.50
23.35

63.52
40.21
72.98
45.42

Panel D: Estimated proportion of using the 1/256ths grid
QE2
MEP
QE3
QE4

19
21
23
23

20.34
39.64
48.68
44.28

16.94
28.77
25.79
18.27

3.22
8.37
14.04
15.44

7.73
18.36
26.55
33.30

15.60
29.59
48.48
41.01

25.20
52.24
66.17
51.79

60.04
100.0099.07
88.00

This table reports the estimated proportions of using the 1/32nds, 1/64ths, 1/128ths, and 1/256ths pricing grids,
for each PD in each sub-period, in the case of offering an off-the-run Treasury note maturing in [5, 10) years.
Therefore, N refers to the number of sample PDs in the sub-period. Note that in this analysis the QE3 period
incorporates the QE-pause period due to a small sample size of the latter. A PD is included in the sample only if
it has at least 100 winning offers in the sub-period. See text for the method of estimating these proportions.

52

Internet Appendix for “Coarse Pricing in QE Auctions”

Table of contents
Internet Appendix A

Fitting a Fed-style yield curve model . . . . . . . . . . . . . . . . . . . . IA.1

Internet Appendix B

Dealer characteristics and coarse pricing . . . . . . . . . . . . . . . . . IA.2

B.1

Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IA.2

B.2

Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IA.5

Internet Appendix A

Fitting a Fed-style yield curve modelSong and Zhu (2018) estimate a yield curve model based on the Fed’s public information. Unless stated otherwise, I replicate their method to the greatest extent possible. The model is a standard cubic spline model.
There are two important choices to consider. First, some prefer setting a positive smoothness parameter,
which effectively prioritizes curve smoothness at the expense of the model’s fit. Following the baseline
approach of Song and Zhu (2018), I set the smoothness parameter to zero. The second consideration is thenumber and location of knots. Again, I employ their choice: 2, 5, 10, 20, and 30 years.
Like Song and Zhu (2018), I exclude Treasury bills and on-the-run securities from the yield curve estimation. One difference between their sample selection and mine is the remaining maturity. Song and
Zhu (2018) discard securities whose remaining maturities are less than one year. This selection makes sensein their setting; Treasury securities nearing maturity occasionally exhibit large, idiosyncratic price fluctuations, and during the period they examine, the Fed did not purchase any securities maturing in less than
1.5 years. In contrast, in my sample period, the shortest maturity purchased by the Fed was 32 days. I
therefore set a minimum remaining maturity restriction of one month.
While Song and Zhu (2018) use the Fed’s internal secondary-market price data (New Price Quote System), my data source is the CRSP Treasury data. For each QE auction, I estimate the yield curve using
the daily closing mid price of the previous trading day. I then measure Treasury securities’ cheapness bycomparing the model-implied prices and their actual closing mid prices on the day of the yield curve estimation. Note that cheapness is measured even for on-the-run securities, which are not included in the yield
curve estimation.

IA.1

Internet Appendix B
B.1

Dealer characteristics and coarse pricing

Data

Does the degree of pricing fineness relate to the PD’s characteristics (other than QE auction market share)?
To explore this question, I look at three dimensions of PD characteristics.
Location of the parent bank: First, I examine whether PDs have a foreign parent company or not,referring to the PD lists of He et al. (2017) and Giannone and Robotti (2022). Foreign takes the value of one
if the PD has a foreign parent bank, and zero otherwise.
Number of years as a primary dealer: Second, To shed light on varying experiences of PDs in the
Treasury market, I measure the number of years as a PD. One challenge is how to handle mergers & acquisitions, which have been pervasive from time to time in this industry. I thus checked the historical primary
dealer lists of the FRBNY. Table IA.1 summarizes the designation dates of domestic PDs, and Table IA.2
those of foreign PDs. The identification of the designation date is particularly challenging for some foreign
PDs. Therefore, for some foreign PDs, I assign both the baseline and earliest start dates. Old takes the valueof one if the (baseline) PD designation date is earlier than the median date, and zero otherwise. The use of
the earliest start dates, instead of the baseline dates, does not much change the result.
Balance sheet size: Third, I hand-collected each PD’s balance sheet information from Form X-17A-5,
as done by Gupta (2021). An advantage of using this form is that I can obtain balance sheet data of dealer
subsidiaries, as opposed to the group-wide consolidated financial data reported in 10-Ks, etc. Note that thisdata is also available for foreign bank-affiliated PDs because they are still incorporated as a U.S. subsidiary,and large U.S. broker-dealers are mandated to file (audited) Form X-17A-5. I was able to locate all mysample PDs’ Form X-17A-5, except for that of the Bank of Nova Scotia, although there exist some gap years
and non-reported items for some PDs.1 Ln(Total assets) is the natural logarithm of the total assets measured
at the beginning of each sub-period.

1 Ownership structures of PDs were examined using the National Information Center repository (https://www.
ffiec.gov/NPW). See Avraham et al. (2012) for its details.

IA.2

IA.3
MF Global filed for bankruptcy on 10/31/2011.

Citigroup was created as a result of a merger between Citicorp and Travelers Group in 1998. Citicorp owned
Citicorp Securities, which was designated as a primary dealer under the name of Citibank on 6/15/1961. Travelers
Group owned Salomon Smith Barney, which had roots in Salomon Brothers. Salomon Brothers was designated asa primary dealer on 5/19/1960.

Merrill Lynch Government Securities was a primary dealer from 5/19/1960 to 2/11/2009. In 2009, Merrill Lynch
was acquired by Bank of America, which also owned a primary dealer, Banc of America Securities. Banc of America Securities had roots in CRT Government Securities, which was designated as a primary dealer on 12/22/1987.

Notepilot program for July 2013–July 2014 are not included. For domestic PDs, the period as a PD is the same whether the start date is based on the baseline or earliest
one. See text for data sources.

This table summarizes domestic primary dealers during my sample period (2010Q3–2020Q2). The four (domestic) dealers that participated in the primary dealer

Morgan Stanley
Wells Fargo

12/4/1974–
6/18/2009–
5/19/1960–
2/2/2011–
10/31/2011
2/1/1978–4/18/2016–

8/1/2006–
5/19/1960–

Cantor Fitzgerald
Citigroup

Goldman Sachs
Jefferies
JP Morgan
MF Global

Period as a PD
5/6/2019–
5/19/1960–

Name
Amherst Pierpont
BofA/Merrill

Table IA.1: Domestic primary dealersIA.4

12/23/1988–

12/11/1986–
12/13/1990–

BNP Paribas

Credit Suisse

Daiwa
Deutsche Bank

12/11/1986–11/30/2007,
7/27/2009–12/28/1989–

12/11/1986–11/30/2007,
7/27/2009–
7/8/2009–
3/6/2000–
2/2/2011–

2/11/2014–
12/7/1989–

MizuhoNomuraRBC
RBS/NatWest

Societe Generale

TD
UBS

UBS Securities was designated as a PD on 12/7/1989. In 1998, SBC and UBS merged, with the newentity name being UBS. Before the merger, UBS Securities, a subsidiary of UBS, had been a primary
dealer since 12/7/1989. SBC also owned S. G. Warburg & Co., which had been a primary dealer
since 6/24/1988, as a result of its acquisition in 1995.

On 3/6/2000, RBS acquired NatWest, which owned a primary dealer Greenwich Capital. Greenwich Capital had been a primary dealer since 7/31/1984, and it was acquired by NatWest in 1996.
For a brief period (7/1/1999–10/31/2001), SG Cowen, a subsidiary of Societe Generale, was a primary dealer. Given the brevity of the period and the time gap, the start date is determined to bethat of SG Americas.

Nomura once quit as a primary dealer on 11/30/2007 and regained a primary dealer status on
7/27/2009. Given the brevity of the discontinuation period, the start date is considered to be thedate on which Nomura was first designated as a primary dealer (12/11/1986).

Deutsche Bank Government Securities was designated as a primary dealer on 12/13/1990. In 1999,
Deutsche Bank acquired Bankers Trust, which had been a primary dealer since 5/19/1960. (Notethat this acquisition occurred well after Deutsche Bank became a primary dealer.)
Marine Midland Bank, whose majority equity stake was owned by the Hongkong and Shanghai
Banking Corporation (current HSBC), acquired Carroll McEntee & McGinley on 12/2/1983. Carroll
McEntee & McGinley had been a primary dealer since 9/29/1976.
Following Fuji Bank’s acquisition, Kleinwort Benson Government Securities was renamed Fuji Securities on 12/28/1989. Kleinwort Benson was designated as a primary dealer on 2/13/1980. (Fuji
Securities was renamed Mizuho Securities on 4/1/2002 due to a merger of its parent bank.)

Barclays De Zoete Wedd was designated as a primary dealer on 12/7/1989.
For a brief period (2/15/2000–3/31/2002), BMO Nesbitt, a subsidiary of BMO, was a primary
dealer. Given the brevity of the period and the time gap, the start date is determined to be thatof BMO.
Paribas was designated as a primary dealer on 5/1/1997. The name was changed to BNP Paribas
on 9/15/2000 as a result of a merger.
On 12/23/1988, First Boston was merged into CS First Boston, which was established as a Londonbased joint venture of Credit Suisse and First Boston in 1978. CS Holding effectively controlled the
new CS First Boston with a 44.5% equity stake. (Credit Suisse’s ownership was further increased to
60% in 1990.) First Boston had been a primary dealer since 5/19/1960.

Note

This table summarizes primary dealers owned by foreign parent companies during my sample period (2010Q3–2020Q2). See text for data sources.

2/11/2014–
6/24/1988–

2/2/2011–

7/8/2009–
7/31/1984–

2/13/1980–

12/2/1983–

HSBC

9/29/1976–

12/11/1986–
5/19/1960–

5/19/1960–

5/1/1997–

5/1/1997–

Bank of Nova Scotia
BarclaysBMO

Name

Period as a primary dealer
Baseline
Earliest
start datestart date
10/4/2011–
10/4/2011–
12/7/1989–
12/7/1989–
10/4/2011–
10/4/2011–

Table IA.2: Primary dealers owned by foreign parent companies

B.2

ResultTable IA.3 shows the correlation matrix of PD-level variables. The results of pooled regressions are reported
in Table IA.4.

Table IA.3: Correlation matrix of PD-level variables
Price grid256
Price grid256

Price
grid128or256

Market share

Ln(Total assets)

ForeignOld

1

Price grid128or256

0.843***[0.000](81)

1Market share

0.457***[0.000](81)

0.428***[0.000](81)

1

Ln(Total assets)

0.254**[0.030](73)

0.271**[0.020](73)

0.615***[0.000](73)1Foreign–0.300***[0.006](81)–0.169[0.132](81)–0.313***[0.004](81)–0.158[0.181]
(73)

1

Old

0.211*[0.059](81)

0.076[0.498](81)0.363***[0.001](81)

0.488***[0.000](73)–0.343***[0.002](81)

1This table presents the correlation matrix of PD-level variables, pooling the data from the four sub-periods. Price grid256 is thepredicted proportion of using the finest 1/256ths pricing grid, and Price grid128or256 is that of using the pricing grid of either
1/128ths or 1/256ths. Market share is the trade amount-based market share of the PD in the sub-period. See Section B.1 for thedefinitions and data sources for the other variables. A PD is included in the sample only if it has at least 100 winning offers
in the sub-period. The number of observations used for calculating the pair-wise correlation is shown in round brackets andp-values in square brackets.

IA.5

Table IA.4: PD characteristics and the predicted proportions of using fine pricing grids
Panel A: Dependent variable: Pricing grid256
(1)
(2)
(3)
Market share
2.393∗∗∗(0.313)(4)(5)
2.084∗∗∗(0.414)7.513∗(4.112)

Ln(Total assets)-16.004∗(7.958)ForeignOld-8.603(7.469)12.025(7.330)1.492(5.738)MEP

18.436∗∗∗(4.625)17.370∗∗∗(5.909)17.700∗∗∗(5.991)17.689∗∗∗(5.999)18.514∗∗∗(4.790)QE329.844∗∗∗(5.301)29.541∗∗∗(6.276)28.799∗∗∗(5.437)28.769∗∗∗(5.505)30.178∗∗∗(5.385)QE426.378∗∗∗(5.230)28.257∗∗∗(6.796)23.528∗∗∗(4.825)24.874∗∗∗(4.981)26.156∗∗∗(5.367)

Constant

7.512(5.202)-169.328(106.113)

30.495∗∗∗(7.406)14.702∗∗∗(4.574)13.731(9.768)N
Adjusted R2

81
0.383

73
0.223

81
0.232

81
0.191

81
0.397

(9)

(10)
1.878∗∗∗(0.490)Panel B: Dependent variable: Pricing grid128or256
(6)
(7)
(8)
Market share
1.836∗∗∗(0.405)Ln(Total assets)

5.826(3.386)

Foreign-8.062(6.335)Old-3.277(6.752)
4.008(5.772)-3.981(5.357)MEP

6.910(5.653)5.591(6.360)

6.259(6.530)

6.199(6.537)6.890(5.751)

QE3

26.165∗∗∗(4.519)25.414∗∗∗(5.537)25.129∗∗∗(4.713)24.967∗∗∗(4.746)26.096∗∗∗(4.517)QE4

11.204∗∗(4.108)11.952∗∗(5.134)9.027∗(4.546)9.482∗∗(4.322)10.812∗∗(4.140)Constant

42.982∗∗∗(4.340)-93.763(86.682)

58.037∗∗∗(6.698)51.106∗∗∗(3.851)46.745∗∗∗(8.699)

81
0.360

73
0.226

81
0.185

81
0.158

81
0.354

N
Adjusted R2This table reports the results of the regressions in which the dependent variable is the predicted proportion of using
the finest pricing grid of 1/256ths (Price grid256 ) or that of using the pricing grids of either 1/128ths or 1/256ths(Price grid128or256 ). A PD is included in the sample only if it has at least 100 winning offers in the sub-period. Market
share is the trade amount-based market share of the PD in the sub-period. MEP, QE3, and QE4 are indicator variables
for the sub-periods. (QE2 is the reference category.) See Section B.1 for the definitions and data sources for the othervariables. Standard errors clustered for PD are reported in parentheses. ***, **, and * indicate significance at 1%, 5%,
and 10% levels.

IA.6