The Linkage between Primary and Secondary Marketsfor Eurozone Sovereign Debt: Free Flow or Bottleneck?

Alexander Eisl ∗ Christian Ochs † Nikolay Osadchiy ‡ Marti G. Subrahmanyam §

March 15, 2019

Abstract

In this paper, we investigate the consequences of interlinked sovereign bond marketsin the Eurozone for the transmission of yields, liquidity and market conditions, specif-ically, the impact of sovereign bond auctions on secondary markets as well as theirpotential feedback to the sovereign’s cost of debt. This linkage is established by theactions of financial intermediaries, the so-called primary dealers, who participate insovereign bond auctions, and are also active as market makers in the secondary mar-kets of multiple countries. We develop a model of financially constrained primarydealers that would like to buy newly issued bonds and may consider selling a propor-tion of their existing inventory, while providing liquidity in the secondary market atthe same time. Our model produces optimal inventory levels for existing- and newlyissued bonds that can be related to predictable price movements around sovereign bondissuances. Empirically, we find support for our model and its resulting hypotheses andpropositions, i.e., primary dealers tend to liquidate more liquid and more risky bondsfrom their inventory in order to be able to participate in sovereign bond auctions andminimize the impact of the auction on their portfolios. The ability of primary dealersto participate in these auctions may depend on different market conditions. Last, wefind that financial constraints impose costs for market participants that are, eventually,borne by the sovereign through the linkage between markets.

Keywords: Eurozone, Financial Intermediation, Sovereign Bond Market

JEL classification: G12, G15, G18, G23, H63

∗We thank Mario Bellia, Kurt Hornik, Rainer Jankowitsch, Stefan Pichler, Davide Tomio and ChristianWagner for their support with the data and helpful discussions on our analysis and interpretation of theresults. In addition, we are grateful to our anonymous interview partners and discussants at sovereigndebt management offices and financial institutions across the Eurozone, as well as the participants anddiscussants of the Annual Meeting of the German Finance Association 2018, the Applied Finance Conference2018 of the Financial Management Association, and the participants of the Debt Management Office of theUnited Kingdom 2018 seminar on sovereign debt. Eisl and Ochs acknowledge the financial support of theOesterreichische Nationalbank via Jubilaumsfondsgrant #15789. Subrahmanyam is grateful for the generoussupport of the NYU Stern Center for Global Economics and Business, the Volkswagen foundation and theAnneliese Maier Award of the Alexander von Humboldt foundation.

∗WU (Vienna University of Economics and Business), Department of Finance, Accounting and Statistics,Welthandelsplatz 1, 1020 Vienna, Austria; email: alexander.eisl@wu.ac.at

†WU (Vienna University of Economics and Business), Department of Finance, Accounting and Statistics,Welthandelsplatz 1, 1020 Vienna, Austria; email: christian.ochs@wu.ac.at

‡Emory University, Goizueta Business School, Department of Information Systems and Operations Man-agement, 1300 Clifton Road, Atlanta, GA 30322; email: nikolay.osadchiy@emory.edu

§New York University, Stern School of Business, Department of Finance, 44 West Fourth Street, Room9-68, New York, NY 10012; email: msubrahm@stern.nyu.edu

1

1 Introduction

Sovereign bonds are an important safe asset class for managing economic risk, and form a

significant proportion of many investor portfolios. At the level of the broader economy, debt

service expenses matter for public finance in determining government spending and the tax

burden for households and firms. For all stakeholders alike, the economic consequences of

the global financial crisis and the recent sovereign debt crisis in Europe have highlighted

the importance of well-functioning sovereign bond markets for the sound management of

government debt.

Sovereign bond markets across the Eurozone (and elsewhere) are designed around the

services of an exclusive group of financial intermediaries, the so-called primary dealers.1

While governments sell their bonds at regular auctions to these selected financial institu-

tions, they, in turn, undertake these purchases for future resale to their clients. In addition,

primary dealers are incentivized, or even required, by the government to maintain secondary

market liquidity. Many of these primary dealers are also active across the Eurozone. While

their activity is intended to facilitate secondary market bond trading and market liquidity,

the dealers’ inventory risk and funding costs determine their willingness and capacity to

participate in sovereign bond auctions, as well as their ability to provide immediacy in the

secondary market. Thus, adverse market conditions can propagate through this channel, po-

tentially affecting the portfolios of market participants, and the cost of debt for the sovereign.

In this paper, we investigate the consequences of interlinked primary- and secondary

markets across the Eurozone for the transmission of market conditions, i.e., the impact

of sovereign bond auctions on secondary markets as well as its potential feedback on the

sovereign’s cost of debt. Since inventory management may be time consuming, dealers may

engage in these activities some time before a bond auction, depending on their capital con-

straints, the market demand for other bonds and the costs for hedging. After the bond

auction, the dealers’ inventory risk may decrease and capital constraints may alleviate again

1Primary dealership is a privileged status for market participants and is part of a bidding system thatis adopted, to different degrees, by countries in the Eurozone. However, we use this term loosely whenreferring to members of a bidding group with special rights, incentives or obligations associated with thesovereign bond auction process.

2

due to inventory sales. As a consequence, bond yields in the secondary market may rise in

the run-up, and decrease again in the aftermath of a sovereign bond auction. Empirically,

these patterns of auction cycles and the associated implicit issuance cost for the sovereign

have been well documented.2

We develop a model of financially constrained primary dealers who would like to buy

newly issued bonds and may consider selling a proportion of their existing inventory in order

to free-up capital and manage their inventory, while providing liquidity in the secondary

market at the same time. Thus, we focus our analysis on the inventory management activ-

ities of market participants in anticipation of the auction, which determine the amplitude

of the auction cycle. Our model produces optimal inventory levels and trading volumes for

existing- and the newly issued bonds that can be related to predictable price movements

in the secondary market, in response to sovereign bond auctions. We find that the optimal

inventory levels depend on the cost of inventory, the funding conditions and the demand

volatility of the existing- and the newly issued bond. In particular, we contribute to the

analysis of auction cycles by modeling how primary dealers would manage their inventory

in anticipation of an auction, i.e., according to different market conditions and the charac-

teristics of the individual bonds in their portfolio.

Based on the theoretical predictions of our model, we empirically study how the relation

between the primary- and secondary markets, in the form of predictable price changes and

traded volumes, is affected by observed market conditions, and the primary dealers’ incen-

tives to participate in the sovereign bond auction, as well as their ability to manage their

portfolio risk and prevalent capital constraints, i.e., by considering their decision to buy or

sell a specific bond position, or engage in other costly inventory management- or funding

activities. Therefore, our empirical contribution extends from the aggregate market to the

level of the individual bond in the primary dealers’ inventory. We, thus, focus on the specific

role of primary dealers in the bond auction process, and draw conclusions regarding their

limited risk bearing capacity and funding capabilities.

2See Lou et al. (2013) for such evidence on US Treasury bond prices. Also, Beetsma et al. (2016) havedocumented predictable bond price changes in the Eurozone, and relate their findings to the limited riskbearing capacity of primary dealers.

3

Intuitively, primary dealers may want to purchase newly issued bonds on the primary

market in order to match the demand of the secondary market and generate a profit from

this transaction. When participating in bond auctions, however, primary dealers may face

different market conditions and uncertain market demand for bonds, as well as funding- and

inventory risk constraints. In order to be able to participate in bond auctions, they may

either have to sell-off part of their existing inventory and free-up liquidity, or borrow against

existing bond positions at a market-determined interest rate. In order to manage their risk

exposure, the primary dealers would also have to engage in costly hedging activities and/or

decide which bonds to sell from the existing inventory, i.e., depending on the anticipated

supply shock due to the auction and the current demand for the issued bond, the incre-

mental risk that the auctioned bond contributes to the primary dealers’ portfolio, and their

preferred level of inventory risk exposure.

According to market participants, derivatives markets do play an important role in man-

aging this risk exposure and for staying within funding constraints. For example, primary

dealers may borrow against existing positions (”repo”) and virtually eliminate the basis risk

of the bond delivery by entering into a futures contract with the same maturity as the repo

transaction at the same time. Due to differences in the liquidity of bonds, some bonds may

be sold at a lower discount than others, or are cheaper to borrow against i.e., there may

be different trading and funding costs associated with the various bonds.3 In principle, if

derivatives markets are well-functioning, risks can be hedged efficiently and funds may be

raised at low cost since government bonds in the Eurozone are perceived to be relatively

secure investments. However, most countries’ derivatives markets for sovereign bonds are

less developed. Thus, since German bonds are widely perceived as almost cash-equivalent

and can be borrowed against, the Bund futures contract is frequently used to hedge a wide

variety of market positions.4 This approach, however, makes hedging imperfect for market

3For funding purposes, general collateral repo (”GC repo”) transactions accept a basket of selected bondsfor delivery of the contract, i.e., based on defined liquidity and risk criteria. This basket of deliverablebonds and the corresponding repo rate determine the secured borrowing cost for primary dealers.

4Besides Germany, other large Eurozone countries such as Spain, Italy, and France also maintain establishedderivatives markets, but the quality and access to these countries’ derivative instruments has varied histor-ically. For example, in Italy, the exchange-traded sovereign bond futures market has been launched severaltimes after periods of unavailability of derivatives instruments for the sovereign bonds.

4

participants since they have to bear basis risk, based on the uncertainty of the spread be-

tween the German Bund and the sovereign bonds of the particular country. The ability to

lay-off risk and borrow against existing bond positions is, hence, dependent on the respective

countries’ derivatives market conditions and also on government bond spreads.5

Generally, primary dealers would only like to participate in the sovereign bond auction

if the purchase of the newly issued bond is lucrative for them. If the dealer purchases too

little at the bond auction, she may forgo the profits of the new issuance, i.e., there are op-

portunity costs associated with the auctioned bond. In addition, the dealer may incur costs

when purchasing and holding the new bond issue. For example, these costs may be due

to margin and risk constraints when funding the position, or due to regulatory constraints

on leverage and liquidity ratios, e.g., as imposed by the European Central Bank under the

Basel framework. However, dealers may already have existing positions in bonds that are

correlated with the auctioned bond. At the same time as they buy sovereign bonds at the

auction, primary dealers are often obliged to make markets for several bonds, and are of-

fered additional incentives to be particularly active and efficient in providing liquidity, i.e.,

their performance as market makers frequently has a direct feedback effect on their status

and benefits or obligations as primary dealers.6 Since primary dealers make markets, they

can use their existing inventory to make a profit by selling bonds at the ask-price to other

market participants. If the dealer does not have the bond required by market participants

available in her inventory, though, she may buy them at the bid price and sell it again at

the ask price. Hence, there are costs of holding too little inventory that arise due to the

reduced ability of the market maker to generate a profit from the existing bonds in terms of

the bid-ask spread. On the other hand, if the dealer would like to participate in the auction

and needs to free-up liquidity or reduce her inventory risk exposure in anticipation of the

new bond position, she has to sell a proportion of the existing inventory at bid-prices. In

the simplest case, a primary dealer may, thus, incur a loss in terms of the bid-ask spread

instead of earning it, i.e., there are also cost of holding too much inventory.7

5See Pelizzon et al. (2016) and Pelizzon et al. (2018) for details.6For example, published performance evaluations and primary dealer rankings are important marketingand cross-selling tools for market participants. In addition, the market makers’ access to primary dealerincentives is often contingent upon secondary market-making performance.

7Alternatively in this example, the primary dealer may again borrow against her existing position or enterinto a futures contract to hedge the risk. Then, the cost of holding ”too much” inventory would be that of

5

From the perspective of the sovereign, auction cycles are relevant to determine the cost

of debt, and the burden incurred to the tax payer. When bond auctions distort secondary

markets and yields go up in the run-up of the auction, this may place an implicit issuance

cost burden on the sovereign. Consequently, the sovereign debt manager may seek a strategy

to reduce these cost. First, the sovereign debt manager may want to maintain some flexibil-

ity when conducting auctions, i.e., set precautionary measures or react to market conditions

when auction cycles are expected to be more pronounced. For example, maintaining a

cash buffer would allow debt managers to smooth the government’s budget needs over time.

Also, debt managers may change the frequency of auctions, i.e., issue smaller volumes more

frequently, or conduct tap-issues in support of an auction, i.e., sell small proportions of

the issuance directly to the secondary market in order to minimize the concentration risk

and impact of sovereign bond auctions. These considerations potentially make a debt man-

ager’s issuance strategy less susceptible to unfavorable market conditions. At the same time,

though, the debt manager may want to refrain from opportunistic behavior. For example,

markets may demand a premium for the increased uncertainty when debt managers start

shifting a proportion of the issuance volume to other auctions if market conditions are not

benevolent for the government. In particular the cancellation of an auction may lead to a

reputation- or commitment cost for the government. Debt managers are often concerned

about the uncertainty in the market and, therefore, usually release detailed issuance cal-

endars, well in advance of sovereign bond auctions. In many cases, debt managers would

like to be as predictable and transparent as possible to market participants and stick to

these calendars even at high cost. Second, the debt manager may maintain flexibility in her

issuance strategy by choosing from a variety of different instruments to be issued. While

the debt manager has to decide on the maturity of the issued instrument, and may also

consider the risk and liquidity of the bond being auctioned, the existence of auction cycles

and prevalent market conditions may have to be taken into account. We aim to contribute to

these considerations by laying out the possible determinants of predictable price movements

in the run-up of an auction.

the secured borrowing rate or the (implicit) cost of hedging.

6

Empirically, we find predictable yield movements to be significant and economically more

pronounced for bonds with certain characteristics. The yield movements are not consistently

observed for all countries but, in line with our hypotheses, seem to depend on measures of

risk and liquidity that can be related to their holding cost of inventory. Furthermore, we

find evidence of a funding liquidity effect during the financial crisis and the sovereign debt

crisis, i.e., lower funding liquidity and tighter capital constraints lead to a higher impact on

secondary market prices. We find that auction cycles may impose an implicit cost to the

participants of the secondary market that are, eventually, borne by the sovereign and the

tax payer. We estimate that these costs range from 2.80 basis points (e.g. in Germany) up

to 20.10 basis points (e.g. in Portugal), and depend on the prevalent market conditions, as

well as the characteristics of the bond being auctioned.

The paper is structured as follows. Section 2 provides a discussion of the relevant liter-

ature and places our contribution in context. In Section 3, we develop a theory model of

the optimal inventory holdings of primary dealers, from which we derive empirical predic-

tions and hypotheses in Section 4. Last, we provide an empirical analysis according to our

hypotheses in Section 5 and conclude with a discussion of our results in Section 6.

2 Discussion of the Related Literature

Primary dealers are the crucial link between the primary and secondary markets for sovereign

debt, deriving compensation for their services and risk bearing from the spreads they charge.

Since it has been documented that secondary market yields exhibit predictable patterns

around sovereign bond auctions, we are particularly interested in how primary dealers and

the market conditions they face may relate to these patterns. Empirically, bond yields tend

to rise prior to an auction and decline again after an auction has taken place (see e.g. Lou

et al. (2013), Beetsma et al. (2016)). Sigaux (2018) provides an extensive discussion of the

relevant literature and the different channels that may lead to this predictable price pattern,

i.e., the potential price impact of liquidity providers, the presence of capital impediments,

the limited risk capacities of market participants, or imperfect information about the future

net-supply of bonds. We relate our research to the overall inventory management problem

of primary dealers, i.e., different aspects of their funding- and risk constraints under future

7

demand uncertainty.

Sigaux (2018) suggests that the gradual increase in yields before an auction may be at-

tributed to market participants’ ex-ante uncertainty about the net-supply that will realize

after an auction has taken place. In the author’s model, risk-averse market participants may

hedge the supply shock by entering a long position in the bond before the auction, i.e., they

may realize a profit in the run-up of the auction when the net-supply and returns after the

auction are low. Conversely, short positions may be entered due to a speculative motive

of market participants. The author finds that, when an auction is approaching and more

information about the future net-supply is revealed, speculation in short positions increases

relative to the hedging motive, i.e., yields increase in the market due to the increased selling

pressure of bonds. By contrast to Sigaux (2018), we do not incorporate the price dynamics

due to the arrival of information but analyze the primary dealers’ inventory management

decision in the cross-section of bonds in their portfolio. In particular, our research focuses

on the uncertainty about the future demand for bonds and their associated cost and profit

opportunities, i.e., whether it is potentially beneficial to sell one particular bond to buy

another one.

As documented by Beetsma et al. (2016), primary dealers in the Eurozone usually have

limited risk bearing capacities, which result in secondary market yields showing auction

cycles around sovereign bond auctions. The authors find that these predictable price pat-

terns exist in the absence of arbitrage opportunities. Also, this temporary impact of bond

auctions on secondary market yields is particularly pronounced when market uncertainty or

risk aversion is high. However, their results vary across bond maturities and go in opposite

directions for different countries. More recently, Beetsma et al. (2018a) control for the effect

of the primary dealers’ clients’ order flows and demand for the bond auction and find support

for the limited risk bearing capacity of primary dealers. Lou et al. (2013) consistently find

similar evidence on auction cycles in US Treasury markets, i.e. rising bond yields before a

Treasury auction and lower yields after the auction. Both these papers, on Eurozone- and

on US sovereign bond markets respectively, imply hidden issuance costs for the government

that arise from auction cycles. However, Cafiso (2015) presents evidence from the Italian

8

sovereign bond market that the observation of auction cycles may emerge because of an

aggregation effect, i.e., particularly pronounced yield movements that occur for specific auc-

tions and bonds, and are, in fact, not frequently observed. We investigate this concern using

a broad panel of different bonds and countries across the Eurozone, and analyze the primary

dealers’ risk management and funding capacities at specific auctions over several time peri-

ods. While Beetsma et al. (2016) suggest that the symmetry and temporary effect of bond

auctions on secondary market yields are due to the increase in dealers’ aggregate inventory

exposure, we present additional channels that may amplify, or even lead to, these temporary

market movements. From a sovereign debt manager’s perspective, secondary markets may,

thus, be driving bond auction prices. Therefore, we attempt to provide a more comprehen-

sive assessment of the implicit transaction costs associated with bond issuances, i.e., the

extent to which the funding and risk capacity of primary dealers and market makers may

affect bond prices and yields. In particular, we investigate the predictable price movements

in the run-up of the auction on the level of the individual bond, i.e., the bond’s lucrativeness,

risk and liquidity, as well the primary dealers’ ability to lay off inventory risk by making use

of derivatives markets, and the dealers’ funding constraints while being active on primary-

and secondary markets.

In addition to their participation in bond auctions, primary dealers have to manage in-

ventory risk in their function as market makers. According to the market micro-structure

models by Stoll (1978) and Ho and Stoll (1981), market makers dynamically manage the

level of their inventory by adjusting bid and ask prices. These dynamic pricing policies

generally imply mean-reverting asset prices according to the market makers’ level of inven-

tory. Most recently, Friewald and Nagler (2016) relate corporate bond inventories to bond

prices and returns, and find that it may take dealers considerable time to periodically re-

duce their inventory under dynamic pricing policies. While the bid-ask spreads of sovereign

bonds appear to be more stable than corporate pricing policies would suggest (see Favero

et al. (2010)), a large increase in inventory due to the participation in bond auctions may

require the adjustment of the market makers’ quotes in a time-consuming process in order

to reduce their inventory exposure to preferred levels. In a study of US Treasury dealers’

inventories, Fleming and Rosenberg (2008) show that primary dealers purchase the majority

9

of sovereign bond positions on their own accounts, but are compensated for this risk by the

bid-ask spread and asset returns they earn. Hence, the observation of auction cycles is gen-

erally in line with market makers’ application of dynamic pricing policies and their demand

for a compensation for the provision of liquidity. In our research, we are interested to study

the asset pricing implications from the active management of a market maker’s inventory,

i.e., the decision to liquidate part of the existing inventory in order to participate in the

sovereign bond auction.

If capital moves slowly, Duffie (2010) suggests that yields may suddenly rise in response to

a supply shock due to the capital constraints of market participants, e.g., when it is costly to

raise immediate capital. When capital impediments are alleviated over time, e.g., some time

after an auction when more buyers arrive in the market, yields may start to decline again.

In a framework of capital-constrained intermediaries, Brunnermeier and Pedersen (2008)

show that initial losses on market maker inventories may lead to vicious liquidity cycles that

move prices away from fundamentals. Following this study, Comerton-Forde et al. (2010)

empirically show that for NYSE stock market makers’ inventory and financing constraints

are a significant and non-linear determinant of market liquidity. While Coughenour and

Saad (2004) find evidence that the prevalence of common market makers for multiple NYSE

stocks leads to commonality in liquidity for those assets, Karolyi et al. (2012) confirm these

findings along with other determinants of market liquidity in an extensive study of stocks in

countries across the globe. Furthermore, Dick-Nielsen et al. (2012) find that funding liquidity

translates into market liquidity for European sovereign bonds due to market makers’ inability

to obtain funding. We, therefore, suggest that hedging facilities are important for market

makers in managing and aligning their inventory risk with their activities on the primary

market. Since bond auctions in the Eurozone are purchased by active market makers in the

current market design, we are interested to study the influence of funding constraints and

supply shocks on market conditions across countries.

10

3 Model

Consider a two period model. There is one dealer and two bonds, indexed with i ∈ {0, 1}.

Time periods are indexed by t ∈ {0, 1}. Denote the dealer’s inventory of bond i at time

t = 1 by Qi and the price by pi. The two bonds are introduced sequentially, thus, at time

t = 0 there is only bond i = 0. At time t = 1, a new bond with index i = 1 is issued. At

time t = 0, the dealer is endowed with an initial inventory Qe0 of the first bond. This initial

inventory is due to previous market making activities, thus, the dealer paid the endowment

price pbid0,e for bonds i = 0 in his inventory. At time t = 1 a second bond i = 1 is issued. The

dealer can participate in this bond auction and decides how much inventory of both bonds

he wants to hold. If it seems beneficial for the dealer, he can decide to actively sell some

quantity of bond i = 0 at the market for pbid0 to be able to buy some quantity of bond i = 1

at the auction price c1. We assume that it is cheaper to buy the bond at the auction than

at the ask price, but there is no direct arbitrage opportunity with the secondary market,

i.e., we assume pbid1 ≤ c1 ≤ pask1 . Thus, dealers would like to participate in the bond auction

and supply new bonds for a marginal compensation. In our model, the demand for market

making of both bonds is stochastic. The dealer chooses the allocation to maximize his ex-

pected profit under uncertain demand for bonds. However, we assume that the dealer has

to finance all his positions at an unsecured financing rate l. In addition, the dealer faces a

capital constraint R with a rate of capital wi, i.e., the opportunity cost of financing a specific

bond.

As demand for the bond is uncertain, the dealer can end up with too little (underage)

or too much (overage or excess) inventory at time t = 1. Both of these situations are costly

as the dealer forgoes profits in the first situation and has unrecoverable costs in the sec-

ond situation. We denote foregone profits of bond i by Ui and the unrecoverable costs by

Ei. Assume demand for bond i is larger than the inventory. Thus, the dealer is not able

to engage in his market making activity to his full potential. In the case of bond 0, this

means that he forgoes the bid-ask spread he would earn as a market maker but, on the other

hand, he would save the financing costs. Hence, the foregone profits of bond 0 are given

by U0 = pask0 − pbid0 − lpbid0 . For bond 1, he forgoes the difference between the auction price

c1 and the ask price of the bond, thus U1 = pask1 − c1 − lc1. If the dealer has too much

11

inventory he can liquidate his excess inventory at market prices. However, as he will not act

as a market maker he has to pay the bid-ask spread. In addition, he has to pay the financing

costs for the position he has. As our model ends after period 1, we assume that the dealer

liquidates his excess position and does not want to carry over inventory.8 For bond 0, he

can recover the same price he paid for the bond but has to bear the financing costs of this

position. Thus, the unrecoverable cost of bond 0 are E0 = pbid0 − pbid0 + lpbid0 = lpbid0 . For

bond 1, he can only recover a part of the purchase price and, in addition, has to pay the

financing costs, i.e., the cost are given by E1 = c1 − pbid1 + lc1.

If Qi is the dealer’s inventory allocation, Di is the realized demand and si = paski − pbidi

denotes the bid-ask spread for bond i, the dealer’s profit from trading bond i is given by:

g(Qi, Di) = si min(Qi, Di)− EiQi

= (Ui + Ei)min(Qi, Di)− EiQi.

Assume that demand for bond i is large so that it can be accurately approximated by a

continuous distribution with the density fi, and cumulative distribution function Fi. Then

the dealer’s problem of expected profit maximization can be written as9

maxQi,i=1..2

∑i

(Ui + Ei)

∫ ∞

0

min(Qi, x)f(x)dx− EiQi, (1)

s.t.∑i

wiQi ≤ R.

The Lagrange function of (1) is given by:

L(Q, λ) =∑i

(Ui + Ei)

∫ ∞

0

min(Qi, x)f(x)dx− EiQi − λ(∑i

wiQi −R).

If Qi, i = 0, 1 is the solution to (1) then there exists λ, such that ∇L((Q1, Q2), λ) = 0.

8We note that, for the implications of our model, the liquidation of the dealer’s position is not required.9This is problem relates to the so-called newsstand or capacitated newsvendor problem first analyzed byHadley and Whitin (1963) and subsequently by Erlebacher (2000).

12

Moreover, because the objective function of (1) is concave and the constraint is linear,

((Q1, Q2), λ) maximizes the objective subject to the constraint, i.e., solves (1) (see Boyd

and Vandenberghe, 2004, Chapter 5). Thus, the unique solution to the problem is stated as

follows:

Qi = F−1i

(Ui − λwi

Ei + Ui

),∑

i

wiQi ≤ R. (2)

If the capital constraint is binding, the Lagrange multiplier λ > 0 represents the shadow

price of the capital constraint and is determined by Equation (3):

∑i

wiF−1i

(Ui − λwi

Ei + Ui

)−R = 0 (3)

Our model produces optimal bond inventories to be held in period 1. For the old issue

bonds, the inventory quantity is related to the depth of auction cycle, i.e., to the magnitude

of price change leading to the new auction. The quantity Qe0 −Q0 represents the amount of

bonds of the existing inventory that need to be sold in advance of the bond issuance. Thus,

smaller Q0 corresponds to a greater sell-out of the old issue bonds, and, correspondingly, to

a stronger auction cycle.

The optimal service level for bond i is Ui−λwi

Ei+Ui, i.e., it is the probability that a market

participant finds a bond available in the primary dealers inventory. A service level can be

thought of as the normalized optimal inventory quantity, i.e., it is adjusted for the shape of

the demand distribution. This optimal quantity balances the risk of turning away a buyer

wanting to purchase bonds with the risk of holding an excess inventory of bonds. We compare

the optimal service level for the bonds of the existing inventory and the newly issued bonds:

∆ =U1 − λw1

E1 + U1

− U0 − λw0

E0 + U0

(4)

In the presence of the budget constraint, Proposition 1 relates the optimal inventory

allocation to the funding costs and the budget constraint of the primary dealer.

13

Proposition 1: Funding Conditions If the funding cost for the newly issued bond w1 >

w0E1+U1

E0+U0, then ∆ ≥ 0, i.e., the new issue bond has a weakly higher optimal service level if

and only if

λ ≤ λ =U1

E1+U1− U0

E0+U0

w1

E1+U1− w0

E0+U0

.

All proofs are available in the Appendix. If the budget constraint is tight, i.e., λ is high,

it is optimal for a dealer to maintain a higher level of inventory of the old bonds. As long as

w1 > w0E1+U1

E0+U0, a smaller w1 increases the threshold λ. Hence, lower funding costs increase

the region where the new issue has a higher optimal level of inventory. If w1 < w0E1+U1

E0+U0, the

new issue has a higher optimal service level for any budget.

Next, we investigate the effect of the cost structure and demand variability on the opti-

mal inventory. To maintain tractability, assume fi = Uniform[ai, ai]. Thus, the expected

demand µi = (ai + ai)/2, and the standard deviation σi = (ai − ai)/(2√3). If the capital

constraint is not binding, Qi = µi + σi

√3( 2Ui

Ei+Ui− 1). Thus, Qi increases with the underage

cost Ui and decreases with the excess cost Ei. The effect of demand variability is propor-

tional to 2Ui

Ei+Ui− 1. That is, the optimal inventory of bond i increases with σi if Ui ≥ Ei,

and decreases otherwise.10

The case of binding capital constraint is more realistic but considerably more complex.

Following Erlebacher (2000) it can be shown that the optimal inventory quantities of bond

i are given by:

Qi = µi + σi

√3Ui − Ei

Ei + Ui

+wi

Ei+Uiσi

w2i

Ei+Uiσi +

w21−i

E1−i+U1−iσ1−i

× (5)

×(R− wiµi − w1−iµ1−i − wiσi

√3Ui − Ei

Ei + Ui

− w1−iσ1−i

√3U1−i − E1−i

E1−i + U1−i

).

The following Proposition evaluates the effect of the overage and underage costs on the

optimal inventory of bond i:

10When the capital constrain is not binding, the assumption fi = Uniform[ai, ai] can be relaxed to asymmetric distribution.

14

Proposition 2: Costs and Optimal Inventory. If fi = Uniform[ai, ai] with the ex-

pected demand µi = (ai + ai)/2, and the standard deviation σi = (ai − ai)/(2√3):

(i) dQi

dUi= 2Eiσi

√3

(Ui+Ei)2− wiσiw

21−iσ1−iRd

(E1−i+U1−i)(w2

i σi+w21−iσ1−i

Ei+UiE1−i+U1−i

)2 − 2w2i σ

2i

√3Ei(

w2i σi+w2

1−iσ1−iEi+Ui

E1−i+U1−i

)(Ei+Ui)2

;

(ii) dQi

dEi= − 2Uiσi

√3

(Ui+Ei)2− wiσiw

21−iσ1−iRd

(E1−i+U1−i)(w2

i σi+w21−iσ1−i

Ei+UiE1−i+U1−i

)2 +2w2

i σ2i

√3Ui(

w2i σi+w2

1−iσ1−iEi+Ui

E1−i+U1−i

)(Ei+Ui)2

;

(iii) dQi

dU1−i=

wiσiw21−iσ1−i(Ei+Ui)Rd

(E1−i+U1−i)2(w2

i σi+w21−iσ1−i

Ei+UiE1−i+U1−i

)2 − 2wiσiw1−iσ1−i

√3E1−i(

w2i σi+w2

1−iσ1−iEi+Ui

E1−i+U1−i

)(E1−i+U1−i)2

;

(iv) dQi

dE1−i=

wiσiw21−iσ1−i(Ei+Ui)Rd

(E1−i+U1−i)2(w2

i σi+w21−iσ1−i

Ei+UiE1−i+U1−i

)2 +2wiσiw1−iσ1−i

√3U1−i(

w2i σi+w2

1−iσ1−iEi+Ui

E1−i+U1−i

)(E1−i+U1−i)2

,

where Rd = R− wiµi − w1−iµ1−i − wiσi

√3Ui−Ei

Ei+Ui− w1−iσ1−i

√3U1−i−E1−i

E1−i+U1−i.

If the capital is budgeted according to the expected demand, the new issue bond is more

lucrative than the old issue bond, and the overage and underage cost for the old issue bond

are equal, then Proposition 2 simplifies to the following statement:

Corollary 1: If R−w1µ1 −w0µ0 = 0, E0 = U0, and U1 ≥ E1, then, for the old issue bonds

(i) dQ0

dU0≥ 0;

(ii) dQ0

dE0≤ 0, if w1σ1 ≤ w0σ0

E1+U1

U1−E1;

(iii) dQ0

dU1≤ 0;

(iv) dQ0

dE1≥ 0,

and for the new issue bonds

(v) dQ1

dU1≥ 0;

(vi) dQ1

dE1≤ 0;

(vii) dQ1

dU0≤ 0;

(viii) dQ1

dE0≥ 0, if and only if E0 ≥ 1

2w0

w1

(U1 − E1 − w0

w1

σ0

σ1(E1 + U1

). The sufficient condition

for dQ1

dE0≥ 0 is w1σ1 ≤ w0σ0

E1+U1

U1−E1.

15

The equivalent condition for E0 = U0 in terms of the bid and ask price is pask0 − pbid0 =

2lpbid0 . For U1 ≥ E1, the equivalent condition is pask1 + pbid1 = 2c1(1 + l).

Parts (ii) and (viii) of Corollary 1 point to the potential heterogeneity of the cost effect

on the optimal bond quantities. Specifically, if w0 = w1, the effect of the overage cost E0

on the old issue optimal inventory Q0 could be reversed if the demand volatility for the old

issue bong is sufficiently large, i.e., σ0 ≥ σ1U1−E1

E1+U1. Under the same condition, the effect of

E0 on the new bond optimal inventory Q1 could similarly be reversed.

Proposition 3 evaluates the effect of own demand volatility σi and cross demand volatility

σ1−i on the optimal inventory for bond i:

Proposition 3: Demand Volatility and Optimal Inventory. If fi = Uniform[ai, ai]

with the expected demand µi = (ai + ai)/2, and the standard deviation σi = (ai − ai)/(2√3),

then:

(i) dQi

dσi=

√3Ui−Ei

Ei+Ui+

wiEi+Ui

w21−iσ1−i

E1−i+U1−iRd(

w2iσi

Ei+Ui+

w21−i

σ1−i

E1−i+U1−i

)2 −wiσi

Ei+Ui

w2iσi

Ei+Ui+

w21−i

σ1−i

E1−i+U1−i

× wi

√3Ui−Ei

Ei+Ui;

(ii) dQi

dσ1−i= −

wiσiEi+Ui

w21−i

E1−i+U1−iRd(

w2iσi

Ei+Ui+

w21−i

σ1−i

E1−i+U1−i

)2 −wiσi

Ei+Ui

w2iσi

Ei+Ui+

w21−i

σ1−i

E1−i+U1−i

× w1−i

√3U1−i−E1−i

E1−i+U1−i

where Rd = R− wiµi − w1−iµ1−i − wiσi

√3Ui−Ei

Ei+Ui− w1−iσ1−i

√3U1−i−E1−i

E1−i+U1−i.

Under the conditions of Corollary 1, Proposition 3 simplifies to the following statement:

Corollary 2: If R− w1µ1 − w0µ0 = 0, E0 = U0, and U1 ≥ E1, then

(i) dQ0

dσ0≤ 0;

(ii) dQ0

dσ1≤ 0;

(iii) dQ1

dσ1≥ 0;

(iv) dQ1

dσ0≥ 0.

First, a higher volatility of demand for the old issue bonds is associated with an increase

in inventory of the new issue bonds and a decrease in inventory of the old issue bonds. The

result for the old issue bonds is due to the budget constraint: absent of the budget constraint,

the optimal order quantity is insensitive to σ0 if E0 = U0. Second, a more volatile demand

16

for the new issue bonds is associated with a decrease of inventory of the old issue bonds and

an increase of inventory of the new issue bonds. The result for the new issue bonds is due

to their greater lucrativeness.

In the next section, we motivate hypotheses that directly follow from the model and

Propositions 1-3, and test them empirically.

4 Hypotheses

In the following, we summarize our theory and derive a number of empirically testable hy-

potheses from our model. The model produces optimal inventory quantities for the old and

new issue bonds, Q0 and Q1, respectively. The lower Q0 corresponds to a greater sell-out

of the existing inventory for the purchase of new issue, Q1. All else equal, an increase in

bond sales leads to a decrease in prices in the run-up of an auction, i.e., the amplitude of

the auction cycle around the new issue must be greater. Hypothesis 1 questions the role

of funding liquidity and constraints as a determinant of the predictable price movements

leading to the auction cycle, Hypothesis 2 is concerned with the cost and lucrativeness of

the inventory, and Hypothesis 3 is focused on the demand uncertainty and inventory risk

management of primary dealers. Last, Hypothesis 4 deals with the increased risk exposure

and inventory management due to the purchase of the auctioned bond.

Hypothesis 1: Funding Conditions. The amount of inventory that is liquidated, as well

as the amplitude of the observed auction cycle will be greater when

A) . . . the shadow cost of capital of the average market participant, λ, as measured by the

funding liquidity, i.e., the spread of the EUR-OIS relative to the German Bund yield,

is lower.

B) . . . the capital rates of the liquidated bond, w0, and the capital rate of the auctioned

bond, w1, both measured by the secured borrowing rate, i.e., the 3M EUR-OIS, are

lower.

17

C) . . . the financing rate, l, as measured by the unsecured borrowing rate for the average

market participant, i.e., the 3M EURIBOR rate, is lower.

Hypotheses 1 on the funding conditions of primary dealers follows directly from Proposi-

tion 1. Empirically, the shadow price of capital, λ, may be measured in terms of the spread

between the EUR-OIS overnight rate, and the German Bund yield as a proxy for the risk

free rate. Dick-Nielsen et al. (2012) find that these funding costs, i.e., the cost of the average

market participant, may affect secondary market liquidity conditions.

We assume that both old and new issue bonds are financed at the same rate, i.e., w0 =

w1.11 The rationale for this assumption is as follows. Primary dealers often enter into a

general collateral (GC) repo agreement on the market, that allow them to deliver a bond

out of a specified basket that is available in their inventory. Therefore, we generally consider

the secured borrowing rate as the opportunity cost of financing a new or existing position.

We proxy the GC repo rate in a specific country by the EUR-OIS rate that is considered

almost risk-free. Furthermore, we control for the level of the unsecured financing rate, l, since

lower borrowing cost of the average market participant would lead to a greater incentive to

purchase new bonds. While a lower financing rate l may also ease pressure on liquidating

old issue bonds, we expect the financing rate to increase the amplitude of the auction cycle

due to present funding constraints and a greater lucrativeness of the new issue.

Hypothesis 2: Costs and Optimal Inventory. The amount of inventory that is liqui-

dated, as well as the amplitude of the observed auction cycle will be greater when

A) . . . the auctioned bond has a greater lucrativeness,pbid1 +pask1

2− c1, as measured by the

spread between the the mid-price quoted closest to the time of the auction and the

auction price.

B) . . . the liquidated bonds have greater liquidity, (pask0 − pbid0 ), as measured by the relative

bid-ask spread of the bond and its average in the market.

Hypothesis 2 follows from Proposition 2 and Corollary 1. If the spread between the

primary- and secondary market price, (pask1 −c1), is greater, the auctioned bond becomes more

lucrative, i.e., the underage/ opportunity cost, U1, increase and the overage/ unrecoverable

11In that case, the new issue bond has a higher optimal service level for any budget.

18

cost, E1, decrease (c1 − pbid1 decreases). Hence, the optimal level of inventory of existing

bonds, Q0, becomes lower and the optimal stock of the newly issued bond, Q1, becomes

greater. The overall effect of an increase in the lucrativeness of the auctioned bond can be

measured in terms of the differencepbid1 +pask1

2− c1, i.e., the difference between the mid-price

and the auction price. If the bid-ask spread, (pask0 − pbid0 ), is lower, the foregone profits

from missing out market making on the existing inventory become lower i.e., the underage/

opportunity cost of the existing inventory U0 decrease. Hence, the optimal level of inventory

of the existing bonds, Q0 becomes lower, and the optimal stock of the newly issued bond,

Q1 becomes greater. Empirically, we are able to observe these underage- and overage cost

and can, thus, test the predictions that derive from our model directly.

Hypothesis 3: Demand Volatility and Optimal Inventory. The amount of inventory

that is liquidated, as well as the amplitude of the observed auction cycle will be greater when

the volatility of demand of the liquidated bonds, σ0, and the volatility of demand of the

auctioned bond, σ1, both measured by the volatility of bond returns, the bonds’ maturity

(duration), the degree of risk-aversion in the market, i.e., the VSTOXX index, and the

aggregate market return volatility, is greater.

We relate Hypothesis 3 to Proposition 3 and Corollary 2. A higher volatility of demand of

the old issue bonds, σ0, may lead to an increase in inventory of the new issue bonds, Q1, and

a decrease in inventory of the old issue bonds, Q0. This may be the case when the primary

dealer is close to, or may even hit, the budget constraint. Conversely, a more volatile demand

of the new issue bonds, σ1, may lead to a decrease of inventory of the old issue bonds, Q0,

due to greater profits earned on the newly issued bond. Empirically, we do not observe the

volatility of demand σ0 and σ1. However, given a fixed supply of bonds, a greater demand

uncertainty results in greater price changes and, thus, also in greater bond return volatility.

Since market participants must be more concerned about their inventory exposure when the

general uncertainty in the market is high, the aggregate market volatility may contribute to

the demand uncertainty. In addition, market participants have to consider which bond to

liquidate from their inventory. Hence, we consider also individual bond characteristics, i.e.,

the return volatility of the individual bonds relative to the market portfolio. Furthermore,

the demand uncertainty may be sensitive to the maturity or duration of a bond, i.e., expected

changes in the interest rate may lead to to greater volatility and uncertainty about demand.

19

Last, we consider that varying demand uncertainty may be due to different degrees of risk

aversion in the market and measure it by the implied volatility index for the Eurozone, i.e.,

the VSTOXX index. An increase in any of these variables may be due, or lead to, greater

demand uncertainty and, thus, a greater optimal service level of the auctioned bond, Q1.

Hypothesis 4: Inventory Risk Management. The amount of inventory that is liqui-

dated, as well as the amplitude of the observed auction cycle will be greater when

A) the anticipated increase in inventory risk exposure due to the auction, Q1, as measured

by the volume issued at the auction, is greater.

B) the hedging capacities are lower, i.e., the basis risk, as measured by the spread between

the issuing country’s benchmark bond yield and the German Bund yield, is greater.

Hypothesis 4 follows from economic considerations that are outside of our model. Fixing

the purchased volume of the auctioned bond, Q1, the optimal service level of the old bonds,

Q0, is lower, i.e., inventory sales in the run-up of the auction are greater. This is due to pri-

mary dealers managing their inventory risk exposure and funding constraints in anticipation

of the bond auction. The volumes and risk profile of the purchased- and liquidated bond

would contribute to the existing portfolio risk of the primary dealer. If derivatives markets

are well functioning and frictionless, there are few serious funding or hedging constraints for

primary dealers with respect to the purchase of the auctioned bond. The ability to lay-off

risk is dependent on derivatives market conditions and government bond spreads, i.e., the

basis risk that dealers face when hedging their positions.

As a summary of the above hypotheses, we expect an impact of sovereign bond auctions

on secondary market prices depending on market conditions and the current risk exposure, as

well as funding and hedging capabilities of primary dealers. Furthermore, market conditions

may affect the primary market price, i.e., the capacity of primary dealers to participate

and bid in sovereign bond auctions. We empirically test these hypotheses in the following

section.

20

5 Empirical Analysis

5.1 Data Description

We consider a panel of Eurozone countries, spanning a time frame from 2003 to 2016. For

our study of the effect of bond auctions secondary market conditions, we analyze MTS inter-

dealer data, i.e., bond trades as well as the best available bid and ask quotes which are

available at the micro-second level. On the MTS platform, bonds may be quoted on domes-

tic and Euro-MTS markets at the same time and liquidity may be fractured between both

markets. According to Dufour and Skinner (2004); Cheung et al. (2005) and Caporale and

Girardi (2011), however, the MTS platform can be seen as one integrated market and price

discrepancies are eliminated almost immediately. We avoid double counting of observations

by always taking the last available, and best quote in a given time interval. On the MTS

platform, quotes are generally binding. Quotes may be submitted to MTS in the pre-market

phase from 07:30 CET to 08:00 CET, in the pre-open phase from 08:00 CET to 08:15 CET

and in the open market from 08:15 CET to 17:30 CET (see Dufour and Skinner (2004)).

However, we only consider trading hours when the market is open. Due to the heterogene-

ity of market conditions across countries and the micro-structure of the market, we sample

this data at half-hour intervals where applicable, e.g., when computing measures of volatility.

We select only bonds that have been issued by the respective country’s debt management

office and for which the government is the direct creditor, i.e., we exclude any government

agency bonds or quasi-government creditors from our analyses. We focus on non-structured

bonds that are quoted on domestic MTS markets and Euro-MTS markets. Table 1 shows our

selection criteria and represented bonds in our data set. In particular, we exclude inflation-

indexed and asset-backed bonds, or bonds that are issued by quasi-government institutions

or municipalities, from our analysis.

[Table 1 about here]

In accordance with our bond selection criteria, we analyze a subset of 8 countries from

the MTS data set, i.e., Austria, Belgium, Germany, Spain, France, Italy, Netherlands and

Portugal, for which reliable data can be obtained from the MTS platform, and for which

21

Thomson Reuters primary bond market data and bond auction results from 2003 to 2016

are also available. We are specifically interested in the bond price, the bond yield, and the

issuance volume. Since many bonds are price quoted, we compute holding period returns

where applicable. Table 2 shows summary statistics of our data set, i.e., on the primary

market and sovereign bond auctions. Germany, Italy and France appear to be the greatest

suppliers of sovereign bonds and also maintain the most developed markets for sovereign

bonds. The average maturity of the auctioned bonds is between two and five years for all

countries except Austria, which has issued mostly long term and issued bonds exceeding

twelve years maturity. On average, auction results across countries have a higher demand

than supplied at the given auction price, i.e., the bid-to-cover ratios are far higher than unity.

Most notably for our analyses, we observe that most countries conduct bond auction every

five business days. While Austria, Germany and Portugal issue less frequently, only France

issues more frequently about every second to third business day. Therefore, we focus our

analyses on a long event window of five days before the auction, i.e., from one auction date

to the next issuance in most countries, and a shorter window of two days before the auction,

i.e., to minimize the interference with other auctions and event windows in the respective

country.12

[Table 2 about here]

For additional information and references, we hand collect data sets from sovereign debt

managers’ and primary dealers’ websites. In particular, we are interested in information

about specific market design, primary dealer incentives and performance measurement. In

addition, we collect information about bond quoting strategies, auction bidding processes

and communication with the government from anonymous market participants, primary

dealers and sovereign debt managers across the Eurozone in a number of extensive interviews.

This information is processed and the formulation of hypotheses, and also quantitatively or

qualitatively enters our data analyses and empirical modeling process.

12We note that the time horizon of our event windows are also in line with the literature and empiricaldocumentation of auction cycles in the Eurozone as in, e.g., Beetsma et al. (2016). Hence, our results arecomparable with the existing empirical research on sovereign bond auction cycles.

22

5.2 Empirical Model and Variable Measurement

5.2.1 Auction Impact and the Optimal Level of Inventory

According to our specified hypotheses, we expect an impact of sovereign bond auctions on

secondary market prices that depends on market conditions and the current risk exposure-,

as well as the funding capabilities of primary dealers and market participants. Empirically,

our main variable of interest is, thus, the optimal level of the existing inventory Q0 that may

be liquidated in order to purchase new bonds and participate in the bond auction.

We generally measure our variables of interest in terms of event time, d, i.e., fractions

of trading days at the microsecond level, for a series of events, a ∈ N. While a is a specific

auction event, d is the time index of the observations centered around the auction time with

index 0, i.e., d represents the time-offset relative to the auction. We look at event windows

beginning five business days before the auction and ending at the time of the auction which,

for most countries, is around the average distance between bond auctions, i.e., d ∈ [−5, 0].

While we are not able to observe Q0 directly, we may observe the liquidation- and pur-

chases of inventory by measures that can be related to the market impact of sovereign bond

auctions. First, we analyze empirical measures of the trading- and liquidation volume of the

existing inventory. We are interested in whether a bond is traded more actively in the run-up

of an auction compared to its usual trading volume. Thus, we define the dependent variable

TV OL as the total daily trading volume per bond in the observation window around an auc-

tion divided by the average daily volume of the same bond in the year. Hence, we measure

the increase in trading activity for a specific bond before an auction relative to the year and

argue that the increase in trading is conducted in favor of the auctioned bond. Furthermore,

we analyze the average trading imbalance, TIMB, i.e., the difference between the aggregate

trading volume initiated from the seller’s side and the aggregate trading volume initiated

from the buyer’s side divided by the aggregate trading volume in the observation window.13

Since market makers are adjusting their quotes when they would like to liquidate part of

their inventory, i.e., so they get to sell-off their inventory to interested buyers, we would

expect so see a an increase in the trading volumes initiated from the buyer’s side before an

13Information about the trade-initiating party is provided by MTS

23

auction.

Second, we study the price impact of bond auctions on the secondary market, i.e., returns

computed from bond prices. We generally assume the normal return for a bond to be close

to zero around the auction and justify this assumption by the very short time horizon

considered, i.e., we consider abnormal returns due to the impact of bond auctions on market

prices. We analyze the mid-price impact, IMPR, as computed by the mean or median of

the best quoted mid price on specific day before the auction, and compute its return with

respect to the best quoted mid price that is observed to be closest to the auction time on

the auction day. The dependent variable y ∈ {TV OL, TIMB, IMPR} is a proxy for the

optimal level of inventory Q0. We explain the dependent variable y by the following linear

model, that is defined over a panel of observed bonds i ∈ I, and a series of observed auction

events a ∈ {1, 2, 3, . . . } in which j ∈ J, J ⊂ I are the bonds being auctioned. The country

of issuance of the observed bond i is always matched with the the auctioned bond j, i.e.,

we only analyze within country effects.14 The panel model for a specific event window is

formulated in Regression (6).

yia = ααα ·Dia + βββ ·Xa + γγγ ·X i

a + δδδ ·Xja + ϵia (6)

The vector ααα measures the different intercepts of the regression line with respect to bond

specific dummy variables Dia. The vector βββ specifies the coefficients with respect to overall

market conditions, X, γγγ denotes the coefficients with respect to characteristics of the bond

being liquidated, X i, and δδδ are the coefficients with respect to the characteristics of the

bond being auctioned, Xj. The term ϵ marks the individual error term for each observation.

We assume ϵ to be uncorrelated with the predictor variables. This assumption may seem

restrictive, because, in general, interest rates and trading volumes can be endogenous to

economic conditions. However, in our case, we focus on the short term abnormal trade

volumes and yields, which suffer less from the endonegeity issues. Standard errors are

14According to Beetsma et al. (2018b), there is evidence of cross-country issuance effects on yields. However,since Favero et al. (2010) indicate that European sovereign bond markets still do not appear to be perfectlyintegrated and market participants generally do not view sovereign bonds as (perfectly) substitutable, wefocus our analysis on the price impact of sovereign bond auctions within a particular country.

24

clustered according to the remaining maturity of the observed bonds to account for different

yield curve dynamics and errors specific to certain bond characteristics.

5.2.2 Determinants of the Optimal Level of Inventory

Empirically, our hypotheses on the optimal level of inventory Q0 translate to the determi-

nants of y in the following ways. Table 3 provides an overview of our empirical measures

used to estimate Regression (6), the corresponding hypotheses, and the variables used in our

model.

[Table 3 about here]

Market conditions, X, are controlled for by the following measures. Market liquidity,

MLIQ, is computed as the average of the relative spread between the best available bid-

and ask price over the event window in the issuing country. Market risk, MRISK, is the

bond return volatility computed from market average trade prices in the issuing country over

sampled, equidistant intervals of 30-minutes per day, and is computed on a rolling window

starting from 5 days before the start of the actual event window. Funding liquidity, FUNDL

is measured the average difference between the 3M EURIBOR and the 3M EUR-OIS, and

the unsecured borrowing rate, and aggregate credit risk, CRISK, is computed as the average

difference between the 3M EUR-OIS and the 3M German Bund rate over the event window,

i.e., we decompose the European TED spread into two variables related to funding liquidity

and credit risk. Furthermore, basis risk, BRISK, is calculated as the average difference

between the 10Y benchmark yield of the issuing country and the 10Y German Bund yield,

and BRATE is measured as the average 3M EURIBOR over the event window. In order to

proxy for the aggregate degree of risk aversion in the market, RISKA, we compute the aver-

age of the VSTOXX index over the event window. Last, we compute a baseline measure for

the capital rate, CRATE, by taking the average of the 3M EUR-OIS over the event window.

Characteristics of the liquidated bonds, X i and Di, are measured as follows. Bond liq-

uidity, LIQ, is computed as the average of the relative spread between the best available bid-

and ask price over the event window for the respective bond, and QLIQ is the bond’s liquid-

ity quantile with respect to this measure, relative to all other bonds in the issuing country.

25

Bond risk, RISK, is the bond’s return volatility computed from best quoted prices over

sampled, equidistant intervals of 60-minutes per day over the event window, and QRISK

is the bond’s risk quantile with respect to this measure, relative to all other bonds in the

issuing country.15 Bond maturity, MAT is the bond’s remaining maturity in years.

Characteristics of the auctioned bond, Xj, are measured as follows. The size of the

auction, ASIZE, is the volume of the auction relative to the total issuance volume in the

respective year. The auction spread, ASPREAD, is the lucrativeness of the auctioned bond

with respect to the secondary market and is measured as the difference between the mid-

price quoted closest to the auction time on the auction day and the auction price. Auctioned

bond liquidity, ALIQ is computed as the average of the relative spread between the best

available bid- and ask price over the event window for the auctioned bond. Auctioned bond

risk, ARISK, is measured as the auctioned bond’s return volatility computed from trade

prices over sampled, equidistant intervals of 30-minutes per day, and is computed on a rolling

window starting from 5 days before the start of the actual event window. Last, we control

for the country issuing the bond, CTRY .

5.3 Discussion of Results

We present our results for y ∈ {TV OL, TIMB, IMPR} over the full period 2003-2016 in

Table 4.

[Table 4 about here]

The explanatory value R2 of our analysis is between 3.5% and 4.5%, and an F -test in-

dicates statistical significance of our model at the highest confidence level. We test our

coefficients after clustering the standard errors according to the bonds’ maturities, i.e., in

order to account for different yield curve dynamics, and find our results to be statistically

significant at different confidence levels. Due to the great number of observations, i.e., more

than 3,000,000 observations of all bonds in our subset traded at each auction, we are careful

15We note that intra-day price jumps exceeding 10% have been eliminated in order to compute the volatilitymeasures. We attribute these implausible intra-day returns to obvious data errors or peculiarities ofmicro-structure and strategic quoting behavior.

26

interpreting the strong statistical significance of the restricted coefficients.16 Therefore, we

pay particular attention to the economic interpretation of our results and their relation to

our model, i.e., to explore whether our results may go in the reverse direction and under

what conditions this may be the case.

In all our regressions, we exclude the explanatory variable CRISK as a proxy for the

aggregate credit risk due to colinearity concerns or singularity issues.17 We reason that

this may be due to the high correlation between the EURIBOR, EUR-OIS and country

benchmark bond rates on which our measures are based, i.e., a measure of aggregate credit

risk is already subsumed in the other variables. Hence, the analyses with the remaining

explanatory variables already provides a good proxy for the aggregate credit risk and our

overall results should qualitatively be left unchanged.

5.3.1 Hypothesis 1: Funding Conditions

Empirically, we find evidence that lower secured borrowing cost contribute to the increase

in trading volumes, y = TV OL, for specific bonds due to the auction, i.e., the coefficient for

CRATE is negative with a value −0.13. We do not, however, find a statistically significant

coefficient for the unsecured capital rate, BRATE, as indicated by a p-value of 0.97. Fur-

thermore, we find that lower shadow cost of capital, FUNDL, affect the trading volume to

increase above average in the run-up of an auction by a coefficient of −0.24, i.e., in line with

our hypotheses.

While we do not find similar results for the trading imbalance, y = TIMB, i.e., the

funding conditions do not seem to play a statistically significant role in determining the

greater sell-off of bonds, there is statistical evidence of a greater price impact, y = IMPR,

when the shadow cost of capital are lower in the run-up of an auction, i.e., the coefficient

of FUNDL is −0.24. In addition, a greater unsecured capital rate seems to increase the

amplitude of the price impact with a weakly significant coefficient of 0.55.

16The applications of clustering raises most of our standard errors and alters the significance of manycoefficients. In general, the majority of the unrestricted coefficients showed statistical significance whereasthe restricted coefficients under clustering are more differentiated with respect to their significance levels.

17We do perform robustness checks using this variable in our online appendix.

27

5.3.2 Hypothesis 2: Costs and Optimal Inventory

We find that a greater lucrativeness of the auctioned bond to a greater increase in trading

activity, y = TV OL, i.e., the coefficient of ASPREAD is positive with a value of 0.01 and

statistically significant. Also, in times when the observed bond has greater liquidity, LIQ,

as well as in times of greater market liquidity, MLIQ, trading and the liquidation of market

participants’ portfolios is facilitated. Even though both coefficients are marginally negative,

they indicate greater trading activity in favor of the auctioned bond when bid-ask spreads

are lower. Furthermore, we find that market participants would first liquidate bonds that

have greater liquidity, i.e., the increase in trading activity before an auction is greater for

bonds in lower quantiles of the bid-ask spread. The coefficient of Q1.LIQ, i.e., the first

quantile the bid-ask spread at a specific auction, has a positive value of 0.39 and Q4.LIQ

shows a negative coefficient of −0.42. Therefore, we see evidence for our hypothesis and

proposition on the cost of inventory and optimal bond service levels.

Similarly, we find statistically significant evidence that the trading imbalance, y =

TIMB, increases before an auction for bonds of greater liquidity. A negative coefficient

of −0.06 of the lowest liquidity quantile, Q1.LIQ, and a positive coefficient of 0.12 of the

highest liquidity quantile, Q4.LIQ, indicate that market participants buy more liquid from

market makers before an auction. We can not find evidence, however, that the increased

sales of more liquid bonds generates a price impact, y = IMPR, in the market for these

bonds.

5.3.3 Hypothesis 3: Demand Volatility and Optimal Inventory

In line with our hypothesis and proposition on the volatility of demand for bonds, the trad-

ing volume of bonds, y = TV OL, increases more greatly in times of greater volatility in

the market. We find that the coefficient for market risk, MRISK, is both statistically and

economically significant with a value of 17.39. Furthermore, we find that there is a greater

increase in trading volume for more risky bonds when an auction is approaching, i.e., bonds

in the highest risk quantile relative to the market portfolio, Q4.RISK, show a statisti-

cally significant coefficient of 0.18. By contrast, the liquidation of less risky bonds seems

to decrease before an auction, i.e., the coefficient for the lowest risk quantile, Q1.RISK, is

28

negative with a value of −0.41. In opposition of our hypotheses, though, we find no evi-

dence that the maturity of the bond, MAT , plays a role for the trading volume before an

auction. Also, in times of greater risk of the individual bonds there seems to be a greater

reluctance to trade the bonds, i.e., the coefficient of RISK is weakly significant at a level

of −11.46. We also do not find evidence that the market risk aversion, RISKA, is affecting

demand volatility and the increased rebalancing of bond portfolios before an auction. On

the other hand, greater demand uncertainty of the auctioned bonds increases the desire to

buy more of the new bond due to its greater lucrativeness, i.e., the coefficient of ARISK

is both economically and statistically significant with a value of 0.11. In contrast to our

hypotheses and theoretical predictions, a greater maturity of the auctioned bond, AMAT ,

seems to decrease the rebalancing of market participants’ portfolios slightly by −0.01. We

argue that this finding may be due to the strategic behavior of the sovereign and the bidding

behavior of dealers that is outside of our model. Endogeneity may arise when the sovereign

issues specific instruments in response to the primary dealers’ demand, i.e., typically long

maturities for which the market has high demand and can absorb large volumes.

In a similar way, the trading imbalance, y = TIMB, is affected by the risk of the bonds

relative to the market. Potential buyers seem to have a greater incentive to buy more risky

bonds from the primary dealers’ portfolios, i.e., because dealers quote more attractive prices

for these bonds. Hence, the coefficient for the highest risk quantile, Q4.LIQ, is statistically

significant and negative at a value of −0.04. Conversely, there are less buyer initiated trades

and volumes for the lowest risk quantile, i.e., Q1.LIQ shows a positive coefficient of 0.09.

For the impact of the sovereign bond auction on prices, y = IMPR, however, there seems

to be no impact regardless of the risk of the bonds in the primary dealers portfolios.

5.3.4 Hypothesis 4: Inventory Risk Management

For the increase in trading volume before an auction, the coefficient for BRISK is both

economically and statistically significant with a value of −0.13. Therefore, we find evidence

that an increase in basis risk makes market participant’s more reluctant to trade in bonds

in favor of the sovereign bond auction, i.e., since they may need to engage in costly hedg-

ing activities when rebalancing their portfolios. By contrast to our expectations, a greater

29

issuance volume decreases the trading activity in the run-up of an auction, i.e., we find a

negative coefficient of −0.08 of ASIZE. We attribute this finding to a potential endogeneity

issue; according to market participants, the sovereign would issue greater volumes when the

market is in greater need of liquidity.

Similarly for the trading imbalance, y = TIMB, we find evidence that an increase in

basis risk makes market participants to buy less from the market makers’ portfolios. The

coefficient of BRISK is positive with a value of 0.04, indicating that market makers attract

less buyers, i.e., they may be reluctant to sell-off their inventory in favor of the auctioned

bond. By contrast, we do not find statistical evidence that the size of an auction or an

increase in basis risk generate an impact on prices, y = IMPR.

5.3.5 Time-period Differences

Tables 5–8 present an analysis of our results over different sub-periods, i.e., before the

financial crisis 2003-2007, during the financial crisis 2007-2009, the begin of the sovereign

crisis 2009-2012, and the post-crisis period 2012-2016.

[Tables 5–8 about here]

Across all sub-periods, we observe similar results and implications for our hypotheses as

described for the full sample period. Most prominently, however, the period of the finan-

cial crisis 2007-2009 and tightened funding conditions for market participants, i.e., greater

shadow cost of capital and a higher secured capital rate, lead to a stronger increase in the

trading activity, y = TV OL, before an bond auction. We find a statistically significant

coefficient of CRATE with a positive value of 0.05 and a significant coefficient of FUNDL

of −1.05 that are both in line with Hypothesis 1. Therefore, we reason that our hypotheses

on the funding condition of primary dealers finds support in the period of the financial cri-

sis when these market conditions were vital. The crisis was also when market risk premia

and the degree of risk aversion were particularly pronounced, i.e., in support of Hypothe-

sis 3, we find a significant and positive coefficient of RISKA with a value of 0.03. We also

see further support of Hypothesis 2 in that there was no economically significant impact

of the primary-secondary market spread, ASPREAD, on the trading volume. We argue

30

that market participants were increasingly driven by risk considerations during the finan-

cial crisis and, hence, were not incentivized by these marginal profits. By contrast, during

the sovereign crisis and its aftermath, the greater lucrativeness of new bonds seems to have

attracted investors’ capital that liquidated their existing bond positions. One possible ex-

planation would be that central bank interventions and quantitative easing programs during

those periods have reduced the liquidation cost and helped to increase the attractiveness of

new bonds.

A significant coefficient of the size of the auction, ASIZE, with a value of −0.04 suggests

a negative impact on the trading volume before an auction during the financial crisis. This

may be an endogeneity issue, i.e., the sovereign may issue greater volumes when the market

is in need of liquidity or, in the case of the financial crisis or sovereign debt crisis, seeks a

safe haven and flight to quality18. Furthermore, the sovereign crisis and its aftermath 2009-

2012 shows the greatest impact of the basis risk on the traded volumes, y = TV OL, and

trading imbalances, y = TIMB. We find evidence that, in times of greater yield differentials,

BRISK, market participants were exposed to greater risk when liquidating their existing

portfolios in favor of the new bond, i.e., the trading volumes would decrease below average

by −0.42 and seller initiated trades would increase by 0.10 relative to buyer initiated trades

before an auction. Hence, we also see stronger support for Hypothesis 4 in times where the

hedging capacities are more limited.

5.3.6 Country Differences

We notice a difference in the amplitude of the auction cycle across countries, as illustrated

in Tables 4–7. The differences are with respect to Germany as a baseline, i.e., we assume the

German market to be most efficient with respect to funding and risk constraints. For the

full-sample period (see Table 4), we observe that all countries except Austria have a greater

amplitude in auction cycles than Germany with respect to the positions that are liquidated,

y = TV OL. We explain this difference by the very close proximity of the Austrian to the

German market, i.e., Austrian bond positions can be hedged almost perfectly against the

German derivatives market. While the sign reverses during the period of the financial crisis

18For example, Beber et al. (2008) shows in an empirical analysis of Euro-area government bonds that, intimes of increased market uncertainty, investors direct their funds to bond markets with greater liquidity.

31

and the sovereign debt crises, the difference stays only minor, i.e., the access to the Ger-

man bond derivatives market and its hedging capabilities have been maintained for Austrian

market participants throughout the time periods.

On a closer inspection, the selling pressure, y = TIMB, is smaller than in Germany

for all countries. We reason that this may be due to the German bonds being almost cash-

equivalent, i.e., existing positions in German bonds can be liquidated easily and at very low

cost. Thus, the greater lucrativeness of a new issue may trigger increased bond sales in the

run-up of the auction even if, e.g., funding constraints are not very pronounced. Overall,

this result holds consistently for the majority of countries and sub-periods and is even more

pronounced in magnitude, i.e., the sell-off of German bond positions has been greater in

times of greater market distress and liquidation cost.

Last, we find the price impact of the auction cycle, y = IMPR, to be heterogeneous

across countries and sub-periods. In Austria, Italy and Spain, the return of trading on the

auction cycle is potentially lower than in Germany and, before the sovereign crisis, also in

France. A possible explanation may be that German bond auctions have been very attractive

for market participants throughout time as a tool for hedging, low-risk investment and safe

liquidity haven, i.e., they may be willing to sell old issue bonds for a new issue even if

the market impact would be greater. In addition, the countries with lower returns have

maintained access to well-functioning bond derivatives markets, i.e., the price impact of

the auction may be more pronounced when the cost of hedging are very high or obtaining

funding is expensive, and less pronounced under usual market conditions.

5.3.7 Economic Cost of the Auction Cycle

Eventually, we assess the economic consequences of the auction cycle for market partici-

pants and the sovereign. In particular, we ask what can be gained from eliminating financial

constraints that determine the auction cycle’s amplitude, i.e., as shown by our model and

empirical findings. We measure the cost of the auction cycle by the average price impact in

basis points (bps), computed from the median quoted mid-price five days before the auction,

32

to the mid-price that is quoted closest to the auction time on the auction day.19 From the

perspective of primary dealers, the auction cycle leads to increased costs when dealers are

liquidating their positions due to the imposed price impact. In terms of our model, pri-

mary dealers incur cost in the secondary market for a potential gain in the primary market.

Absent financial constraints and market inefficiencies, there would be no such auction cycle

according to our findings.

From the perspective of the sovereign, the auction cycle may lead to greater yields and

greater debt service expenses. We assume that the primary- and secondary market are, on

average, separated by a constant spread, i.e., the costs of the auction cycle in the primary

market are similar to the costs in the secondary market. We rationalize this assumption by

the following argument. If primary dealers are financially constrained, they may liquidate

their positions to participate in the sovereign bond auction due to a greater lucrativeness of

the auctioned bond. However, if financial constraints are tight and the auction cycle is more

pronounced, the cost of liquidation are also high. Hence, at some point, primary dealers will

adjust their bidding behaviour in addition to their inventory sales, i.e., tighter financial con-

straints would also lead to a lower auction price and greater costs to the tax payer.20 Thus,

the sovereign may ultimately bear the cost of market inefficiencies and financial constraints

of primary dealers.

[Table 9 about here]

Table 9 illustrates the discussed cost of the auction cycle in the primary- and secondary

market for different countries and bond maturities. On average, the cost range from 2.80bps

in Germany to 20.10bps in Portugal and we see that the cost are strongly varying across

countries. Furthermore, the cost of the auction cycle seems to be increasing in the maturity

of the bonds with only a few exceptions. We reason these cases, i.e., longer maturities with

19This measure is the net-return version of the price impact measure y = IMPR, given in basis points.20We note that, in theory, a high degree of competition among primary dealers and great lucrativeness ofthe auctioned bond may leave the primary market prices unaffected by the auction cycle, i.e., primarydealers would always choose to liquidate their existing positions in favor of the auctioned bond. However,debt managers as well as market participants have pointed out to us that the secondary market servesas a reference point for the sovereign bond auction, i.e., we have no reason to believe that these marketconditions are common.

33

lower associated costs, are mostly observations of benchmark bonds in the respective country

such as the 10Y German Bund. Germany is, by far, the country with the lowest economic

cost due to the auction cycle. It is followed by Italy and Spain which both have maintained

a well-functioning derivatives market in most of the time periods considered.

At first glance, the economic cost for primary dealers may seem to be negligable in

countries such as Germany where the cost only amount to a couple of basis points. However,

bond auctions are conducted on a frequent basis. While, in Austria, bond auctions are held

at least once a month, bond auctions in e.g. France may happen every three business days.

In fact, most of the countries hold bond auctions around once a week. Hence, the cost

are not limited to a short time horizon but extend to a very present situation for primary

dealers throughout the entire trading year. Therefore, we argue that the auction cycle is an

economically significant effect that is relevant to market participants and policy makers.

6 Conclusion

Our results emphasize the importance of well-functioning secondary markets for sovereign

bonds. As already document by Beetsma et al. (2016) and Sigaux (2018), many European

sovereign bond markets exhibit auction yield cycles, i.e., a predictable rise of yields before,

and lowering of yields after auction dates. In this paper, we explore this effect further and

explain what drives the yields in the run-up of a bond auction in a cross-section of six Eu-

rozone sovereign bond markets.

We develop a model that solves for the optimal inventory levels of the existing- and the

newly issued bonds that can be related to predictable price movements around sovereign

bond issuances. The optimal inventory levels depend on the cost of inventory and regula-

tory capital, prevalent funding conditions, and the demand volatility of the existing- and the

newly issued bond. From the solution of our model, we derive empirically testable hypothe-

ses and predictions.

First, our empirical results support the previous papers, i.e., we find evidence of increased

34

trading activity in the run-up of an auction. We find that, according to our hypotheses, the

liquidation of bonds seems to depend on individual bond characteristics, i.e., their risk and

liquidity profile. Intuitively, more liquid and more risky bonds would be sold-off first by pri-

mary dealers, in order to minimize the impact of sovereign bond auctions on their portfolios.

Even more, we see that there is a stronger increase in trading activity for these bonds before

an auction.

Second, there is evidence that the risk and liquidity of the bond being auctioned have an

influence on the extent that primary dealers have to adjust their positions. We infer that a

higher risk and greater lucrativeness of the issued bond requires the dealers to sell-off more

of their risky inventory and, thus, increases the amplitude of the auction cycle. By contrast,

a higher liquidity of the auctioned bond potentially allows dealers to pass on inventory risk

to the market more easily, thereby decreasing the amplitude of auction cycles.

Third, we find that overall market conditions themselves determine auction cycles. We

construct a funding liquidity proxy by computing the difference between the 3M EURIBOR

and the 3M EUR-OIS swap rate. We find that this funding liquidity proxy, as well as

the basis risk of hedging instruments are a significant determinant of the trading activity

in the run-up of an auction, especially during the financial crisis and the sovereign debt crisis.

Last, we quantify the implicit cost of the auction cycle and argue that these cost are

eventually borne by the sovereign. We find the cost between 2.80 and 20.10 basis points,

depending on the country of issuance and the maturity of the bonds. Furthermore, we find

the cost to be economically significant due to the relatively large number of bond auctions

across countries throughout the year.

35

References

Beber, A., Brandt, M. W., and Kavajecz, K. A. (2008). Flight-to-Quality or Flight-to-

Liquidity? Evidence from the Euro-Area Bond Market. The Review of Financial Studies,

22(3):925–957.

Beetsma, R., Giuliodori, M., De Jong, F., and Hanson, J. (2018a). Bid-to-Cover and Yield

Changes around Public Debt Auctions in the Euro Area. Journal of Banking and Finance,

87:118–134.

Beetsma, R., Giuliodori, M., De Jong, F., and Hanson, J. (2018b). Cross-Border Auction

Cycle Effects of Sovereign Bond Issuance in the Euro Area. Journal of Money, Credit and

Banking, 50:1401–1440.

Beetsma, R., Giuliodori, M., De Jong, F., and Widijanto, D. (2016). Price Effects of

Sovereign Debt Auctions in the Euro-zone: The Role of the Crisis. Journal of Finan-

cial Intermediation, 25:30–53.

Boyd, S. and Vandenberghe, L. (2004). Convex Optimization. Cambridge University Press.

Brunnermeier, M. K. and Pedersen, L. H. (2008). Market Liquidity and Funding Liquidity.

Review of Financial Studies, 22(6):2201–2238.

Cafiso, G. (2015). Treasury Auctions and Secondary Market Dynamics. An Analysis Based

on the MTS Market for Italy. CESifo Working Paper, 1(5357).

Caporale, G. and Girardi, A. (2011). Price Formation on the EuroMTS Platform. Applied

Economics Letters, 18(3):229–233.

Cheung, Y.-C., de Jong, F., and Rindi, B. (2005). Trading european sovereign bonds: The

microstructure of the mts trading platforms. ECB Working Paper Series, 432.

Comerton-Forde, C., Hendershott, T., Jones, C. M., Moulton, P. C., and S., S. M. (2010).

Time Variation in Liquidity: The Role of Market-Maker Inventories and Revenues. Journal

of Finance, 125(1):295–331.

Coughenour, J. F. and Saad, M. M. (2004). Common Market Makers and Commonality in

Liquidity. Journal of Financial Economics, 73(1):37–69.

36

Dick-Nielsen, J., Gyntelberg, J., and Lund, J. (2012). From Funding Liquidity to Market

Liquidity: Evidence from Danish Bond Markets. Working Paper.

Duffie, D. (2010). Asset price dynamics with slow-moving capital. Journal of Finance,

65:1237–1267.

Dufour, A. and Skinner, F. (2004). MTS Time Series: Market and Data Description for the

European Bond and Repo Database. Icma centre discussion papers in finance.

Erlebacher, S. J. (2000). Optimal and Heuristic Solutions for the Multi-Item Newsvendor

Problem with a Single Capacity Constraint. Production and Operations Management,

9(3):303–318.

Favero, C., Pagano, M., and Von Thadden, E.-L. (2010). How Does Liquidity Affect Gov-

ernment Bond Yields? Journal of Financial and Quantitative Analysis, 45(1):107–134.

Fleming, M. J. and Rosenberg, J. V. (2008). How Do Treasury Dealers Manage Their

Positions? Federal Reserve Bank of New York Staff Reports, 1(299).

Friewald, N. and Nagler, F. (2016). Dealer Inventory and the Cross-Section of Corporate

Bond Returns. Working paper.

Hadley, G. and Whitin, T. M. (1963). Analysis of Inventory Systems. Technical report.

Ho, T. and Stoll, H. R. (1981). Optimal Dealer Pricing under Transactions and Return

Uncertainty. Journal of Financial Economics, 9(1):47–73.

Karolyi, G. A., Lee, K.-H., and Van Dijk, M. A. (2012). Understanding Commonality in

Liquidity Around the World. Journal of Financial Economics, 105(1):82–112.

Lou, D., Yan, H., and Zhang, J. (2013). Anticipated and Repeated Shocks in Liquid Markets.

The Review of Financial Studies, 26(8):1891–1912.

Pelizzon, L., Subrahmanyam, M. G., Tomio, D., and Uno, J. (2016). Sovereign Credit

Risk, Liquidity, and European Central Bank Intervention: Deus Ex Machina? Journal of

Financial Economics, 122(1):86–115.

Pelizzon, L., Subrahmanyam, M. G., Tomio, D., and Uno, J. (2018). Central Bank-Driven

Mispricing. SAFE Working Paper Nr. 226.

37

Sigaux, J.-D. (2018). Trading ahead of treasury auctions. ECB Working Paper Series,

(2208).

Stoll, H. R. (1978). The Supply of Dealer Services in Securities Markets. Journal of Finance,

33(4):1133–1151.

38

Tables

Country MTS Code Bond Type Description

Austria ATS Bills, Bonds Government BondsBelgium BTC ZCB Zero Coupon BondsBelgium OLO Bonds Obligations lineaires ordinairesFrance BTA Bills Bons du TresorFrance OAT Bonds Obligations assimilables du TresorFrance FCO Bonds Coupon BondsFrance FTB ZCB Zero Coupon BondsFrance TEC Floater Floating rate bond, 10-year OAT par yieldGermany DEM Bonds Government Bonds (Bobls, Bunds)Germany GTC Bills BubillsItaly BOT Bills Buoni Ordinari del TesoroItaly BTP Bonds Buoni del Tesoro PoliannualiItaly CTZ ZCB Certificati del Tesoro Zero CouponNetherlands DSL Bonds Dutch State LoanNetherlands DTC ZCB Dutch Treasury CertificatesPortugal PTC Bills Portugese Treasury CertificatesPortugal PTE Bonds Portugese Government BondsSpain BON Bonds Bonos del EstadoSpain OBE Bonds Oblicaciones del EstadoSpain LET Bills Letras del Tesoro

Table 1: This table shows the bonds represented in our sample. We report the issuing country ofthe bond, the internal code of the bond in the MTS data set, and a short description of the bond if available.We collected data on government bonds from the MTS data set description and DMO websites.

Country Vol BA SD AY Mat B2C #Auctions #Bonds #DaysAustria 0.87 0.50 0.39 1.52 12.41 2.40 153 25 27.82Belgium 1.12 0.08 0.08 0.17 3.03 2.18 840 177 5.16France 2.44 0.09 0.10 0.06 2.99 2.75 1817 527 2.40Germany 4.79 0.08 0.16 0.31 4.98 1.76 446 252 9.07Italy 4.44 0.16 0.22 0.17 4.89 1.68 882 397 4.88Netherlands 1.67 0.05 0.08 0.18 2.18 2.76 868 186 4.83Spain 1.79 0.34 0.27 0.22 4.73 2.83 794 167 5.50Portugal 0.83 0.36 0.16 0.65 2.39 2.69 352 98 12.02

Table 2: This table shows summary statistics for our sample. We report statistics of theprimary and secondary market for each country in our sample. Vol is the average auction volume,BA is the average bid/ask spread of the issued bond at the auction day, SD is the standard deviation of thedaily returns, AY is the average yield, Mat is the average maturity of the auctions, B2C is the bid-to-coverratio, #A is the number of auctions, is the number of bonds auctioned, and #Days is the average numberof business days in between auctions. Secondary market data is taken from MTS for the time-period from2003 to 2016. We collected data on sovereign bond auctions from Thomson Reuters and DMO websites.

39

Dependent VariablesModel Empirical Measures

Variable Description Component Description Variable Description Measurement Expected Sign Hypothesis

Q0Optimal inventory of old bonds

(liquidation volume)/ /

TV OL Trading volumeBond trading volume/total trading volume

/ /

TIMB Trading imbalanceAggregate of

(SELL trading volume - BUY trading volume)/ (BUY trading volume)

/ /

IMPR Price impactMean (median) quoted mid price /

quoted mid price closest to auction time/ /

Explanatory VariablesModel Empirical Measures

Variable Description Component Description Variable Description Measurement Expected Sign Hypothesis

Q1Optimal inventory of new bonds

(demand for auction)/ / ASIZE Auction size

Auction volumerelative to countrystotal issuance volume

+ 4

U0Underage cost(old bond)

pask − pbid

Foregone market-making profits,bid-ask spread

LIQ LiquidityAverage of

(best quoted ask pricebest quoted bid price)

- 2

QLIQ Relative liquidity Quantile of bid-ask spread - 2

MLIQ Market liquidityAverage of best bid-ask spread

across bonds in the issuing country- 2

−l · pbid Financing costs,savings

BRATEUnsecured

borrowing rateAverage 3M EURIBOR + 1

U1Underage cost(new bond)

pask − cForegone marginal

auction profitASPREAD Auction Spread

Best quoted mid price closest toauction time - auction price

+ 2

−l · c Financing costs,savings

BRATEUnsecured

borrowing rateAverage 3M EURIBOR + 1

E0Overage cost(old bond)

+l · pbid Financing costs,expenses

BRATEUnsecured

borrowing rateAverage 3M EURIBOR + 1

E1Overage cost(new bond)

c− pbid Unrecoverable cost ASPREAD Auction SpreadBest quoted mid price closest toauction time - auction price

+ 2

+l · c Financing costs,expenses

BRATEUnsecured

borrowing rateAverage 3M EURIBOR + 1

w0 Rate of capital(old / new bond)

/Opportunity cost

of financingCRATE Secured borrowing rate Average of 3M EUROIS - 1

w1

λ / / Shadow price of capital FUNDL Funding liquidityAverage of

(3M EURIBOR - 3M EUR-OIS)- 1

σ0 / /Demand volatility

(old bond)

RISK Return volatility Return volatility per bond + 3MAT Maturity Maturity in years + 3

QRISK Marginal risk Quantile of return volatility + 2

MRISK Market riskAverage of return volatility

across bonds in the issuing country+ 3

RISKA Risk aversionAverage of

VSTOXX index+ 3

σ1/ /

Demand volatility(new bond)

ARISK Auctioned bond riskAverage return volatility of

auctioned bond+ 3

AMAT Auctioned bond maturity Auctioned bond maturity + 3

MRISK Market riskAverage of best bid-ask spread

across bonds in the issuing country+ 3

RISKA Risk aversionAverage of

VSTOXX index+ 3

/ / / / BRISK Basis riskAverage of

(10Y benchmark yield -10Y Bund yield)

+ 4

/ / / / CRISK Credit riskAverage of

(3M EUR-OIS -3M Bund yield)

+ /

/ / / / CTRY Country Country dummy / /

Table 3: This table provides an overview of the variables in our model, the derived empiricalhypotheses and the corresponding empirical measures used to estimate Regression (6), i.e., thedependent and explanatory variables. Column Variable contains the variable names used in the modelequation and regression, Description explains the economic interpretation of the variables and measures,column Measurement states how the variables are computed, column Expected Sign reports the expectedsign of the regression coefficients, and column Hypothesis indicates which empirical hypothesis the modelvariables and empirical measures are corresponding to.

40

Panel Regression

TVOL TIMB IMPR

Variable Coefficient p-value Coefficient p-value Coefficient p-value

Intercept 0.85 0.00 0.58 0.00 99.68 0.00Austria -0.01 0.82 -0.03 0.49 -0.07 0.03Belgium 1.08 0.00 -0.23 0.00 0.07 0.19France 1.79 0.00 -0.09 0.02 0.02 0.60Italy 1.79 0.00 -0.49 0.00 -0.04 0.49Netherlands 1.54 0.00 -0.08 0.04 0.07 0.03Portugal 0.51 0.00 -0.26 0.00 -0.06 0.35Spain 0.92 0.00 -0.23 0.00 -0.20 0.07Q1.LIQ 0.39 0.00 -0.06 0.00 -0.00 0.83Q4.LIQ -0.42 0.00 0.12 0.00 0.00 0.93Q1.RISK -0.41 0.00 0.09 0.00 0.01 0.60Q4.RISK 0.18 0.02 -0.04 0.14 -0.15 0.01LIQ -0.00 0.46 -0.00 0.32 0.00 0.41RISK -0.11 0.07 0.02 0.08 0.26 0.00CRATE -0.13 0.00 -0.01 0.25 0.05 0.23MAT 0.00 0.29 0.00 0.40 0.01 0.00ASIZE -0.08 0.00 -0.00 0.47 0.00 0.47ASPREAD 0.01 0.05 -0.00 0.66 0.00 0.29ALIQ 0.00 0.96 0.00 0.99 0.00 0.17ARISK 0.11 0.00 -0.01 0.20 -0.01 0.53AMAT -0.01 0.00 -0.00 0.04 0.00 0.32MLIQ -0.00 0.00 0.00 0.85 -0.00 0.32MRISK 0.17 0.05 -0.02 0.14 0.06 0.28FUNDL -0.24 0.07 0.01 0.59 -0.24 0.02BRATE -0.00 0.97 -0.02 0.75 0.55 0.11BRISK -0.13 0.00 0.04 0.00 0.06 0.28RISKA 0.00 0.23 0.00 0.00 0.01 0.04

Table 4: This table shows the panel regression results for the period 2003-2016 (full sample).The regression variables are the trading volume before the auction relative to the average trading volume ofthe bond, TV OL, the trading imbalance based on seller initiated bond trades before the auction relative tobuyer initiated bond trades, TIMB, and the price impact based on returns calculated from the mid-pricebefore the auction to the time of the auction on the auction day and is given in price points, IMPR. Thetable shows the estimated coefficients for different explanatory variables and corresponding t-statistics inparentheses. For the fixed effects, the following factors are defined as a baseline and are included in theintercept term: Germany (country), the risk quantiles (Q2.RISK, Q3.RISK), and the liquidity (Q2.LIQ,Q3.LIQ) for specific bonds. Our sample comprises auctions during the time period from 2003 to 2016 (fullsample). Reported standard errors are clustered around the remaining maturity of the observed bonds, sortedin buckets: [0, 3) years, [3, 10) years, [10, 20) years and [20, Inf) years. The R2 of the regressions is between3.5% and 4.5%. The F -test is significant at the 99% confidence level for all regression models. Secondarybond market data is provided by MTS markets, and sovereign bond auction data has been collected fromThomson Reuters and DMO websites.

41

Panel Regression

TVOL TIMB IMPR

Variable Coefficient p-value Coefficient p-value Coefficient p-value

Intercept 0.82 0.00 0.32 0.00 100.05 0.00Austria -0.13 0.03 -0.04 0.22 -0.05 0.49Belgium 0.52 0.00 -0.08 0.28 -0.09 0.05France 1.21 0.00 0.03 0.12 -0.04 0.22Italy 1.28 0.00 -0.30 0.00 -0.02 0.46Netherlands 0.34 0.00 -0.04 0.18 0.03 0.41Portugal -0.10 0.29 -0.19 0.02 -0.06 0.02Spain 0.64 0.00 -0.07 0.05 -0.09 0.00Q1.LIQ 0.18 0.04 -0.03 0.24 -0.03 0.00Q4.LIQ -0.17 0.02 0.05 0.01 -0.01 0.87Q1.RISK -0.48 0.00 0.11 0.00 -0.00 0.63Q4.RISK 0.21 0.06 -0.12 0.00 -0.00 0.95LIQ -0.00 0.00 0.00 0.00 0.00 0.00RISK 0.18 0.06 -0.09 0.22 0.06 0.58CRATE -0.12 0.00 0.05 0.00 0.02 0.50MAT -0.01 0.03 0.01 0.00 0.00 0.23ASIZE -0.04 0.00 0.00 0.48 -0.00 0.50ASPREAD 0.00 0.00 0.00 0.33 0.00 0.03ALIQ 0.00 0.57 0.00 0.79 0.00 0.14ARISK -0.05 0.41 -0.01 0.14 0.04 0.20AMAT -0.00 0.00 -0.00 0.00 0.00 0.15MLIQ -0.00 0.43 -0.00 0.04 0.00 0.05MRISK 0.29 0.26 -0.03 0.07 -0.25 0.12FUNDL -5.38 0.00 0.51 0.03 0.26 0.52BRATE -0.50 0.70 -0.14 0.01 0.15 0.45BRISK 1.21 0.20 -0.67 0.01 0.16 0.53RISKA 0.04 0.00 0.00 0.38 -0.01 0.06

Table 5: This table shows the panel regression results for the period 2003-2007 (beforefinancial-crisis-period). The regression variables are the trading volume before the auction relative tothe average trading volume of the bond, TV OL, the trading imbalance based on seller initiated bond tradesbefore the auction relative to buyer initiated bond trades, TIMB, and the price impact based on returnscalculated from the mid-price before the auction to the time of the auction on the auction day and is givenin price points, IMPR. The table shows the estimated coefficients for different explanatory variables andcorresponding t-statistics in parentheses. For the fixed effects, the following factors are defined as a baselineand are included in the intercept term: Germany (country), the risk quantiles (Q2.RISK, Q3.RISK), andthe liquidity (Q2.LIQ, Q3.LIQ) for specific bonds. Our sample comprises auctions during the time periodfrom 2003 to 2016 (full sample). Reported standard errors are clustered around the remaining maturity ofthe observed bonds, sorted in buckets: [0, 3) years, [3, 10) years, [10, 20) years and [20, Inf) years. The R2

of the regressions is between 3.5% and 4.5%. The F -test is significant at the 99% confidence level for allregression models. Secondary bond market data is provided by MTS markets, and sovereign bond auctiondata has been collected from Thomson Reuters and DMO websites.

42

Panel Regression

TVOL TIMB IMPR

Variable Coefficient p-value Coefficient p-value Coefficient p-value

Intercept 0.48 0.07 0.65 0.00 99.83 0.00Austria 0.10 0.55 -0.09 0.26 -0.26 0.02Belgium 1.16 0.00 -0.20 0.00 0.40 0.18France 1.28 0.00 0.02 0.74 -0.06 0.46Italy 1.51 0.00 -0.45 0.00 0.19 0.31Netherlands 1.83 0.00 -0.03 0.35 0.27 0.06Portugal 0.00 0.99 -0.19 0.09 -0.25 0.16Spain 0.27 0.11 0.01 0.82 0.02 0.87Q1.LIQ 0.42 0.00 -0.09 0.00 0.03 0.27Q4.LIQ -0.25 0.37 0.15 0.01 -0.08 0.59Q1.RISK -0.42 0.00 0.10 0.00 0.05 0.07Q4.RISK 0.35 0.04 -0.05 0.00 -0.07 0.55LIQ 0.00 0.65 -0.00 0.00 -0.00 0.07RISK -0.06 0.73 -0.10 0.18 0.62 0.01CRATE 0.05 0.17 -0.02 0.07 -0.02 0.00MAT -0.00 0.84 0.00 0.62 0.01 0.42ASIZE -0.04 0.00 -0.00 0.71 -0.01 0.11ASPREAD -0.00 0.61 -0.00 0.71 -0.00 0.24ALIQ -0.00 0.00 -0.00 0.55 0.00 0.15ARISK -0.13 0.00 0.01 0.62 0.04 0.05AMAT -0.01 0.00 -0.00 0.63 -0.00 0.00MLIQ -0.00 0.11 -0.00 0.62 0.00 0.52MRISK -0.22 0.00 0.01 0.88 0.45 0.02FUNDL -1.05 0.00 0.09 0.34 0.22 0.09BRATE -0.63 0.07 -0.11 0.02 0.15 0.26BRISK -0.13 0.50 0.03 0.38 -0.06 0.72RISKA 0.03 0.01 0.00 0.56 -0.01 0.20

Table 6: This table shows the panel regression results for the period 2007-2009 (financial-crisisperiod). The regression variables are the trading volume before the auction relative to the average tradingvolume of the bond, TV OL, the trading imbalance based on seller initiated bond trades before the auctionrelative to buyer initiated bond trades, TIMB, and the price impact based on returns calculated from themid-price before the auction to the time of the auction on the auction day and is given in price points, IMPR.The table shows the estimated coefficients for different explanatory variables and corresponding t-statisticsin parentheses. For the fixed effects, the following factors are defined as a baseline and are included in theintercept term: Germany (country), the risk quantiles (Q2.RISK, Q3.RISK), and the liquidity (Q2.LIQ,Q3.LIQ) for specific bonds. Our sample comprises auctions during the time period from 2003 to 2016 (fullsample). Reported standard errors are clustered around the remaining maturity of the observed bonds, sortedin buckets: [0, 3) years, [3, 10) years, [10, 20) years and [20, Inf) years. The R2 of the regressions is between3.5% and 4.5%. The F -test is significant at the 99% confidence level for all regression models. Secondarybond market data is provided by MTS markets, and sovereign bond auction data has been collected fromThomson Reuters and DMO websites.

43

Panel Regression

TVOL TIMB IMPR

Variable Coefficient p-value Coefficient p-value Coefficient p-value

Intercept -0.45 0.03 0.85 0.00 99.75 0.00Austria 0.26 0.01 0.00 0.93 0.09 0.66Belgium 2.17 0.00 -0.31 0.00 0.22 0.18France 2.16 0.00 -0.12 0.07 0.38 0.06Italy 2.13 0.00 -0.58 0.00 -0.14 0.49Netherlands 1.91 0.00 -0.18 0.00 -0.04 0.81Portugal 1.79 0.00 -0.48 0.00 -0.34 0.34Spain 1.05 0.00 -0.36 0.00 -0.40 0.08Q1.LIQ 0.68 0.00 -0.12 0.00 -0.08 0.00Q4.LIQ -0.40 0.00 0.07 0.09 0.01 0.84Q1.RISK -0.85 0.00 0.18 0.00 -0.05 0.26Q4.RISK 0.28 0.19 0.02 0.55 -0.08 0.53LIQ -0.00 0.00 0.00 0.02 0.00 0.11RISK 0.05 0.53 -0.01 0.42 0.30 0.03CRATE 0.67 0.03 -0.10 0.10 -0.09 0.38MAT 0.01 0.52 -0.00 0.23 0.01 0.11ASIZE -0.10 0.00 -0.00 0.30 -0.00 0.42ASPREAD 0.63 0.00 0.01 0.15 0.51 0.04ALIQ -0.00 0.08 0.00 0.39 -0.00 0.02ARISK -0.03 0.64 0.00 0.91 0.04 0.13AMAT -0.03 0.00 -0.00 0.05 -0.01 0.06MLIQ -0.00 0.58 -0.00 0.04 -0.01 0.03MRISK 0.22 0.00 0.00 0.91 0.18 0.10FUNDL 1.63 0.00 -0.43 0.00 0.31 0.42BRATE 0.11 0.79 -0.04 0.84 -0.34 0.09BRISK -0.42 0.00 0.10 0.00 0.19 0.23RISKA 0.01 0.01 0.00 0.02 0.00 0.68

Table 7: This table shows the panel regression results for the period 2009-2012 (sovereign-crisisperiod).The regression variables are the trading volume before the auction relative to the average tradingvolume of the bond, TV OL, the trading imbalance based on seller initiated bond trades before the auctionrelative to buyer initiated bond trades, TIMB, and the price impact based on returns calculated from themid-price before the auction to the time of the auction on the auction day and is given in price points, IMPR.The table shows the estimated coefficients for different explanatory variables and corresponding t-statisticsin parentheses. For the fixed effects, the following factors are defined as a baseline and are included in theintercept term: Germany (country), the risk quantiles (Q2.RISK, Q3.RISK), and the liquidity (Q2.LIQ,Q3.LIQ) for specific bonds. Our sample comprises auctions during the time period from 2003 to 2016 (fullsample). Reported standard errors are clustered around the remaining maturity of the observed bonds, sortedin buckets: [0, 3) years, [3, 10) years, [10, 20) years and [20, Inf) years. The R2 of the regressions is between3.5% and 4.5%. The F -test is significant at the 99% confidence level for all regression models. Secondarybond market data is provided by MTS markets, and sovereign bond auction data has been collected fromThomson Reuters and DMO websites.

44

Panel Regression

TVOL TIMB IMPR

Variable Coefficient p-value Coefficient p-value Coefficient p-value

Intercept 0.45 0.01 0.64 0.00 99.63 0.00Austria 0.17 0.00 -0.04 0.39 -0.11 0.11Belgium 1.23 0.00 -0.28 0.00 0.02 0.27France 2.20 0.00 -0.14 0.00 -0.06 0.30Italy 2.34 0.00 -0.52 0.00 -0.14 0.34Netherlands 1.47 0.01 -0.05 0.04 0.04 0.04Portugal 1.59 0.00 -0.36 0.01 -0.12 0.44Spain 1.61 0.00 -0.29 0.00 -0.16 0.21Q1.LIQ 0.40 0.00 -0.04 0.03 0.02 0.07Q4.LIQ -0.38 0.00 0.14 0.00 -0.20 0.04Q1.RISK -0.32 0.01 0.06 0.10 0.00 0.32Q4.RISK 0.02 0.85 -0.00 0.89 -0.12 0.00LIQ -0.00 0.20 0.00 0.93 0.01 0.00RISK -0.05 0.11 0.02 0.00 -0.01 0.82CRATE 0.39 0.25 0.04 0.75 0.57 0.00MAT 0.00 0.62 0.00 0.48 0.00 0.80ASIZE -0.13 0.00 -0.00 0.65 0.01 0.34ASPREAD 0.20 0.00 -0.00 0.46 0.09 0.10ALIQ -0.00 0.00 0.00 0.83 -0.00 0.07ARISK 0.16 0.00 -0.01 0.31 -0.01 0.00AMAT 0.00 0.56 -0.00 0.00 0.01 0.17MLIQ -0.00 0.05 0.00 0.79 -0.00 0.31MRISK 0.09 0.41 -0.00 0.92 -0.03 0.01FUNDL 0.71 0.02 -0.22 0.00 -1.01 0.02BRATE 0.09 0.78 0.14 0.10 1.11 0.06BRISK -0.27 0.00 0.05 0.01 0.03 0.53RISKA 0.01 0.37 0.00 0.00 0.02 0.02

Table 8: This table shows the panel regression results for the period 2012-2016 (after sovereign-crisis period). The regression variables are the trading volume before the auction relative to the averagetrading volume of the bond, TV OL, the trading imbalance based on seller initiated bond trades before theauction relative to buyer initiated bond trades, TIMB, and the price impact based on returns calculated fromthe mid-price before the auction to the time of the auction on the auction day and is given in price points,IMPR. The table shows the estimated coefficients for different explanatory variables and correspondingt-statistics in parentheses. For the fixed effects, the following factors are defined as a baseline and areincluded in the intercept term: Germany (country), the risk quantiles (Q2.RISK, Q3.RISK), and the liquidity(Q2.LIQ, Q3.LIQ) for specific bonds. Our sample comprises auctions during the time period from 2003 to2016 (full sample). Reported standard errors are clustered around the remaining maturity of the observedbonds, sorted in buckets: [0, 3) years, [3, 10) years, [10, 20) years and [20, Inf) years. The R2 of theregressions is between 3.5% and 4.5%. The F -test is significant at the 99% confidence level for all regressionmodels. Secondary bond market data is provided by MTS markets, and sovereign bond auction data hasbeen collected from Thomson Reuters and DMO websites.

45

Economic Cost of the Auction Cycle

Country / Maturity 0-50Y 0-3Y 3-10Y 10-20Y 20-50Y

Austria 11.20 4.30 7.70 13.70 29.40Belgium 12.00 2.00 14.40 36.80 34.80France 12.80 2.00 10.30 30.30 53.70Germany 2.80 1.70 1.60 2.70 11.30Italy 8.20 1.70 13.90 20.60 23.20Netherlands 7.10 1.10 10.40 31.70 12.60Portugal 20.10 5.20 33.80 21.40 49.30Spain 6.70 1.70 5.90 9.00 30.70

Table 9: This table shows the average cost of the auction cycle for bonds of different maturitiesin basis points for the period 2003-2016 (full sample). The cost of the auction cycle are computedas the average price impact derived from the median quoted mid price five days before the auction to themid price that is quoted closest to the auction time. Bonds are sorted into buckets according to theirmaturity. Secondary bond market data is provided by MTS markets, and sovereign bond auction data hasbeen collected from Thomson Reuters and DMO websites.

46

Proofs

Proof of Proposition 1. The new issue bonds have higher service level if and only if

∆ =U1 − λw1

E1 + U1

− U0 − λw0

E0 + U0

≥ 0.

Solving for λ and noting that w1 > w0E1+U1

E0+U0yields the result.

Proof of Proposition 2. Taking derivatives of (5) with respect to Ui, Ei, U1−i, E1−i

yields the result.

Proof of Corollary 1. Consider the old issue bonds. Under conditions of the Corollary,

Proposition 2.(i) simplifies to

dQ0

dU0

=σ0

√3

2U0

+w0σ0w

31σ

21

√3(U1 − E1)

(E1 + U1)2(w2

0σ0 + w21σ1

2U0

E1+U1

)2 − w20σ

20

√3

w20σ0 + w2

1σ12U0

E1+U1

1

2U0

≥ σ0

√3

2U0

(1− w2

0σ0

w20σ0 + w2

1σ12U0

E1+U1

)≥ 0.

Hence, (i) is proved.

From Proposition 2.(ii),

dQ0

dE0

= −σ0

√3

2U0

+w0σ0w

31σ

21

√3(U1 − E1)

(E1 + U1)2(w2

0σ0 + w21σ1

2U0

E1+U1

)2 +w2

0σ20

√3

w20σ0 + w2

1σ12U0

E1+U1

1

2U0

.

Thus, dQ0

dE0≤ 0 if and only if

w1σ1

w0σ0

U1 − E1

E1 + U1

1

1 +w2

1σ1

w20σ0

2U0

E1+U1

− 1 ≤ 0,

which holds if w1σ1 ≤ w0σ0E1+U1

U1−E1. Hence, (ii) is proved.

From Proposition 2.(iii),

dQ0

dU1

= −w0σ0w

31σ

212√3U0

(U1−E1)E1+U1

(E1 + U1)2(w2

0σ0 + w21σ1

2U0

E1+U1

)2 − w0σ0w1σ1

√3

w20σ0 + w2

1σ12U0

E1+U1

2E1

(E1 + U1)2≤ 0.

Hence, (iii) is proved.

47

From Proposition 2.(iv),

dQ0

dE1

= −w0σ0w

31σ

212√3U0

(U1−E1)E1+U1

(E1 + U1)2(w2

0σ0 + w21σ1

2U0

E1+U1

)2 +w0σ0w1σ1

√3

w20σ0 + w2

1σ12U0

E1+U1

2U1

(E1 + U1)2.

Thus, dQ0

dE1≥ 0 if and only if

U1 −w2

1σ1U0

w20σ0 + w2

1σ12U0

E1+U1

U1 − E1

E1 + U1

≥ 0.

The latter can be simplified to the following quantity, non-negative on the feasible set of

parameters:

U1

(1− w2

1σ1U0

2w21σ1U0 + w2

0σ0(E1 + U1)

)+

w21σ1U0

w20σ0 + w2

1σ12U0

E1+U1

E1

E1 + U1

≥ 0.

Hence, (iv) is proved.

Now, consider the new issue bonds. Proposition 2.(i) simplifies to

dQ1

dU1

=σ12

√3E1

(U1 + E1)2

(1− w2

1σ1

w21σ1 + w2

0σ0E1+U1

2U0

)+

w20σ0w

21σ

21

√3(U1 − E1)

2U0(E1 + U1)(w2

1σ1 + w20σ0

E1+U1

2U0

)2 ≥ 0.

Hence, (v) is proved.

From Proposition 2.(ii),

dQ1

dE1

= − σ12√3U1

(U1 + E1)2+

w20σ0w

21σ

21

√3(U1 − E1)

2U0(E1 + U1)(w2

1σ1 + w20σ0

E1+U1

2U0

)2 +w2

1σ21

√3

w21σ1 + w2

0σ0E1+U1

2U0

2U1

(U1 + E1)2.

Simplifying, dQ1

dE1≤ 0 if and only if

U1 − E1

U1 + E1

w21σ1

2w21σ1 + w2

0σ0E1+U1

E0

− 1 ≤ 0.

The latter quantity is non-positive on the set of feasible parameters. Hence, (vi) is proved.

48

From Proposition 2.(iii),

dQ1

dU0

= − w20σ0w

21σ

21

√3(U1 − E1)

4U20

(w2

1σ1 + w20σ0

E1+U1

2U0

)2 − w1σ1w0σ0

√3

w21σ1 + w2

0σ0E1+U1

2U0

1

2U0

≤ 0.

Hence, (vii) is proved.

Finally, from Proposition 2.(iv)

dQ1

dE0

= − w20σ0w

21σ

21

√3(U1 − E1)

4U20

(w2

1σ1 + w20σ0

E1+U1

2E0

)2 +w1σ1w0σ0

√3

w21σ1 + w2

0σ0E1+U1

2E0

1

2E0

.

Thus, dQ1

dE0≥ 0 if and only if

1−w0

w1(U1 − E − 1)

2E0 +w2

0σ0

w21σ1

(E1 + U1)≥ 0.

Solving for E0, get the necessary and sufficient condition:

E0 ≥1

2

w0

w1

(U1 − E1 −

w0

w1

σ0

σ1

(E1 + U1)

).

Note, that the corresponding sufficient condition is U1 −E1 − w0

w1

σ0

σ1(E1 +U1) ≤ 0, or w1σ1 ≤

w0σ0E1+U1

U1−E1. Hence, (viii) is proved, completing the proof of the corollary.

Proof of Proposition 3. Taking derivatives of (5) with respect to σi, σ1−i yields the

result.

Proof of Corollary 2. Consider the old issue bonds. Under conditions of the Corollary,

Proposition 3.(i) simplifies to

dQ0

dσ0

= −w0

2E0

w21σ1

E1+U1w1σ1

√3U1−E1

E1+U1(w2

0σ0

2E0+

w21σ1

E1+U1

)2 ≤ 0.

Hence, (i) is proved.

From Proposition 3.(ii),

dQ0

dσ1

=w0σ0

2E0

w21

E1+U1w1σ1

√3U1−E1

E1+U1(w2

0σ0

2E0+

w21σ1

E1+U1

)2 −w0σ0

2E0

w20σ0

2E0+

w21σ1

E1+U1

× w1

√3U1 − E1

E1 + U1

49

Simplifying, dQ0

dσ1≤ 0 if and only if

w21σ1

w21σ1 + w2

0σ0E1+U1

2U0

− 1 ≤ 0

The latter quantity is non-positive on the set of feasible parameters. Hence, (ii) is proved.

Now, consider the new issue bonds. Proposition 3.(i) simplifies to

dQ1

dσ1

=√3U1 − E1

E1 + U1

+w1

E1+U1

w20σ0

2E0w1σ1

√3U1−E1

E1+U1(w2

1σ1

E1+U1+

w20σ0

2E0

)2 −w1σ1

E1+U1

w21σ1

E1+U1+

w20σ0

2E0

× w1

√3U1 − E1

E1 + U1

Simplifying, dQ1

dσ1≥ 0 if and only if

w1w20σ0 + 2E0

(w2

1σ1

E1 + U1

+w2

0σ0

2E0

)[w2

1σ1

E1 + U1

+w2

0σ0

2E0

− w21σ1

E1 + U1

]≥ 0

The latter quantity is non-negative on the set of feasible parameters. Hence, (iii) is proved.

Finally, from Proposition 3.(ii)

dQ1

dσ0

=w1σ1

E1+U1

w20

2E0w1σ1

√3U1−E1

E1+U1(w2

1σ1

E1+U1+

w20σ0

2E0

)2 ≥ 0.

Hence, (iv) is proved, completing the proof of the corollary.

50