Sovereign Debt, Default Risk, and the Liquidity ofGovernment Bonds
Gaston Chaumont∗
University of Rochester
May, 2024

Abstract
Secondary markets for sovereign bonds are illiquid because of trading frictions.
I build a framework with endogenous illiquidity to study its implications on credit
spreads and default risk. The model integrates directed search in secondary markets
into a macro model of sovereign default. In equilibrium, investors face a state dependent trade-off between transaction costs and trading probabilities that generates atime-varying liquidity premium. With trading frictions demand and supply flows in
the secondary market are important drivers of bond prices, while they are irrelevant
and indeterminate in standard sovereign default models. I also use the model to study
the effects of bond purchasing policies in secondary markets. I find that trading frictions significantly tighten the financial constraint of the government and that policy
interventions that reduce the sell flows in the secondary markets can partially revert
the effect of trading frictions.
Keywords: Sovereign Debt; Default; Intermediation costs.
JEL classification: D83, E32; E43; F34; G12.

∗ Email address: gaston.chaumont@rochester.edu. Postal address: 229 Harkness Hall, Department of
Economics, University of Rochester, Rochester, NY, 14627.

1

Introduction

Governments often issue long-term bonds to sell in the international credit market. Before
sovereign bonds mature, they are traded in over-the-counter (OTC) secondary markets wheretransactions are decentralized, costly and time consuming.1 Because investors value liquidity,
trading frictions in the secondary market affect not only the price of outstanding bonds, butalso the price of newly issued bonds and, hence, the government’s decision on whether todefault on its debt. In turn, the liquidity of sovereign bonds endogenously depends on thestate of the economy in addition to trading frictions in the secondary market. In this paper,
I study sovereign defaults incorporating the role of trading frictions in the secondary market,
which has been largely ignored in the sovereign default literature originated in the seminal
work of Eaton and Gersovitz (1981).2
Recent empirical work has emphasized the role of liquidity on sovereign bond markets.
For example, Nguyen (2014) reports that during the 2010-2012 European debt crisis evencountries with very liquid bonds faced illiquidity periods. She documents that the relative
bid-ask spread - a standard measure of liquidity - of Italian bonds reached 667 basis points,3an unprecedented level for Italian bonds which bid-ask spreads are usually below 50 basis
points.4 Large bid-ask spreads were also observed in Ireland and Portugal. However, themost extreme example is Greece. Bloomberg data shows that bid-ask spreads for 10 year
Greek bonds were about 2, 000 basis points, on average, in the fourth quarter of 2011, a
couple of months before the debt restructuring of March, 2012. However, the standard
Eaton and Gersovitz (1981) framework used in the sovereign debt literature cannot be used
to understand these issues.
To endogenize the liquidity of bonds, I integrate search frictions in the secondary market
into a general equilibrium model of sovereign debt with default risk. Because of search
frictions, the flows of demand and supply for bonds are important determinants of bondprices in the secondary market, while quantities traded in the secondary market are usuallyindeterminate and irrelevant for prices in the sovereign debt literature. Using this model I
can qualitatively and quantitatively address the first question of the paper. Namely, how
do demand and supply flows in the secondary market for sovereign bonds affect the price
1 See, for example, Duffie (2012) for details on OTC markets for bonds and World-Bank and IMF (2001)

for details structure of sovereign debt markets.
2 The only exception in the literature is Passadore and Xu (2022), who impose exogenous trading frictions
on individuals selling sovereign bonds in the secondary market.
3 See equation (12) for a formal defintion of the relative bid-ask spread.
4 For more details on the importance of liquidity shock in the Eurozone debt crisis, see, for example,
Nguyen (2014), and Pelizzon et al. (2016). For emerging market bonds, see Hund and Lesmond (2008).

1

schedule for newly issued bonds and how they interact with default risk?
In addition, because demand and supply flows in the secondary market have an effecton sovereign credit spreads, I can study the effects of policy interventions in the secondary
market for sovereign bonds. An example of such intervention in the context of default risk is
the Securities Market Programme (SMP) of the European Central Bank (ECB) implemented
in 2010-2011 in the context of the European debt crisis. Under the SMP, the ECB directly
purchased sovereign bonds in secondary markets to improve liquidity conditions and help
stabilize distressed sovereign bond yields.5 Between May and June, 2010, the ECB purchasedaround 10% of the outsdaing stock of Greek debt. Trebesch and Zettelmeyer (2018) findthat, after the intervention, the yields of Greek bonds purchased by the ECB decreased
significantly and persistently. The effects of such are interventions are puzzling from astandard sovereign default model’s perspective. This is because bond purchases in secondary
markets do not affect the usual determinants of credit spreads in sovereign default models,i.e. the debt-to-GDP ratio of the issuing country, bonds maturity, fiscal deficit, or aggregate
production. How do then secondary market interventions reduce bond yields? When are suchinterventions effective? By endogenizing liquidity in sovereign bonds’ market, I highlight therole of such policies in affecting demand and supply flows and contribute to understanding
how secondary market interventions affect bonds’ price, debt issuance and default.
To model endogenous liquidity, I incorporate frictions in bond markets in the same spirit
as Shi (1995) and Trejos and Wright (1995), for fiat money, and Duffie et al. (2005), forcorporate bonds. More specifically, in the model the sovereign government sells its debt todealers in a centralized primary market.6 Dealers trade bonds with foreign investors in adecentralized secondary market, acting as intermediaries between the government and foreign
investors. In the model, dealers do not have reasons to hold bonds other than re-selling themto investors. For their intermediation service, dealers charge investors a transaction fee. On
the other side of the market, investors demand sovereign bonds to maximize expected returns
and optimize their portfolio composition. In order to be able to trade a bond, investors
need to meet dealers in OTC markets, which are subject to search frictions. An investor’svaluation for a bond incorporates the cost of intermediation fees, the expected time to trade,
5 As stated by the ECB Press Release of May 10th, 2010, one of the goals of the SMP interventions was"to ensure depth and liquidity in those market segments which are dysfunctional." See the press release athttps://www.ecb.europa.eu/press/pr/date/2010/html/pr100510.en.html.
6 In reality, only a few large banks can trade in primary bond markets. All other investors, such asindividual investors, institutional investors, and investment funds, need to buy bonds in OTC markets. In
the model, dealers represent agents of those banks. For the case of Greece, the list of primary dealers canbe found at https://www.bankofgreece.gr/Pages/en/Markets/HDAT/members.aspx.

2

and default risk.
In the secondary market, search is competitive (or directed). Every period, dealers andinvestors choose to visit one specific submarket to search for a trading counterpart. Each
submarket is characterized by a transaction fee that the investor needs to pay to the dealer
if they trade. A matching technology determines the number of trades in each submarket
given the numbers of dealers and investors. In equilibrium, investors and dealers face a tradeoff between the intermediation fee and the trading probability. For an investor, the higher
the intermediation fee the investor pays, the higher the investor’s probability of trading.
For a dealer, the higher the intermediation fee the dealer charges, the lower the probabilityof trading. Investors’ and dealers’ entry decision into submarkets endogenously determine
transaction fees, trading volumes, and trading frequencies.7
Qualitatively, the model generates two main new insights. First, trade flows between
investors and dealers in the secondary market affect the price of newly issued government
bonds in the primary market. For example, if the flow of demand orders from investors inthe secondary market increases, the demand for bonds by dealers in the centralized primarymarket increases too. In this case, the bond price must increase to clear the centralized
primary market and restore equilibrium. Even if the government does not change debt
issuances. Second, default risk and illiquidity can in theory be positively or negatively
correlated. In a simplified version of the model I show that the sign of this correlation
crucially depends on the relative size of potential demand and supply for bonds in thesecondary market. All else equal, the higher the default probability, the lower the incentives
for investors to purchase bonds and the higher the incentives for bond holders to resale them.
This results in less buyers and more sellers matching with dealers in the secondary market,
which reduces the net demand for bonds in the primary market. To restore equilibrium
prices must fall, and whether the fall is proportionally larger or smaller than the requiredcompensation for the increase in default risk depends on how sensitive demand and supply
flows are to changes in prices. When the potential world demand is larger than potential
supply of bonds, as is the case for small open economies often studied in the literature, there
is a positive correlation between default risk and illiquidity that amplifies the effect of shocks
on bonds’ interest rates.
To quantify the effects of liquidity frictions and secondary market policy interventions in
7 Consistent with directed search, Li and Schurhoff (2019) document that, for the case of US municipal

bonds, there is a systematic price dispersion in fees charged by different dealers and that those dealers
charging higher fees provide more liquidity immediacy to investors.

3

an equilibrium model of sovereign debt, I calibrate the model to Greece.8 I find that trading
frictions are quantitatively important both for prices and quantities of sovereign bonds.
The model produces an average spread 75 basis points larger than a standard model withfrictionless secondary market under the same parameters. In addition, the debt to output
ratio is 124% in the baseline model while it would be 139% in the model with frictionlesssecondary markets. That is, trading frictions significantly tighten the borrowing constraintof the government. In addition, I decompose Greek credit spreads during the debt crises andfind that without trading frictions in the secondary market the interest rate paid by Greek
bonds would have been about 140 basis points lower.
I also use the model to study the role of sale flows in the secondary market and the effects
of policy interventions like the ECB’s Securities Market Programme of 2010-2011. I findthat the size of sale flows are also quantitatively important for bond prices and debt to GDP
ratios. Reducing the volume of sell orders to a half reduces the average spreads by 40 basis
points and increases the average debt to GDP ratio by 10 percentage points. If instead all
investors buy bonds and hold it until maturity, the average spread falls from 3.42% to 2%
and the average debt to output ratio increases from 124% to 137%. This result highlights theimportance of the type of lenders that hold sovereign bonds. If sovereign bonds are held by
investors that rarely adjust their portfolio then bond prices are larger than if bonds are heldby investors who adjust their portfolio frequently. In this way, policy interventions like the
SMP can increase sovereign bond prices by affecting the frequency at which bonds are tradedin the secondary market. If the SMP commits to hold purchased bonds until maturity, as it
did, the present and future sell flows are smaller and bond prices increase. Using the model,
I find that the SMP reduced sovereign spreads by around 15-35 basis points. If the ECB had
purchased twice as many bonds the spreads would have fallen by 32-72 basis points while if
the ECB had bought the whole stock of Greek debt spreads would have been 139-313 basis
points lower.
Relation to the literature. This paper contributes to the large body of research on
sovereign debt with strategic default originated in Eaton and Gersovitz (1981), with a strong
quantitative focus after the work by Neumeyer and Perri (2005), Aguiar and Gopinath(2007), and Arellano (2008). Because debt is long term, my work is closer to Hatchondo
and Martinez (2009) and Chatterjee and Eyigungor (2012). I contribute to this literature byendogenizing liquidity in the secondary market. The framework provides a tool to understand
8 Greece is a good case study because of data availability as it is one of the few countries with a default

episode since sovereign bonds trade in OTC markets.

4

how liquidity and default risk interact in equilibrium. In addition, I use information on bidask spreads and volumes traded in the secondary market to quantify the importance ofliquidity frictions for sovereign bonds’ credit spreads, government revenues from new debt
issuance, and incentives to default. Most papers in the literature on sovereign default haveabstracted from the secondary market completely. Exceptions are Broner et al. (2010) andBai and Zhang (2012), who consider frictionless secondary markets. Chaumont et al. (2024)
considers a short-term debt model where OTC markets for sovereign bonds and credit default
swaps interact in equilibrium but there is no role for sale flows as bonds mature in one period.
Moretti (2020) uses search frictions in a sovereign debt model to quantify the role of frictions
upon introducing GDP-linked bonds as a new asset.
The most closely related paper is Passadore and Xu (2022) who incorporates an exogenous trading frictions in the secondary market into a standard model of sovereign default.
They assume that investors that aim to re-sell bonds in the secondary market face an exogenously fixed probability of trading lower than one. Although both papers feature somesource of trading friction their work has a different focus. They study how this infrequent
selling opportunity interacts with default risk how default risk changes expected maturityof bonds and outside options of sellers. Because trading frictions are exogenously imposed,
their model is unable to capture how demand and supply flows in the secondary market
endogenously responds to changes in the state of the economy. To capture endogenous liquidity, I assume that investors buy and sell bonds in the secondary market and use directed
search to endogenize the trading probabilities in this market. Endogenous liquidity is important to assess the effects of policy interventions in the sovereign bond market, such as
the ECB’s securities market programme (SMP) of 2010-2011. In my model, interventions of
this kind directly affect the price, the bid-ask spreads, and the liquidity premium of newly
issued bonds, by changing the net demand for bonds in the secondary market. In contrast,
if trading probabilities are exogenous, such interventions may not affect the price of newly
issued bonds nor the terms of trade in bilateral meetings, because the interventions do notaffect the outstanding level of debt or default probabilities.
This paper is also related to the literature following Duffie et al. (2005) where investors
with heterogeneous valuations trade assets in OTC markets and liquidity is modeled using
search frictions. To execute mutually beneficial transactions, investors who have relatively
high and low valuations for an asset search for each other in frictional markets. In mymodel, the trading structure in the secondary market is closer to Lagos and Rocheteau(2009) and Lester et al. (2015), where (i) trades occur only between a dealer and an investor

5but not directly between investors; and, (ii) dealers do not need to hold inventories as they
have permanent access to a centralized primary market that serves as a clearing system.
The main contribution of my paper to this literature is to build a tractable equilibrium
model with endogenous liquidity over the business cycle and strategic default decisions,while this literature usually focuses on steady states and no defaults.9 Another contribution
is to endogenize the supply of assets. Supply’s response to market conditions is critical forunderstanding endogenous liquidity, but it is absent in this literature.
Previous studies also consider endogenous default decisions on corporate bonds. One
example is He and Milbradt (2014). However, the model assumes a stationary environment
where both the characteristics and the supply of assets are fixed over time. Chen et al. (2018)
extend the analysis to allow the aggregate state to take two values and show how exogenous
reductions in trading probabilities increase credit spreads. As a result, their model misses
two critical features of my model: an endogenous supply of bonds and endogenous trading
probabilities in the secondary market. As explained above, both features are necessary tounderstand how the liquidity of sovereign bonds responds to changes in the state of theeconomy and affects the price of newly issued bonds.
Finally, this paper complements a growing body of empirical work that investigates how
liquidity explains sovereign credit spreads during the European debt crisis and the effects of
ECB’s policy interventions in the secondary market. See, for example, Beber et al. (2009),
Calice et al. (2013), Nguyen (2014), Pelizzon et al. (2016), Eser and Schwaab (2016), Trebesch
and Zettelmeyer (2018), and De-Pooter et al. (2018).
Layout. The remainder of the paper is organized as follows. Section 2 describes the model
economy and defines an equilibrium. Section 3 illustrates the new channels in the model.
Section 4 calibrates the model and provides quantitative results. Section 5 studies the Greek
debt crisis. Section 6 concludes.

2

Model

Time is discrete and infinite. There are three types of agents: (i) a sovereign government;(ii) dealers; and (iii) investors. The country of interest faces a random endowment process
yt , distributed according to F (yt+1 |yt ). Its sovereign government maximizes the lifetime
9 One exception is De-Pooter et al. (2018) who incorporate an exogenous default probability into a Duffie

et al. (2005) model and studies the differences across steady states with different default probabilities andan exogenously determined supply of assets.

6

utility of the domestic representative household, given by∞X

β t U ( ct ) ,t=0

where U (·) is strictly increasing and concave, ct is household’s consumption, and β ∈ (0, 1) isthe discount factor. The government can save or borrow from international credit markets,described later in this section, and cannot commit to repay its debt obligations.
At the beginning of each period the stock of debt is Bt . The government chooses whether
to default or not on its debt obligations. There are two costs of defaults. First, the government is temporarily excluded from financial markets and cannot borrow or save underfinancial autarky. While the government is in default, every period it can re-gain access tofinancial markets with exogenous probability φ ∈ (0, 1).10 Upon re-gaining access to credit
markets, the government starts with no outstanding debt. Second, there is an output cost.
Under default the endowment is given by h(yt ) ≤ yt , for all yt .
If the country repays its debt and is in good credit standings, the government chooses next
period’s debt, Bt+1 . Newly issued bonds are sold to dealers in a Walrasian primary market.11
The sovereign government takes as given the price schedule of bonds q (yt , Bt , Bt+1 ), which
is determined in equilibrium. The arguments of this pricing schedule are the current levelof endowment, which allows investors to forecast next period’s endowment, the current levelof debt, Bt , and next period’s debt, Bt+1 . The price of bonds depend on the outstanding
stock of debt because it provides information on the flows of bonds traded in the secondary
market and the liquidity premium, as I it describe in section 2.4.
The maturity of bonds is determined by a parameter λ. Every period, each unit of debt
matures with probability λ ∈ [0, 1], independently of when that unit of debt was issued.
Thus, the average time to maturity of each bond is λ1 periods. Each unit of unmatured debt
pays a coupon z ≥ 0 every period. Therefore, the period t budget constraint of a sovereign
government that is in good credit standings is given by
ct + [λ + (1 − λ) z ] Bt ≤ yt + q (yt , Bt , Bt+1 ) [Bt+1 − (1 − λ) Bt ] .
The left hand side adds expenditures on consumption, coupon payments, and repayment of
matured bonds. The right hand side adds incomes from endowment and newly issued debt.
10 For models with endogenous market re-access see Yue (2010) and Benjamin and Wright (2013).
11 The government could organize an auction to sell the bonds. Since in this model there is complete andperfect information about valuations of the bonds, the auction could be designed to extract all the surplus
from dealers. That is, dealers would be acting as if there is perfect competition among them.

7

Dealers are risk-neutral. They can access the primary market without any cost andpurchase bonds issued by the government at the competitive price, q (yt , Bt , Bt+1 ). There
are no frictions in the primary market. Also, I assume that dealers have permanent accessto the primary market so that they only purchase bonds when they want to re-sell them toan investor.12 In other words, the primary market in the model merges the market for newly
issued bonds together with an interdealer market where dealers can trade with each other.
In addition, dealers have access to a frictional secondary market where they can trade withinvestors. The secondary market is characterized by directed search. Specifically, there isa continuum of submarkets that are labeled by the transaction fee that dealers charge toinvestors if they trade. Entry into submarkets is competitive. To enter any submarket, adealer needs to pay a per-period flow cost, γ.13 A dealer compensates the entry cost by
the bid-ask spread between submarkets. As the intermediaries between the primary and thesecondary market, dealers collect the orders from the secondary market and clear the net
demand (or supply) in the primary market at the end of each period.
There is a a fixed measure I¯ < ∞ of foreign investors in the secondary market. They
can trade bonds only by meeting a dealer. In order to trade, investors choose a submarket
to enter. That is, investor’s search for dealers is also directed. For simplicity, I assume
that investors can hold either zero or one unit of the bond. I denote an investor’s bondholdings a ∈ {0, 1}.14 Thus, the mass I¯ represents the maximum available wealth that could
be allocated to purchase sovereign bonds. To capture that the sovereign government isfrom a small open economy, I assume that I¯ is large and the amount of debt issued by thegovernment is not constrained by the total wealth of investors.
To parsimoniously generate incentives to trade sovereign bonds in the secondary market,
I assume that there are two types of investors, denoted ` and h. Type i ∈ {`, h} investors
have preferences ui over the bond, with uh > u` , in addition to other consumption c.15 To
12 This simplification avoids working with dealers that hold bond inventories and the need to solve addi-

tional dynamic optimization problems. Notice that this assumption replaces a dynamic cost for a static cost.
The model does not consider the dynamic cost of maintaining inventories and bearing the associated default
risk. However, this cost is replaced by the cost of creating enough transactions to re-allocate to investors all
bonds purchased by dealers, within each period.
13 This cost can be interpreted as a constant marginal cost of allocating a dealer into a submarket, for abank that participates in the primary market.
14 Although at the investor’s level the demand/supply of bonds in the secondary market is a discrete choice,at the aggregate level the investor’s net demand for bonds is continuous on the bond’s price and default
risk. This is because in equilibrium the trading probabilities are continuous in those variables (see lemma 3
in appendix A). Since the model has aggregate shocks and rational expectations, it would be unfeasible tosolve the model with divisible bond holdings because it requires keeping track of the distribution of bondholdings across all investors.
15 As described by Duffie et al. (2005), these simple differences in preferences for holding bonds are a8

fix ideas, one can think of type ` investors as liquidity constrained individuals that have alower than average valuation for holding the bonds. Thus, type ` investors are the naturalsellers of bonds.
Investors enter the economy as type h and without bonds. In equilibrium, type h investors
are the natural buyers of sovereign bonds. Once a type h investor acquires a bond, every
period the investor is exposed to a transitory preference shock that arrives with probabilityζ ∈ (0, 1), which is i.i.d. across periods and investors. The preference shock changes theinvestor type to ` during the current period. In the equilibrium, type ` investors wish tosell the bonds in the secondary market. For simplicity, I assume that once an investor sells
the bond, the investor leaves the economy and is replaced by a new type h investor without
bonds. An investor can get rid the bond in two ways: (i) by selling it to a dealer in thesecondary market, or (ii) if the bond matures, which occurs every period with probability
λ. If the investor does not get rid of the bond, next period it will be a type h investor, withprobability 1 − ζ, or a type ` investor, with probability ζ. Finally, investors have access to arisk-free, perfectly liquid, one period zero-coupon bond that pays an exogenous return r > 0.
In each submarket in the secondary market, there is a constant returns to scale orderprocessing (or matching) technology denoted M(d, n), where d is the number of dealers andn is the number of investors. Each investor’s order is equally likely to be executed at any
time. The probability of an order being executed is given by α(θ ) ≡ M(nd,n) = M(θ, 1), with
θ ≡ nd . The number of orders executed by a dealer in a period of time is then ρ(θ ) ≡ M(dd,n) .
I assume that M(·, ·) is the same across submarkets and satisfies that α(0) = 0, α(∞) = 1,
ρ(∞) = 0, and α(·) is strictly increasing and concave.
The timing of actions within each period is as follows:
1. Endowment yt is observed. The government decides whether or not to default. If
the government defaults the bond is not available as a possible investment choice forinvestors, and investor’s and dealer’s problems are irrelevant.
2. If the government repays, it chooses next period debt, Bt+1 , optimally.
3. Investors’ preference shock is realized and a fraction ζ of bond holders become type `investors.
4. A fraction λ of Bt matures. Their owners are replaced by h investors without bonds.
Principal of matured bonds is paid to current bond owners. Unmatured bonds pay
coupon z and yield utility ui to investors of type i ∈ {`, h}.
reduced form to capture that investors face different liquidity needs, financing costs, hedging reasons, tax
advantages, and/or personal use of the asset.

95. The centralized primary market and the decentralized secondary markets open. In thecentralized primary market, the government and dealers trade at a competitive price.
Investors and dealers decide optimally which submarket to visit and, those who meeta counterpart, trade in the secondary market.
6. The government and the dealers derives utility from consumption while the investors
derive utility from consumption and from their bond position according to their types.
At the beginning of each period, once the endowment shock is realized, the aggregate stateof the economy is xt ≡ (yt , Bt ). By the time investors make their submarket choices in step5, government’s debt choice Bt+1 (xt ) is already known. Thus, if the government is undergood credit standings, the relevant state of the economy for investors is st = (yt , Bt , Bt+1 ).
In what follows I formulate the government’s, investors’, and dealers’ problems in recursive
form and eliminate time period subindices.

2.1

Government

At the beginning of each period, the government chooses whether to default, δ = 1, or repay,
δ = 0, and the optimal debt issuance in case of repayment. For each state x = (y, B ), itsvalue function is:
V (x) = maxδ∈{0,1}no(1 − δ ) V R (x) + δV D (y ) ,(1)

where V R (·) is the value of repaying debt obligations and V D (·) is the value of default. The
value of defaulting is
h





 i

V D (y ) = U (h (y )) + βEy0 |y φV y 0 , 0 + (1 − φ) V D y 0.

(2)

That is, if the government decides to default, it does not repay its outstanding debt andconsumes the total output of the current period. However, there is an output loss associatedto the default decision and today’s consumption is given by the function h(y ). In addition,
the continuation value is a weighted average of the value of re-gaining credit access, which
occurs with probability φ, and starting next period in default, with probability (1−φ). If thegovernment re-gains access to credit, it starts with zero outstanding debt, with all investors
being type h and holding no bonds. In the case of debt repayment, the government chooses
consumption of the domestic household, c, and the new stock of debt, B 0 . The value of
repaying isnoV R (x) = max0 U (c) + βEy0 |y V (x0 ) , s.t. :c,B10

[BCG ]:hic + [λ + (1 − λ) z ] B = y + q (x, B 0 ) B 0 − (1 − λ) B .

The price schedule q (y, B, B 0 ) is determined endogenously in the primary debt market anddepends on the amount of newly issued government bonds. The government internalizes
the effect of changes in the stock of debt on bonds’ price, but it takes as given the pricing
function. As described in more details in section 2.4, the price schedule has the current stockof debt as a state variable. This is because, in equilibrium, ζB investors are sellers of thebond in the secondary market and reduce the net demand for newly issued bonds. The total
amount of bonds sold in the secondary market is determined by the mass of investors thatexogenously change types from h to ` as well as by the endogenous mass of investors thatare able to trade bonds in the decentralized market. The government internalizes that bychanging the supply of bonds it will affect investors’ policy functions and the evolution of
trade flows in the secondary markets. Using the budget constraint to substitute for c, theobjective function becomes:


h

i



V R (x) = max{U y + q (x, B 0 ) B 0 − (1 − λ) B − [λ + (1 − λ) z ] B + βEy0 |y V (x0 )}. (3)0B

As it shall be described later, the price of newly issued bonds, q, is determined by thegovernment supply of bonds and the dealers’ net demand for bonds in the primary market.
Liquidity frictions in the secondary market and the state of economy affect the dealers’
net demand for bonds and, thus, q. Therefore, liquidity frictions affect the price of newly
issued bonds, the optimal debt issuance, B 0 , the value of repayment, V R , and eventually the
optimal choice of repayment versus default, δ.

2.2

Investors

Investors trade bonds in the secondary market. I denote the set of submarkets as F̄ =[fmin , fmax ]. I let the lower bound be fmin ≡ γ, since no dealer will be willing to enter asubmarket that does not pay enough to cover the entry cost. In addition, the price fora bond that matures with probability λ, with a flow return (z + uh ) every period until
maturity, and that never defaults would beλ+(1−λ)(z +uh ).
λ+rThus, we can define the upperλ+(1−λ)(z +uh )bound fmax ≡since no investor would be willing to pay a higher intermediation
λ+rfee, even if the trading probability is equal to one and q = 0.
Denote the value functions of an investor holding a units of the bond at the beginning of
the period as I a , with a ∈ {0, 1}. Denote Ji1 the value function for an investor of type i ∈ `, hwith bond holdings a = 1 after the realization of the preference shock determines the type11

of the investor in the current period.
2.2.1

Investors without a bondFor each state s = (y, B, B 0 ), the value for an investor with a = 0 is given byhi1
I (s) = max{α (θ (f )) −q (s) − f + uh +Ey0 |y 1 − δ (x0 ) I 1 (s0 )
1+r
f ∈F̄
h
i
1 − α (θ (f ))+Ey0 |y 1 − δ (x0 ) I 0 (s0 )}.
1+r


0



(4)

The investor chooses optimally which submarket to visit (how much transaction fee, f , topay) in order to purchase a unit of the sovereign bond. In submarket f , the investor will beable to trade with a dealer with probability α (θ (f )). Once matched, the investor purchasesa unit of the bond after paying its price in the primary market, q (s), plus the transactionfee, f . In addition, holding a bond derives a continuation value of I 1 (s0 ) in the next period,
provided that the government does not default on the bond when next period starts. The
continuation value is discounted at rate r, which is the rate of return on the perfectly liquid,risk-free bond. If the investor is not matched with a dealer, which happens with probability
1 − α (θ (f )), the investor receives the discounted continuation value of not holding a bondat the beginning of the next period, conditional on the government not defaulting. If thegovernment defaults the continuation values are zero.
Notice that the value for an investor is computed for all B 0 , including those B 0 6= B 0 (x)that are off the equilibrium path because they are not optimal choices for the government.
This is because the government internalizes how the amount of newly issued debt affects
investors incentives to buy and sell bonds.
2.2.2

Investors with a bond

The value of holding a bond at the beginning of the period is,I 1 (s) = λ + (1 − λ) [z + ζJ` + (1 − ζ )Jh ] .

(5)The investor obtains the face value, 1, if the bond matures, which occurs with probability λ.
If the bond does not mature, the bond pays the coupon z and, depending on their individual
realization of the preference shock, they obtain the value Ji , i ∈ {`, h}, with
Ey0 |y [1 − δ (x0 )]I 1 (s0 )J` = max α (θ (f )) [q (s) − f ] + [1 − α (θ (f ))] u` +
1+r
f ∈F̄
(

"

12

#)

(6)

Ey0 |y [1 − δ (x0 )]I 1 (s0 )Jh = max α (θ (f )) [q (s) − f ] + [1 − α (θ (f ))] uh +
1+r
f ∈F̄
(

"

#)

(7)

The value of holding a bond for a type ` investors is the maximized value of participating into
the secondary market for sovereign bonds. Thus, investors choose optimally a submarket,
f , to sell their bond and trade with associated probability α (θ (f )). If they match with adealer, they sell their bond at the interdealer price q (s) minus the intermediation fee f . Ifthey do not match with a dealer, type ` investors receive a flow value u` < 0 for holding thebond and the expected discounted continuation value of holding a bond in the next period,
I 1 (s0 ), conditional on the government deciding to repay its debt obligations.
Similarly, the value of holding a bond for a type h investor is derived from the optimized
value of participating in the secondary market for sovereign bonds. In equilibrium, it isusually the case that type h investors do not wish to re-sell their bonds and choose and fconsistent with α (θ (f )) = 0. They obtain the flow value uh > 0 for holding a bond plus the
expected continuation value of being a bond holder, I 1 (s0 ), conditional on the government
not defaulting. However, if the price of sovereign bonds, q, in the interdealer market is highenough, even type h investors may decide to pay a fee f > γ and have a positive probability ofselling their bond. For a type h investor with a = 1 to chose to enter a submarket f > γ, theprice q (s) should be relatively large, as these investors like holding the bond. In addition,
since I 0 (s0 ) ≥ 0 and the type h investor’s revenue of buying a bond when holding a = 0 is
lower than the value of keeping a bond for investors with a = 1, i.e.,
h
ih
i
h
i
1
1
Ey0 |y 1 − δ (x0 ) I 1 (s0 ) − I 0 (s0 ) < uh +Ey0 |y 1 − δ (x0 ) I 1 (s0 ),1+r
1+rthen, whenever type h investors wish to sell their bond, no investor is willing to purchasea bond. Thus, only investors trying to sell will participate in the secondary market, which
can only occur in the case that the sovereign government decides to retire a large amount ofbonds from the market. That is, bond buybacks are expensive for the government when it
needs to induce type h investors to sell.

2.3

Dealers

Dealers participate competitively in debt markets. Each dealer chooses a transaction fee tocharge investors for the intermediation service. To enter any given submarket a dealer needs13to pay a flow cost γ > 0. A dealer posts the intermediation fee to maximize expected profits:Π = max{ρ (θ (f )) f − γ},

(8)

¯
f ∈F

where ρ (·) represents the probability of being able to execute an order, derived from thematching technology M described earlier. Competitive entry of dealers implies that
Π (f ) ≤ 0 and θ (f ) ≥ 0,

(9)

holds with complementary slackness. Whenever expected profits for dealers are negative ina submarket f , the associated market tightness θ (f ) is zero, since no dealers have incentives
to enter. On the other hand, whenever the market tightness is positive, a positive mass ofdealers enter the submarket until expected profits are zero.
Condition (9) provides a mapping from each submarket intermediation fee, f , to thetightness in that submarket. This mapping is given byθ (f ) =  ρ−1 γif Π (f ) = 0,0otherwise

f



.

(10)

In turn, market tightness determines trading probabilities for investors as a function ofintermediation fees, α(θ (f )).

2.4

Market clearing

The primary market is Walrasian and only government and dealers can access it. In eachstate s, and conditional on the government being in good credit standings, the price q (s)must clear the bonds primary market.
Recall that I¯ is the total mass of investors. Each period, after the realization of idiosyncratic preference shocks, let H0 = I¯ − B be the mass of type h investors with a = 0,
H1 = (1 − ζ )B the mass of type h investors with a = 1, and L1 = ζB the mass of type `investors with a = 1. Then, in any given period, I¯ = H0 + H1 + L1 . In addition, since outstanding bonds are held by investors, B = H1 + L1 .16 Using this notation, the total supply
of bonds in the primary market is given by the newly issued debt by the government plus
16 Notice that knowing the outstanding stock of debt, B, is enough to know the distribution of investors

over their types and bond holdings, which significantly reduces the dimensionality of the state spaces andsimplifies the computation of the model.

14the selling orders received by dealers from investors in the secondary market. That is,
n

o









max B 0 , 0 − (1 − λ) B + α θ`1 (1 − λ) L1 + α θh1 (1 − λ) H1 .
|

{z

|

}

Government’s supply

{z

|

}

Sellers’ supply{z}

Potential type h sellers

The first term is the government’s new bond issuances. The operator max {B 0 , 0} captures
the possibility that the government chooses B 0 < 0, in which case the government can atmost demand the (1 − λ) B outstanding bonds in the primary market before turning its net
savings positive. The second term is the supply of bonds from "sellers", i.e., bond holders
who are hit by the preference shock to become type ` investors. They wish to sell their bondholdings. Since a fraction λ of them will see their bond mature, only a fraction 1 − λ of them

will attempt to sell the bond to a dealer. Among the sellers, only a fraction α θ`1 will get
matched with a dealer, where θ`1 is the tightness in the submarket optimally chosen by type
` investors. Finally, the third term is the potential supply of bonds by type h investors witha = 1, which is positive only if the government is buying back bonds at a sufficiently highprice, as discussed in section 2.2.2.
On the other side of the primary market, the demand for bonds, are the buying ordersα θh0 λB.

received by dealers from the fraction of type h investors, α θh0 :α θh0 H0
|

{z



+

}

{z

|

Old Buyers’ demand}

New Buyers’ demand

The first term represents "old buyers," i.e. type h investors with a = 0 who are in themarket from the last period. The second term, represents the "new buyers," i.e. those typeh investors who entered the economy in the current period to replace the investors that leftthe economy after their unit of the bond matured. The tightness θh0 corresponds to thesubmarket optimally chosen by type h investors with a = 0.
Combining the demand and the supply for bonds in the primary market, I define theexcess demand function for each state s = (y, B, B 0 ) asED (s) ≡ α θh0 (s)|hI − (1 − λ) B − max B 0 , 0 − (1 − λ) B

o

i

{z

{z

}

Buyers’ demand



i

h

}

|





n

Government’s supply−α θ`1 (s) (1 − λ) ζB − α θh1 (s) (1 − λ) (1 − ζ ) B .
|

{z

}

Sellers’ supply

15

|

{z

Potential type h sellers}

Thus, the price schedule in this economy is implicitly defined by the following conditions
ED (s; q (s)) ≤ 0 and q (s) ≥ 0,

(11)

which hold with complementary slackness. As I show in appendix A.1, this excess demand
function is consistent with only one price q (s) clearing the market, for each state s. In the
appendix, Lemma 3 states that, given a government’s default policy function and a choice fornext period’s debt, the dealers’ net demand for bonds in the primary market is decreasingin the price of the bond, and strictly decreasing in most of the cases. The result follows
because: (i) the fraction of type h investors without bonds that purchase one from a dealer,α(θh0 ), is decreasing in the price of the bond, and (ii) the fraction of investors holding a bondthat sell it to a dealer, α(θ`1 ), is increasing in the price of the bond. α(θh0 ) is decreasing in q
because the dealer charges the investor q + fh0 (s), and so the investor pays more for the same
expected return. Therefore, the investor responds by optimally reducing the intermediation
fee fh0 (s). As a result, dealers earn lower expected profits and there is less entry into thesecondary market, which reduces the matching probability of investors trying to purchasebonds. As each investor’s trading probability decreases, a smaller mass of them trade. A
similar argument explains why the α(θ`1 ) increases with q.
Appendix A.2 shows that there is only one market clearing price q (s) for each s. To be
more precise, this uniqueness statement is conditional on a given government’s default andbond supply policies, investors value functions, and future expected prices. As it is standard
in sovereign default models of long term debt, I cannot guaranteed that the solution of theprice schedule is unique in general equilibrium. However, the result highlights the parallelismof the pricing schedule to the standard no-arbitrage condition that maps future prices anda default policy function into current prices. In this sense, solving this model is not harderthan other models of sovereign default. Instead of having a closed form expression for the
price as in the standard no arbitrage condition, I need to find the price consistent with (11).

2.5

Equilibrium

The equilibrium concept used here is recursive competitive equilibrium.
Definition 1. A Recursive Competitive Equilibrium (RCE) in this economy consists of a set
n

o

n

o

of value functions V , V R , V D , I 0 , I 1 , J` , Jh , Π , a set of policy functions δ, B 0 , fh0 , fh1 , f`1 ,a tightness function θ, and a pricing function q, such that for all x = (y, B ): (i) givenfunctions q (x, B 0 ), f`1 (x, B 0 ), θ (x, B 0 ), the functions V (x), V R (x), V D (y ), δ (x), B 0 (x),solve the sovereign government’s problem in (1)-(3); (ii) for all s = (x, B 0 ), given q (s),16δ (x), B 0 (x), θ (s), the functions I 0 (s), I 1 (s), J` (s), Jh (s), fh0 (s), fh1 (s), f`1 (s) solve theinvestor’s problem in (4), (5), (6) and (7); (iii) the tightness function θ (x, B 0 ) is consistent
with free entry of dealers according to (10); and (iv ) the function q (x, B 0 ) clears the primarymarket for bonds.

3

Main mechanisms and the role of liquidity

Before turning to a quantitative evaluation of the role of trading frictions on sovereign bondmarkets, I discuss the most novel qualitative features of the model. First, I provide a
simplified example where I can show in closed form how trading frictions affect the bondprice schedule and compare it to standard no arbitrage pricing. Second, I provide a two
period example that shows how trading frictions can contain or amplify the fall in bondprices when default risk increases.

3.1

The role of trading frictions

To build intuition, I consider a particular case in which I can write a closed form expression
for the pricing schedule implicitly defined in (11). In particular, I assume that the orderprocessing technology (matching function) is the "Telephone Line" function, given byM(n, d) =n×dn+d

In addition, I assume that high type investors never become low type, ζ = 0. Under theseassumptions, the masses of investors buying, keeping and selling their bond are, respectively,hi

H0 ≡ I − (1 − λ) B , H1 ≡ (1 − λ) B, L1 ≡ 0.
For this example I define ∆B ≡ B 0 − (1 − λ)B, the amount of newly issued bonds. After
solving for q in the market clearing condition, I can write the price in the primary market as
nh
ih
io
1
q (s) = uh +Ey0 |y I 1 (s0 ) − I 0 (s0 ) 1 − δ (y 0 , B 0 ) −
1+r
{z
}
|
{z
}|
Default Risk

Value of holding bond

2



1 
γ
1 − ∆B
H0
|

{z

.
}

Liquidity Component

Some remarks are in order. The price is divided into three component: (i) investor’s expected discounted value of acquiring a bond, (ii) an adjustment for default risk, and (iii)17a liquidity component.17 The first term in the right hand side corresponds to components(i) and (ii) and is very similar to the standard no arbitrage condition of sovereign default
models. The only difference is that q (s0 ) is replaced by the value of becoming bond holder,hi

I 1 (s0 ) − I 0 (s0 ) , and that having possession of the bond gives investors utility uh . The

second term in the right hand side is the liquidity component, which contains the following
ingredients. First, γ represents the importance of intermediation frictions. The more dealers
have to pay to participate in the secondary market (higher γ), the larger is the price discount from the liquidity component. Second, the ratio ∆B/H0 (≤ 1), represents the size of
new issuances relative to potential investors’ demand for bonds. The larger is the amountof newly issued debt the larger is the price discount from the liquidity component. This isbecause the larger is the debt issuance, the more investors need to be matched with dealers
in the secondary market, which is only possible if more dealers enter. Entry is larger only if
investors visit a submarket where they pay a larger intermediation fee. Therefore, since thetotal amount paid by investors is q + f , to induce investors to pay a higher intermediation fee
and attract more dealers, the price in the primary market has to fall. This term highlights
how the flows of bonds traded affect the bond’s price in the primary market. Finally, thelast ingredient of the liquidity component is the maturity probability λ. Because of the small
open economy assumption, I¯ > B 0 , it is always true that the longer the maturity of the bond(smaller λ), the lower is the price discount due to the liquidity component. Mechanically,
this is because a smaller λ implies a smaller ∆B/H0 ratio. The reason is that, in order toachieve certain new stock of debt B 0 , a smaller flow of debt issuance is needed when a smaller
fraction of bonds mature every period.

3.2

The amplification mechanism

Section 3.1 provides some intuition on how trading frictions affect the level of bond’s price.
To do so, the example eliminates type ` investors’ bond re-sales in the secondary market,
which compete for buyers with government’s newly issued bonds. In the data re-sales usually
represent the largest fraction of bonds supplied. I re-incorporate investors’ sales in thesecondary market to turn to the question of whether the bond price’s response is more orless than proportional to changes in default risk. To simplify the exposition, I use a simplified
two-period version of the model. Although simple, this example misses part of the dynamic
effects of changes in default risk on investor’s utility.
 1 0
I (s ) − I 0 (s0 ) which takes
into account future liquidity conditions and their effects on the value for holding the bond. The purpose ofthis example is to build some intuition.
17 Notice that some part of the effects of liquidity are hidden inside the term

18

Assume that there are only two periods, t = {1, 2}. The government has an initial outstanding debt level B1 > 0. L = ζB1 of the bonds are held by type ` investors while B1 − L
are held by type h investors. In addition, there is a mass H of type h investors that do
not hold a bond and could potentially buy a unit. At t = 2, if there is no default, low type
investors receive a value U` < 1 per unit of bond held and high type investors receive a value
Uh > 1 per unit of bond in their portfolio.
At t = 1, the government issues new debt to be paid at the final period, ∆B ≡ B2 − B1 ,where B2 is the stock of debt at the beginning of t = 2. All bonds mature in period 2 andthe government defaults with probability δ̄.
Investors decide whether to buy or sell a unit of a bond in the secondary market, and picka submarket to do so. Type h investors holding no bonds (buyers) maximize()(1 − δ̄ )Uh
I 0 (B1 , B2 ) = max α(θh ) −q +− γθh ,
θh1+rwhere I use the conditions ρ(θ )f = γ and αρ((θθ)) = θ to solve for θh instead of fh . Similarly,
type ` investors holding a bond (sellers) maximize()(1 − δ̄ )U`J` (B1 , B2 ) = max α(θ` ) q −− γθ` .
θ`
1+r
At an interior optimum18 , we have that(

(1 − δ̄ )Uh[θh ] : γ = α0 (θh ) −q +1+r(

(1 − δ̄ )U`[θ` ] : γ = α0 (θ` ) q −
1+r

)

)



=⇒ θh = α0−1 =⇒ θ` = α0−1 γδ̄ )Uh−q + (1−1+rγδ̄ )U`q − (1−1+r

 , and.

The market clearing condition at t = 1 is given by∆B = α(θh )H − α(θ` )L.
Assuming the same functional form of the matching function as in section 3.1, and using
18 Whenever investors have strictly positive gains from trade there is an interior optimum in this example.

If gains from trade are weakly negative the optimal tightness is zero and investors do not participate of thesecondary market. In this example I focus on interior optima and assume that gains from trade are strictly
positive for both types of investors.

19

investors’ optimality conditions, the price of the bond solves1/2 

1/2 






γ
γ



−
L
1
−
∆B = H 1 − 







(−q + (1−δ̄ )Uh )(q − (1−δ̄ )U` )






1+r

1+r

I define M ≡ H −L− ∆B, which I assume to be positive, as it will be the case in the calibrated
model in section 4.19 Using this definition and the optimality conditions, I express the market
clearing condition asHq(− 1−δ
+ 1U+hr )1/2

−

L
q
( 1−δ
− 1U+`r )1/2=

M
(1 − δ )1/2
γ 1/2

After changes in the default probability, if the price changes in the same proportion as theprobability of repayment, the above condition does not hold. This is because the left handside remains unchanged while the right hand side decreases (because M > 0). Therefore, theleft hand side must decrease to restore equilibrium. The left hand side decreases if and onlyq
if the ratio 1−δ
falls. That is, to clear the bond market, the price of bonds has to fall morethan the probability of repayment. This implies that the price of the bond gets relatively
closer to valuation of low type investors. I provide a formal proof in appendix A.3.
Intuitively, when the default probability increases, low valuation investors are willing to
pay higher intermediation fees in order to sell their bonds faster. Therefore, more investors
sells their bonds. But, since high valuation investors also trade bonds in the secondary
market, more buyers have to meet dealers to acquire the larger amount of bonds sold by
low type investors. That can only happen if more dealers enter submarkets with high type
investors, which requires that high type investors pay higher intermediation fees to attract
them. Thus, to induce high type investors to pay higher intermediation fees, the price ofthe bond in the centralized primary market, has to fall more than what would compensate
investors for higher default probability. The mechanism highlights the importance of allowing for endogenous trading probabilities as a determinant of secondary market liquidity inequilibrium.
The small open economy assumption that the mass of potential buyers, H, is large relative
to the debt levels is key for this result. It implies that M > 0. If M = 0 market clears when
19 In the calibrated model M > 0, reflecting the assumption that we are working with an small open
economy. Therefore, the worlds’ potential demand for sovereign bonds from the small open economy islarger than the newly supplied bonds and the bonds held by low type investors together. The effective
demand for bonds, α(θh )H, depends on how intensively each investor aims to trade a bond, which in turns
is endogenously determined by the expected return of the bond.

20

q
the ratio 1−δremains unchanged. If M < 0 the result reverses and an increase in the default

probability results in a decrease in the price of the bond that is less than proportional to thefall in the probability of repayment.
As a final remark, notice that this simple example assumes that the amount of newly
issued bonds ∆B is exogenous and constant. In the full model, the government responds
to changes in the state of the economy by adjusting the supply of bonds. If the supply ofbonds increases (M decreases) in states in which the probability of default increases theamplification effect is stronger. If the supply of bonds decreases when default risk increases
the amplification effect is weaker and can even be reverted.

4

Quantitative analysis

In this section I assess the quantitative importance of trading frictions in the secondary
market for sovereign bonds and conduct exercises to measure some of the effects of policy
interventions. It is well known that long-term debt models have convergence issues as clearly
documented by Chatterjee and Eyigungor (2012). To obtain convergence, I follow the literature and solve an approximate model by introducing preference shocks to the default andnext period’s debt choices of the government. Preference shocks are as small as needed toget convergence and do not significantly change government optimal choices.20 The maindifference from a standard long-term debt sovereign default model is that the price scheduleis determined by market clearing condition (11) instead of the standard no-arbitrage condition. Updating the pricing schedule requires solving for investors’ optimal choices and thenet demand from investors for each solution of the government problem. Then, given thesolution of the government problem and investors’ net demand for bonds, I can solve forthe market clearing price, q (s), for each s = (y, B, B 0 ). Although the solution requires somemore steps than solving standard models, it does not significantly increase the computationalburden. I describe the model with preference shocks and the solution algorithm in appendix
B.

4.1

Interest rate spreads and liquidity measures in the model

The model produces trading probabilities for dealers and investors as well as intermediation
fees. In this section I show how trading probabilities and intermediation fees in the model
can be mapped to the measures of liquidity observed in the data such as the bid-ask spread,volume traded, and the turnover rate of bonds.
20 See Dvorkin et al. (2021) and Gordon (2019) for further details.

21

I first consider the bid-ask spread. The bid-ask spread is defined as the difference of theask price, q A , that an investor pays to buy a bond and the bid price, q B , that an investor
gets for selling a bond. This spread is measured as a proportion of the mid price, q M . In
the model, for each s = (y, B, B 0 ), I defineq A (s) ≡ q (s) + fh0 (s), q B (s) ≡ q (s) − f`1 (s), and q M (s) ≡q A (s) + q B (s).
2

So the model counterpart of the bid-ask spread is given by the sum of the intermediation
fees divided by the mid-price and multiplied by 10, 000 to measure it in basis points, i.e.,
S B−A (s) ≡q A (s) − q B (s)fh0 (s) + f`1 (s)
×
10,
000
=
× 10, 000.
q M (s)q M (s)(12)

Using the model, I also construct the traded volume in secondary markets, given by 

V ol (s) ≡ α θ fh0 (s)

 h

i

 

I − (1 − λ) B (s) + α θ f`1 (s)(1 − λ) [ζB (s) + (1 − ζ ) L1 (s)] .

Finally, I define the turnover rate for bonds as
T urnover (s) ≡V ol (s)
.
B (s)

(13)

In addition, I compute the interest rate spread of the risky sovereign bond over a perfectly
liquid risk-free bond that pays an interest rate r every period. To compute this credit spread,denoted S R (s), I calculate the return rate rg (s) that equates the present discounted value of
λ+(1−λ z

the promised sequence of future payments on a bond to its price. That is, q (s) = λ+rg (s)) .
Then, the credit spread is"

λ + (1 − λ) z
S (s) ≡ (1 + r (s)) − (1 + r ) = 1 +
−λ
q (s)
R

4

4

#4

4

− (1 + r ) .

(14)

The power 4 in (14) it to calculate annualized spreads because I calibrate the model at thequarterly frequency. In subsection 4.2, I use available information on data counterparts for
S B−A (s), S R (s), and T urnover (s), together with standard variables used in the sovereigndefault literature, to calibrate the parameters of the model.

22

4.2

Functional forms and parameters

Functional forms. The utility function of the government is a standard CRRA and theoutput cost of default has the same functional form as in Chatterjee and Eyigungor (2012),
U (c) =noc1−σ, and h(y ) = y − max 0, d0 y + d1 y 2 .
1−σ

The stochastic process for the endowment, F (yt |yt−1 ), is given byyt − ȳ = ρy (yt−1 − ȳ ) + εt , with εt ∼ N (0, ηy ),where the ȳ is the average endowment and is normalized to 1. I use a standard Dagum (1975)
matching function, also known as the telephone line matching function, or the Den Haan
et al. (2000) matching technology in the labor search literature. Trading probabilities forinvestors and dealers are given byα (θ ) =α (θ )
θand ρ (θ ) =,1+θ
θ

where market tightness θ ≡ nd . So, if a submarket has measure n of investors orders andtightness θ, then the measure of dealers is θn and the measure of matches is
M (n, θn) = α (θ ) n =n × (θn).
n + (θn)Some convenient properties of this matching function are that α (θ ) ∈ [0, 1] and that α (·) is
twice continuously differentiable for all θ ∈ R+ .
Parameters. The model has in total 17 parameters and I calibrate it at a quarterly frequency using data moments for Greece. To assign parameter values I take some of themfrom the literature and calibrate some others using quarterly Greek data from 1995Q1 until the default in 2012Q1. Table 1 shows all parameter values. Appendix B describes thesolution algorithm and the simulation of moments to calibrate some parameters for Greece.
Appendix C describes all data sources to calibrate the model to the Greek economy.
Calibration strategy. Some of the parameters require simulating the model to match
some moments in the data. These parameters can be divided into two groups. The firstgroup of parameters consist of {β, d0 , d1 , z}, which are standard to most sovereign default
23

models. To calibrate those parameters, I follow the strategy of Chatterjee and Eyigungor(2012) and target the average credit spread, the volatility of credit spreads, the debt-tooutput ratio, and choose the coupon z to target an average price for bonds equal 1, whichmeans that bonds are traded at par value on average. The second group consists of the newparameters that govern trading frictions and activity in the secondary market for sovereign¯ The entry cost for dealers, γ and thebonds. These parameters are: uh , u` , ζ, γ and I.
difference between uh and u` are key determinants of the bid-ask spread. I calibrate γ tothe minimum bid-ask spreads observed in the data and calibrate uh and u` to match theaverage bid ask spread in the data. In addition, the parameters uh , u` , are set such thatthe ex-ante expected flow utility of holding the bond is zero so that the price for a buyer
is not artificially distorted by flow utilities. Investors holding a bond become low type withprobability ζ. In equilibrium, low type investors have incentives to sell their bonds. Thus,
the fraction of bond holders that become low type is a key determinant of the turnover rate
for bonds in secondary markets (defined in equation (13)). Therefore, the parameter ζ isthe one consistent with the turnover rate in secondary markets. Finally, I set the measureof investors I¯ to be significantly larger than the stock of debt issued in equilibrium so that
it is never the case that the government would want to choose a debt level in the grid where
there are not enough investors in the economy to purchase such amount.
Table 1: Calibration
Symbol
σ
φ
ρy
ηy
λ
γ
r
β
d0
d1
z
uh
u`
ζ
I¯
σ εD
σm

Description
Intertemporal elasticity of substitution
Probability of re-gaining market access
Persistence of endowment process
Variance of endowment innovations
Bond’s maturity probability
Dealers’ entry cost
Risk-free rate
Government’s discount factor
Slope of output cost of default
Curvature of output cost of default
Bond’s coupon rate (%)
Type h utility for holding bonds
Type ` utility for holding bonds
Investor’s preference shock probability
Mass of investors in the economy
Std. dev. default preference shocks
Std. dev. issuance preference shocks

Greece2.000
0.050
0.95270.02030.039
0.000250.010
0.976−0.522
0.650
1.133
0.001
−0.1600.315
5.000
0.00010.0001

Note: The table lists all parameters of the models and the assigned values.

24

Greece. The risk aversion parameter is set to σ = 2. The probability of regaining access tointernational credit markets after a default is chosen to be φ = 0.05, which implies an average
exclusion length of 5 years, in line with evidence in Cruces and Trebesch (2013) 21 The output
process is discretized using Rouwenhorst’s method. After discretization, income follows ay , for all
Markov process with transition probabilities given by Pr (yt+1 = ym |yt = yn ) = πn,mn, m ∈ {1, 2, ..., N }, I choose N = 51. I then calibrate ρy and ηy to match the persistence for
the AR(1) process for the GDP cycle in Greece for the period 1995Q1 − 2023Q4, and ηy tomatch the standard deviation of the residuals. The maturity parameter is set to λ = 0.0385,
which represents an average expected time to maturity of 6.5 years. This corresponds to theaverage time to maturity of outstanding bonds before the debt restructuring as reported by
Mihalache (2020). The parameters {β, d0 , d1 } are calibrated to target an average spread of
4.43%, a standard deviation of spreads of 4.65% and a debt-to-output ratio of 115% observed
on average between 1995Q1 − 2012Q1. The quarterly coupon rate z = 1.13% to obtain an
average price for bonds of 1. Dealers’ entry cost, γ, determines the minimum transaction feethat can arise in the model. Since I do not have direct estimations of the intermediation cost
I set γ = 0.00025, which is 2.5 basis points of the average price. This is consistent with thevery low bid-ask spreads of around 5 − 10 basis points observed around 2006 after Greecejoined the Euro and before US subprime crisis. Finally, r = 0.01 corresponds to the averageinterest rates for 3-month German bonds. In the model the risk free rate is assumed tocorrespond to perfectly liquid bonds. German bonds are almost perfectly liquid with bid-ask
spreads below 5 basis points in most of trading dates, including the period of the European
debt crisis. I set I¯ = 5, which is more than two times larger than the stock of debts observedin equilibrium. Appendix D.3 shows that increasing I¯ further does not significantly change
the results.

4.3

Model fit and implications for business cycle moments

Table 2 shows data moments for Greece, the model fit and its implications for standard
business cycle moments that are not targeted in the calibration. The model calibration
provides a close fit to the data, except that it produces a samller volatility of spreads.
Relative to a model with frictionless secondary markets, trading frictions add more slope tothe price schedule as illustrated by the example in section 3.1. As a response the government
implements a relatively less volatile debt issuance policy. In net, it the government behavior
keeps spreads volatility low.
21 This number is also within the 2 year median duration calculated by Gelos et al. (2011) since 1990 andthe more than 7 year median duration computed by Benjamin and Wright (2013).

25

Table 2: Targeted and untargeted moments
Targeted moments
Average bond spread (%)
Std. dev. bond spread (%)
Debt to output (%)
Average bid-ask spread (bps.)
Bonds turnover rate (%)
Untargeted moments
Ratio of std. dev. of consumption and output
Corr. between bond spread and trade balance to GDP
Corr. between bond spread and consumption
Corr. between bond spread and output
Corr. between output and trade balance to GDP

Data
3.43
4.65
115
75
78
Data
0.98
0.71
−0.45
−0.56
−0.59

Model
3.42
2.18
124
76
78
Model
1.07
0.58
−0.77
−0.75
−0.43

Note: Data moments for Greece are calculated using data from 1995Q1 until 2012Q1, when Greece defaulted
on its sovereign debt. The data sources are described in details in appendix C. The moments from the model
are constructed from over 1, 000 simulations of the same length as the data, computing the moment for eachsimulation and averaging across simulations, as described in appendix B.2.

Table 2 also reports the implications of the model for business cycle moments often analyzed in the sovereign default literature. The model captures well the correlations between
the main variables of the model. However, as it is standard in sovereign debt models, consumption is more volatile than output, while in the data output is slightly more volatile thanconsumption.

4.4

The role of trading frictions and secondary market flows

In this section I explore quantitatively the role trading frictions and net demand flows frominvestors in the secondary market for sovereign bonds. To quantify the role of trading
frictions I compare the baseline model to a counterfactual model with frictionless secondary
market. The goal of this exercise is to understand how secondary market frictions affect thefinancial constraint of the government. Thus, I keep the parameters of the baseline model
but I consider the case in which γ = 0 and the price of bonds is determined by a standard
no-arbitrage condition. Table 3 reports the results from the frictionless model in the thirdcolumn. Compared to the benchmkark model, the frictionless model produces a spreadthat is 75 basis points lower on average, reducing the borrowing cost for the government.
Moreover, the average debt to output ratio becomes 139%, which is 15 percentage points
larger than the baseline model. That is, trading frictions significantly tighten government’s
financial constraint, reducing its borrowing capacity and increasing is borrowing costs.

26

Table 3: The role of secondary market’s trade frictions and flows
Key Parameters
Low type probability: ζ
Dealer’s entry cost: γ × 100
Moments
Mean bond spread (%)
Std. dev. bond spread (%)
Debt to output (%)
Mean bid-ask spread (bps.)
Bonds turnover rate (%)

Baseline
0.315
0.025
Baseline
3.42
2.18
124
76
78

Frictionless—
0.000
Frictionless
2.67
2.67
139
—
—

Longer holding
0.15750.025
Longer holding
3.03
2.81
134
74
42

Hold to maturity
0.000
0.025
Hold to maturity
2.00
1.83
137
71
6

Note: The table describes the key parameters that differ across counterfactual scenarios and the moments
obtained from simulations after changing only the key parameters. Baseline refers to the model under thebaseline calibration of 1. Frictionless refers to a counterfactual model in which there are no trading frictionsin the secondary markets and bonds are priced by a no-arbitrage condition like the one in equation (15).
Longer holding refers to a counterfactual scenario in which the average holding horizon of investors is twiceas large as in the baseline model. This is obtained by reducing the probability of becoming ` by a half. Holdto maturity refers to the counterfactual scenario in which investors buy a bond and never become ` type
investors, so they hold the bond until they mature. The turnover rate is defined as in (13) and the bid-ask
spread is defined as in (12) and measured in basis points.

To understand the quantitative importance of investors trade flows in the secondary market, I compare the baseline results to two alternative counterfactuals. In these two counterfactuals I keep all the same parameters of the baseline calibration but change the probability
ζ of becoming a type ` investor. Reducing ζ increases the holding horizon of investors andreduces the supply of bonds in the secondary market. Table 3 shows the results for thesetwo counterfactuals in columns 4 and 5, labeled "longer holding" and "hold to maturity".
The longer holding scenario considers a probability ζ that is a half of the baseline parameter
while the hold to maturity case eliminates the possibility of becoming ` type and investors
do not supply bonds in the secondary market.22 Compared to the baseline model, the longer
the holding horizon of the investors the lower is the average credit spread and the larger isthe stock of debt to output in equilibrium. In fact, if investors held bond to maturity theaverage spread would be about 140 basis points smaller than in the baseline and the stockof debt to output would be on average 13 percentage points larger.
In sum, trading frictions and sell flows in the secondary market are important determinants of the financial constraint of a government. Trading frictions in the secondary market
may be more of a technological constraint outside the reach of policy but the design ofsecondary market may help improve in this margin. However, average holding horizons ofinvestors can generate important changes in the financial constraint of the government, which
22 In the hold to maturity scenario there is still activity in the secondary market because investors buy

newly issued bonds from dealers.

27can more easily be address by policy. For example, quantitative easing programs of central
banks change the lender composition of sovereign governments and affects the average holding horizon. By committing to keep bonds in their balance sheet for long enough, central
banks can relax the borrowing constraint of the government. I explore the effects of suchinterventions in section 5.4. Alternatively, sovereign governments could target lenders who
keep bonds for longer horizons in their portfolio to relax their financial constraints.

5

The Greek debt crisis

5.1

Greek time series

The Greek sovereign debt crisis of 2010-2012 resulted in a debt restructuring in the firstquarter of 2012. Figure 4 shows the time series for macroeconomic variables from 2006Q1
until 2012Q4. The left panel shows Greek GDP cycle (blue dashed line with scale in left axis),
the interest rate spread of Greek long-term bonds compared to same maturity German bonds
as defined in equation (14) (red solid line with scale in right axis), and the bid-ask spreadfor Greek bonds in secondary markets as defined in equation (12) (red dashed-dotted linewith scale in right axis). The right panel shows the negative of net international investment
position (NIIP) as percentage of GDP23 (blue solid line) together with total public andpublicly guaranteed (PPG) debt as percentage of GDP (red dashed line). Appendix C
describes the variables and data sources.
Between 2006Q1 and 2010Q4 Greece experienced sustained economic growth, only temporarily interrupted in 2009Q1 during the sub-prime crisis. However, during 2010 and 2011
the GDP gap decreased sharply taking the economy into a deep recession.
Before the subprime crisis, the Greek government was able to take debt at very low interest
rates of 20 − 50 basis points above German rates. Cheap interest rates allowed Greece toaccumulate debt over the years until 2008, when U.S. crisis hit international financial markets
and Greek interest rate spreads began to increase, reaching a first spike at 250 basis points
in the first quarter of 2009. As interest rate started increasing, Greece began reducing its net
international investment positions during 2008 and the beginning of 2009. In 2009 output
recovered and interest rate spreads went down to stay around 150 basis points during theentire year. By the end of 2009 there was a fast increase in interest rate spreads of Greek
bonds, moving from 170 basis points in 2009Q4 to around 900 basis points during 2010Q2,
amid a political crisis and the revelation that Greece had been understating its debt and
23 A positive number means a negative NIIP with the rest of the world.

28

Figure 1: Greek time series 2006Q1 − 2012Q4.
(A) GDP Cycle, Credit Spread, and Bid-Ask Spread

180

15

(B) (-) NIIP (%GDP) and Public Debt (%GDP)

2500
160
2000
1500
0
1000
-5
-10
2006 2007 2008 2009 2010 2011 2012

140

% GDP

5

Basis Points% Dev. from Trend

10

120
100

500

80

0

60
2006

Quarter(-) NIIP (%GDP)
Public Debt (%GDP)

2007

2008

2009

2010

2011

2012

Quarter

Note: The figure shows quarterly data for Greece from 2006Q1 to 2012Q4. The left panel shows the evolutionof the GDP cycle (blue dashed line) together with the interest rate spread (red solid line) and the bid-ask
spread (red dashed-dotted line). The right panel plots the negative of the net international investment
position (NIIP) as percentage of GDP (blue solid line) together with total public debt as percentage of GDP.
Source: Bloomberg and Eurostats.

deficit figures for years. Attempts to stop the crisis during 2010 were not successful andby the end of 2011 interest rate spreads were above 2500 basis points. In 2012Q1 Greek
debt was restructured involving bond swaps and a 65% haircut on investors’ bond holdings.
On March of 2012 the International Swaps and Derivatives Association declared a triggering
credit event. In other words, a default. During 2012 interest rate spreads decreased due to thedebt relief on Greek bond in 2012Q1 following the restructure but remained high for the restof the year. In 2012Q4 Greece bought back a large fraction of the newly issued bonds over the
debt restructuring, which increase the market price of bonds on 20%, significantly reducing
the interest rate spreads. See Zettelmeyer et al. (2013) and Trebesch and Zettelmeyer (2018)
for detailed and clear exposition of events in during Greek debt crisis.
The bid-ask spreads, defined as in (12), remained below 50 basis points during the period
2006Q1 − 2010Q4. However, bid-ask spreads sharply increase from 165 basis points during
2011Q1 to about 2000 basis points before the debt restructure of March in 2012, when totalinterest rate spreads where between 2600 − 3000 basis points. The model presented in section
2 can be used to assess how the endowment process affects the rest of variables depicted in
Figure 4, and determine how much movements in interest rate spreads reflects changes in
the probability of default and liquidity frictions.

29

5.2

Model time series

In this section, I simulate the model for Greece. I feed the GDP cycle plotted in theleft panel of Figure 4 into the model as the realizations of the endowment process fort ∈ {1995Q1, ..., 2011Q4}. I stop the time series in 2011Q4 because Greece defaulted in
2012Q1 and the model shuts down the secondary market after a default because the recovery value is zero. I let the government, investors, and dealers react optimally by choosing
debt issuances, defaulting or not, and optimal submarket to visit. Using optimal choices I
compute the model implied interest rate spreads, StR , and the bid-ask spreads, StB−A , asdefined in (14) and (12), respectively. Figure 2 shows model’s predictions compared to thedata. The blue solid lines represent model’s predictions while red dashed lines correspondto data.
Panel (A) shows the evolution of credit spreads. The model does a good job to capture
the dynamics of spreads and it is able to explain a large fraction of the changes in magnitudein the data. However, it is not able to account for the exact magnitudes. This is not
surprising given that the model is calibrated to match the average spreads in Greece in atime period with very large changes. Before the subprime crisis in the US, Greece was able totake debt at German rates. After the subprime crisis interest rate spreads behaved as those
ones in emerging economies. In addition, the model abstracts from bailout expectations,time varying changes in investors discount factors and redenomination risk, which have beenshown to be important determinants of interest rate spreads in European countries in the
period of analysis.24
Panel (B) shows the evolution of bid-ask spreads. The figure shows that the model isable to capture the fact that bid-ask spreads increase when the economy gets closer to adefault episode and the qualitative dynamics over the business cycle. However, the model(blue solid line with scale in left axis) is not able to account for the extremely large increasein bid-ask spreads in the data (red dashed line with scale in right axis). In the model, thebid-ask spread at the peak of the crisis are around 65% higher than pre-crisis levels while inthe data bid-ask spreads at the peak of the crisis are 5000% larger. Again, the calibration of
the model targets average bid-ask spreads in the sample. Before the crisis Greek bonds were
almost perfect substitutes to German bonds and very liquid. When the crisis hit, Greek
bonds became extremely illiquid.
24 See Bocola and Dovis (2019) for a decomposition of interest spreads that accounts for a time varying

investor’s discount factor and Dovis and Kirpalani (2018) for the effects of bailout expectations on interest
rate spreads dynamics. See Krishnamurthy et al. (2018) for the role of redenomination risk.

30

Figure 2: Interest Rate Spreads and Bid-Ask Spreads.
(A) Interest Rate Spread Model vs Data

130

Basis Points

Basis Points

1500

1500
1000

110
100

1000

90
500

500

80

2007

2008

2009

2010

70
2006

2011

2007

Quarter(C) Debt to GDP Dynamics

3000

1.8

2500

1.6

2000

1.4
1.2
1
0.8
2006

2008

2009

2010

2011

0

Quarter

Basis Points

Basis Points

2

2000

120

2000

0
2006

(B) Bid-Ask Spread Model vs Data

Basis Points

2500(D) Interest Rate Spread in the Model

1500
1000
500

2007

2008

2009

2010

0
2006

2011

2007

2008

Quarter

2009

2010

2011

Quarter

Note: In panels (A)-(C) the blue solid lines show model equilibrium outcomes while the red dashed lines
show data counterparts for interest rate spreads, bid-ask spreads and debt-to-output ratios, respectively.
Panel (D) shows the interest rate spread from the model (blue solid line) as in panel (A) together with theequilibrium interest rate spread that the model would produce if the government issued debt as in the data
(red dashed line).

Panel (C) shows the dynamics of debt to GDP in the model (blue solid line) and the
dynamics of Public and Publicly Guaranteed (PPG) debt as percentage of GDP in the data
(red dashed line). Panel (D) shows the spreads in the model when debt dynamics are those
produced by the government’s optimal policy in the model (blue solid line) and the spreads
that would arise in the model if debt to GDP ratio was as the PPG debt to GDP ratio in thedata. Tracking debt stocks in the data generates a spread of 2500 basis points in 2011Q4 as
observed in the data in panel (A). However, incorporating actual debt dynamics is still notenough to match the low spreads observed in the data before 2009.

31

5.3

Spread decomposition

How large is the liquidity premium? To answer this question, I use the structure of themodel to decompose the predicted interest rate spreads, StR , into a default risk component
and a liquidity component. To decompose interest rate spreads I do the following exercise. I
take optimal government policies from the model and calculate the bond’s price that would
reflect the probability of default in an alternative model in which bonds are perfectly liquid.
That, is I solve the following Bellman equation for pricesq̃ (y, B ) =

nh

i h


io
1
Ey0 |y 1 − δ ∗ y 0 , B 0∗ λ + (1 − λ) z + q̃ y 0 , B 0∗
,
1+r

(15)

where B 0∗ ≡ B 0 (y, B ) is the optimal policy of the government in the model with liquidity frictions. Using the counterfactual price for bonds we can calculate the interest rate consistent
with default risk. For each pair (y, B ), this interest rate is given byrd (y, B ) =λ + (1 − λ) z− λ.
q̃ (y, B )Then, for each state (y, B, L1 ) I decompose interest rate spreads, S R (y, B, L1 ), into a default
risk component, S d (y, B, L1 ), and a liquidity component, S ` (y, B, L1 ), which are given by
S R (y, B, L1 ) = S d (y, B, L1 ) + S ` (y, B, L1 ) ,S d (y, B, L1 ) ≡

1 + rd (y, B )44− (1 + r ) ,S ` (y, B, L1 ) = S R (y, B, L1 ) − S d (y, B, L1 ) .
As a residual, the liquidity component captures both pure liquidity frictions plus the feedback interactions between liquidity risk and default risk. Figure 3 shows the results of thisdecomposition. In the left panel, the black solid line is the total interest rate spread generated by the model, StR , the red area represents the amount of the spread representing default
risk while the blue area is the liquidity component. The right panel show the shares of total
spread that are due to default risk and liquidity in red and blue, respectively.
The model interpretation of the data is that liquidity frictions can significantly contributeto interest rate spreads. According to this decomposition trade frictions in the secondary
market add about 140 basis points paid by Greek government bonds.

32

Figure 3: Interest Rate Spreads and Bid-Ask Spreads.
(B) Spreads Decomposition Including SMP

2500

2500

2000

2000

Basis Points

Basis Points(A) Spreads Decomposition

1500
1000

1500
1000
500

500
0
2006

2007

2008

2009

2010

0
2006

2011

2007

Quarter
Total Spread

Liquidity Premium

2008

2009

2010

2011

Quarter
Default Risk

Total Spread

Liquidity Premium

Default Risk

Note: The left panel shows the interest rate spread predicted by the model assuming that the government
issues the amount of debt observed in the data as show in panel (C) of Figure 2 (black line). The totalinterest rate spread is decomposed into a default risk component (red area) and a liquidity component (bluearea). The right panel shows the same decomposition of interest rate spreads but it also includes the effectsof the Securities Market Programme (SMP) that the ECB implemented on Greek bonds during 2010.

5.4

Liquidity and secondary market interventions

One of the key contributions of this paper is to propose a framework in which default risk andthe liquidity of bonds in the secondary market are endogenously determined in equilibrium.
This has important implications for policy analysis that do not arise in other frameworks
in the literature. In a simple way, the model captures the fact that heterogenous investors
have different trading needs. Thus, how bonds are distributed across investors is a key
determinant for bond’s price because it affects the magnitudes of demand and supply flows.
Therefore, any intervention in the economy that affects the distribution of bond holdings
across different types of investors would have an impact on the equilibrium bond price. This
result relies in the endogeneity of trading probabilities and intermediation fees.
During the European debt crisis, the European Central Bank (ECB) directly purchased
sovereign bonds in the secondary market in 2010 under the Securities Market Programme(SMP). Interventions of this kind would affect liquidity if the ECB trading behavior isdifferent from investor’s behavior. Interventions in the secondary market can potentiallyhave long lasting consequences in bond prices depending on future trading decisions. In thecase of ECB’s intervention in 2010, the European Central Bank committed to keep bondspurchased under the SMP in its balance sheet until maturity. Therefore, in addition to anincrease in net demand at the time of intervention, there is an increase in future net demand33

too. This is because ECB’s behavior in the secondary market differs from type ` investor’s
behavior. Instead of reselling bonds, the ECB kept the bonds in its balance sheet until
maturity reducing the supply of those bonds in the secondary market, relative to what type
` investors would have done.
Under the SMP, the ECB purchased about 10% of Greek outstanding debt within the firstmonth of the program, launched in May 2010. Trebesch and Zettelmeyer (2018) study theeffects of such intervention on Greek bonds’ price. They conduct a difference-in-difference
excersise and compare the change in bonds’ price before and after the intervention for thegroup of Greek bonds that the ECB bought in the secondary market relative to the groupof Greek bonds that the ECB did not purchase. Consistent with the implications of themodel, Trebesch and Zettelmeyer (2018) find that the SMP has a significant and persistentpositive impact on the price of bonds purchased by the ECB. Eser and Schwaab (2016) and
De-Pooter et al. (2018) reach similar conclusions after evaluating the effects of the SMP incountries targetted by the program.
Figure 4: The effect of secondary market interventions on interest rate spreads.
0

Secondary Market Policies and Spreads

Basis points

-50
-100
-150
-200
-250
-300
2009

10% less sellers (20% less sellers
50% less sellers
No sellers

SMP)

2010

2011

Quarter
Note: The figure reports the difference in interest rate spreads between the baseline each counterfactual
scenario and the baseline. Each of the counterfactuals considers the effects of a reduction in the probabilityof becoming ` type investors (ζ) from its baseline value zeta = 0.315 to 0.02835 in blue solid line, to 0.252
in red dashed line, to 0.1575 in yellow dashed-dotted line, and to zero in purple dashed-circled line.

Using the model, I compute the effects of interventions like the SMP. I model such anintervention as a permanent change in the fraction of bonds that are in hands of type `investors. That is I reduce the probability of becoming type ` investors by 10% to capture
the fact that the ECB purchased about 10% of the outstanding Greek sovereign debt and34

kept the bonds until maturity. That is the SMP policy is modeled as an unexpected andpermanent fall in the parameter ζ to 0.2835 in the second quarter of 2010. In Figure 4 I also
plot the effects of reducing the probability of becoming type ` investors by 20% (ζ = 0.252),
to a half (ζ = 0.1575) and to zero. In all cases, I keep the debt issuance policy as in thebaseline scenario without intervention. That is, all the effects in the spread are purely comingfrom the amount of sell orders in the secondary market and not from the response of thegovernment to changes in prices.25
Panel (B) of Figure 3 shows the decomposition of credit spreads taking into account theeffects of the SMP. However, the effects of the SMP on credit spreads can be appreciatedmore clearly in Figure 4. The blue solid line shows the effects of the policy on Greek interest
rate spreads relative to no intervention in the blue solid line. The model predicts that the
SMP reduced Greek spreads by around 15 to 35 basis points.
Figure 4 shows the effects of reducing the fraction of sellers by alternative magnitudes. I
find that reducing ζ by 20% generates a fall in spreads of 32 to 72 basis points. Reducing ζ
to a half of the baseline produces a fall in spreads of 73 to 173 basis points, while bringing ζto zero reduces the spread by 139 to 313 basis points. In all cases, the maximum reductionin spreads is observed when default risk is the highest, i.e. in the last quarter of 2011.

6

Conclusions

I incorporate endogenous liquidity frictions into a standard quantitative model of sovereigndefault.

To model liquidity I introduce directed search into the secondary market forsovereign bonds, where investors need to meet dealers in order to trade. Since search is
directed, investors and dealers face a trade-off between the intermediation fee and the trading probability. For investors, the higher the intermediation fee that they choose to pay,
the higher the probability of trading. For dealers, the higher the intermediation fee thatthey choose to charge, the lower the probability of trading. In addition, the optimal balanceof this trade-off varies with the state of the economy. Thus, as trading probabilities andintermediation fees are endogenous and time varying, the liquidity of the secondary market
for bonds is also endogenous and time varying over the business cycle.
The model provides a micro-foundation for transactions of bonds in the secondary market
that highlights the importance of taking into account the size of trade flows to determine
the price of both outstanding and newly issued bonds. I find that trading friction are
significantly tighten the financial constraint of the government. After calibrating the model
25 The results in table 3 incorporate the response of the government.

35

to the recent debt crisis in Greece, the model based decomposition of interest rates shows that
trade frictions in the secondary market significantly contributed to explain credit spreads.
Between 2006Q1 − 2011Q4, they increased credit spreads by around 140 basis points.
The model presented in this paper is also useful to understand the effects of policy interventions in the secondary market. In section 5.4, I study the effects of the ECB’s Securities
Market Programme. I model the policy as a permanent 10% fall in the flow of sell orders
in the secondary market because the purchased around 10% of the oustanding stock of debt
and committed to keep it until maturity. The model predict that the SMP reduced interest
rate spreads by 15 to 35 basis points. Policies that reduce the magnitudes of sell volume inlarger fractions could achieve significantly stronger falls in credit spreads. For examples, if
sell orders fell to zero instead, the credit spreads would have by 139 to 313 basis points.
The analysis can be extended in several ways. One important dimension that is notconsidered in this framework is maturity choice. There seems to be a trade-off between
offering a wide set of maturities that satisfy the needs of different types of investors andthe liquidity of each of the alternative bonds issued. In the data, governments tend tooffer a wide range of maturities but usually a couple of them are much more liquid thanthe others. The model also abstracts from the effects of changes in the risk free rate on
liquidity conditions of risky bonds due to changes in investors’ discount factor. Such changes
would affect interest rate spreads by the usual channels studied in the literature26 but could
also generate interesting amplification dynamics in liquidity premium. In addition, themodel highlights that different types of investors can generate different demand and supply
flows and affect prices. It would fruitful to understand and incorporate richer heterogeneityof investors and their behavior in sovereign debt markets. Finally, the model abstracts
from government bonds held by domestic households. As pointed out by Broner et al.
(2010), bonds transactions in secondary markets may rule out default episodes. This isbecause a benevolent government does not want to default on domestic households. Then,in equilibrium foreign investors sell bonds to domestic households before a default is declared.
However, my model predicts that secondary markets endogenously become more illiquid in
bad times, exactly when foreign investors have incentives to transfer bonds to domestic
ones. Thus, the composition of debt holdings across foreign and domestic investors may
significantly affect the cost of bonds re-allocation. I leave these extensions for future research.

26 See Lizarazo (2013) and Bocola and Dovis (2019).

36

Acknowledgements
I am greatly indebted to Shouyong Shi for his continuous advice and encouragement. I
would like to thank Jonathan Eaton, Neil Wallace, Rishabh Kirpalani, Ignacio Presno, Federico Mandelman, Alessandro Dovis, Yan Bai, George Alessandria and seminar participants
at Penn State, Atlanta Fed, St. Louis Fed, Penn, Rochester, Bank of Canada, Western
Ontario, FGV-Rio, FGV-Sao Paulo, PUC-Rio, Central Bank of Chile, Carlos III, Bank of
Mexico, ITAM-Business, Richmond Fed, IADB, IMF, EGSC at WUSTL, Midwest Macro,
7th UdeSA Alumni Conference, 3rd Fordham Economics Conference, NASMES, SED, 9th
Annual European Search and Matching Conferece, LACEA-LAMES, RIDGE, 8th UTDTAnnual Conference, Spring 2020 NBER (IFM), the 2020 Workshop on Money, Payments,
Banking and Finance, and the 2021 Asian Meeting of the Econometric Society for valuablecomments and suggestions. In addition, I thank the Federal Reserve Banks of Atlanta and
St. Louis for hosting me during the writing of this paper. Finally, I thank Bates and White
fellowship for financial support. All errors are of my own.

37

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40

Appendix (For Online Publication)A

ProofsA.1

Investors Problems

Definition 1. I say that for a given bond price q, an investor participates in the secondary market
for bonds if there exists an intermediation fee f ≥ γ at which gains from trade are positive. I
denote Qai (s) the set of prices at which investors with holdings a ∈ {0, 1} and type i ∈ {`, h}participates in secondary markets when state of the economy is s ∈ S.
For each state s ∈ S I define the expected continuation value of holding a bond as
h



i h

I 1 ( s0 ) − I 0 ( s0 )

h



i h

I 1 ( s0 ) ,E 0 (s) ≡ Ey0 |y 1 − δ y 0 , B 0
E 1 (s) ≡ Ey0 |y 1 − δ y 0 , B 0iiand the gains from trade, before including transaction costs, for each type of investor as1
E 0 (s),1+r
1
Rh1 (s) ≡ q (s) − uh −E 1 (s),1+r
1R`1 (s) ≡ q (s) − u` −E 1 (s).
1+r
R0 (s) ≡ −q (s) + uh +Before proving lemma 3 it is handy to show some intermediate results. First, I show that,for each state s ∈ S, we can find price thresholds that determine the set of investors thatparticipate in the secondary market. In addition, I show that there might be a price regionin which the secondary market completely shuts down.
Lemma 1. For any state s ∈ S, If the maximum gain for a type ` seller is smaller than the
maximum gain for a type h buyer, i.e.
u` +

1
1
E 1 (s) + γ < uh +E 0 (s) − γ,
1+r
1+r

(A.1)

there exist three price thresholds {q̃1 (s), q̃2 (s), q̃3 (s)} such that(i) If q (s) < q̃1 (s), q (s) ∈ Q0h (s) and q (s) 6∈ Q1` (s), Q1h (s). Only type h investors with a = 0
participate in secondary markets;(ii) If q (s) ∈ (q̃1 (s), q̃2 (s)), q (s) ∈ Q0h (s), Q1` (s) and q (s) 6∈ Q1h (s). Both type h investors
with a = 0 and type ` investors with a = 1 participate in secondary markets;(iii) If q (s) ∈ (q̃2 (s), q̃3 (s)), q (s) ∈ Q1` (s) and q (s) 6∈ Q0h (s), Q1h (s). Only type ` investors
with a = 1 participate in secondary markets; and1

(iv ) If q (s) > q̃( s), q (s) ∈ Q1` (s), Q1h (s) and q (s) 6∈ Q0h (s). Both type ` investors with a = 1
and type h investors with a = 1 participate in secondary markets.
If the inequality in (A.1) is reversed, there exist three price thresholds {q̃1 (s), q̃2 (s), q̃3 (s)}such that(i) If q (s) < q̃1 (s), q (s) ∈ Q0h (s) and q (s) 6∈ Q1` (s), Q1h (s). Only type h investors with a = 0
participate in secondary markets;(ii) If q (s) ∈ (q̃1 (s), q̃2 (s)), q (s) 6∈ Q0h (s), Q1` (s), Q1h (s). No investor has incentives to participate and secondary markets shut down;(iii) If q (s) ∈ (q̃2 (s), q̃3 (s)), q (s) ∈ Q1` (s) and q (s) 6∈ Q0h (s), Q1h (s). Only type ` investors
with a = 1 participate in secondary markets; and(iv ) If q (s) > q̃3 (s), q (s) ∈ Q1` (s), Q1h (s) and q (s) 6∈ Q0h (s). Both type ` investors with a = 1
and type sh investors with a = 1 participate in secondary markets.
Proof. Notice that investors with a = 0 can only participate in trades in which they purchasea bond and investors with a = 1 can only participate in trades where they sell a bond. We
start by analyzing the case in which
u` +

1
1
E 1 (s) + γ < uh +E 0 (s) − γ.
1+r1+rHere we define1
E 1 (s) + γ,
1+r1
q̃2 (s) ≡ uh +E 0 (s) − γ,1+r1q̃3 (s) ≡ uh +E 1 (s) + γ,
1+rq̃1 (s) ≡ u` +

(A.2)
(A.3)

with q̃1 (s) < q̃2 (s) < q̃3 (s). Notice that if
q (s) < q̃1 (s) < q̃2 (s),then no type ` investor is willing to participate in secondary markets since it will imply
a negative gain from trade for all f ≥ γ. In addition, since Rh1 (s) < R`1 (s), also type hinvestors with a = 1 have no incentives to sell their bonds. Finally, if q̃2 (s) > q (s) implies
that R0 (s) > γ. Therefore, type h investors with a = 0 participate on secondary markets.
Next, if the price is such thatq̃1 (s) < q (s) < q̃2 (s),we get that R`1 (s), R0 (s) > γ. So, both of these types of investors participate on secondary
markets. Also, since I 0 (s0 ) ≥ 0, we have thatq (s) < q̃2 (s) ≤ q̃3 (s),2which implies that Rh1 (s) < γ, and hence type h investors with a = 1 do not participate insecondary markets.
Now, lets consider the case in which the price is such thatq̃2 (s) < q (s) < q̃3 (s).
Here we get that R`1 (s) > γ > Rh1 (s), R0 (s). Thus, type ` investors with a = 1 are the only
ones that participate in secondary markets. Finally, if q̃3 (s) < q (s), we get that R`1 (s) >Rh1 (s) > 0 > R0 (s), and all investors with a = 1 participate in secondary markets.
It remains to analyze the case in which
u` +

1
1
E 1 (s) + γ ≥ uh +E 0 (s) − γ.
1+r1+rIn this case q̃2 (s) < q̃1 (s) < q̃3 (s). So, if q (s) < q̃2 (s), we get that R0 (s) > γ > R`1 (s), Rh1 (s),and only type h investors with a = 0 participate in secondary markets. If q (s) ∈ (q̃2 (s) , q̃1 (s)),
then we have that Rh0 (s), Rh1 (s), R`1 (s) < γ. Therefore, secondary markets shut down sinceno investor has incentives to participate. In a similar way as before, it can be checkedthat if q (s) ∈ (q̃1 (s) , q̃3 (s)) we get that R`1 (s) > γ > Rh0 (s) , Rh1 (s), so only type ` investors
with a = 1 participate in secondary markets. Finally, since the definition of q̃3 (s) has notchange, it is straightforward to see that q (s) > q̃3 (s) implies that all investors holding a bondparticipate in secondary markets. This completes the proof.
Next, I show that whenever investors participate in the secondary market, optimal transaction fee fh0 (s) is decreasing and f`1 (s) , fh1 (s) are increasing in q (s).
Lemma 2. Given an aggregate state and a debt issuance choice s = (y, B, B 0 ) and taking
government policy functions δ (y, B ), B 0 (y, B ) as given:fh0 (s) is unique, continuous, and strictly decreasing in(i) The optimal submarket choiceq (s), for all q (s) ∈ int Q0h (s) .

(ii) The optimal submarket choice f`1 (s) is unique, continuous, and strictly increasing in
q (s), for all q (s) ∈ int Q1` (s) .
(iii) The optimal submarket choice fh1 (s) is unique, continuous, and strictly increasing in
q (s), for all q (s) ∈ int Q1h (s) .
(iv ) fia (s) = 0 is optimal for all q (s) 6∈ int (Qai (s)), all i ∈ {`, h}, and all a ∈ {0, 1}.
Proof. Let the aggregate state of the economy and debt issuance choice be an arbitrarys = (y, B, B 0 ) ∈ S. In all cases we focus on the price region in which investors are willing toparticipate in secondary markets, characterized in proposition 1.

(i) Using the free entry condition (10) we have that in any active submarket ρ (θ ) f = γ and,α θby properties of the matching function we have that ρ((θ )) = θ. So, we can re-write the3problem of a type h with a = 0 as if the investor chooses θ instead of f . That is,

I 0 (s) = max α(θ ) −q (s) + uh +
θ

+

h

i  
1
Ey0 |y 1 − δ y 0 , B 0 I 1 s0 − γθ
1+r


h

i  
1 − α (θ )Ey0 |y 1 − δ y 0 , B 0 I 0 s0 .
1+rNow, since α (·) is differentiable, we can take first order condition with respect to θ to
get
h

i h
 i
1
= γ.
Ey0 |y 1 − δ y 0 , B 0 I 1 (s0 ) − I 0 s0
[θ ] : α (θ ) −q (s) + uh +
1+r
0





We definedR0 (s) ≡ −q (s) +

h

i h
 i
1
Ey0 |y 1 − δ y 0 , B 0 I 1 (s0 ) − I 0 s0 ,1+rwhich is independent of θ and decreasing in q (s). Remember we focus in the region ofprices in which R0 (s) is positive, else investors would prefer not to purchase the asset
and optimal tightness with be zero. Thus, we have that the optimal choice of θ in states is given by!
γ.
θh0 (s) = α0−1
R0 ( s )Next, notice that since α (·) is strictly concave, α0 (·) is strictly decreasing in its argument. Therefore, its inverse is also strictly decreasing. So, since R0 (s) is decreasing in
q (s), γ/R0 (s) > 0, and α0−1 (·) is strictly decreasing, we have that θh0 (s) is decreasing
in q (s). Finally, since in any open submarket ρ (θ ) f = γ and ρ (·) is strictly decreasing,
we have that fh0 (s) is strictly decreasing in q (s). Continuity follows from continuityof R0 (s) on q (s) and by continuity of α0 (·).

(ii) Similarly, we can write the first order condition for a type ` investors with a = 1 ashi  
1
Ey0 |y 1 − δ y 0 , B 0 I 1 s0 = γ.
[θ ] : α (θ ) q (s) − u` −
1+r
0





So, defininghi  
1
Ey0 |y 1 − δ y 0 , B 0 I 1 s0 ,1+rwhich is positive and increasing in q (s), we can find that the optimal submarket choice
is given by!
γθ`1 (s) = α0−1.
R`1 (s)R`1 (s) ≡ q (s) − u` −Using similar arguments than in (i) since R`1 (s) is strictly increasing in q (s) we get thatθ`1 (s) is strictly increasing in q (s) and so is f`1 (s). Continuity follows from continuityof R`1 (s) on q (s) and by continuity of α0 (·).

4

(iii) Finally, using the definition
Rh1 (s) ≡ q (s) − uh −

h

i
1
Ey0 |y 1 − δ y 0 , B 0 I 1 (s0 ),1+r

we can write the first order condition for a type h investors with a = 1 as[θ ] : α0 (θ ) Rh1 (s) = γ.
So, we have that
θh1 (s) = α0−1

!

γ
,
1
Rh (s)and noticing that Rh1 (s) is strictly increasing in q (s) and similar arguments as before
we get the proposed result.

Proof for Lemma 3
Lemma 3 is an important (partial equilibrium) intermediate result that characterises theslope of the inverstor’s net demand for bonds as a function of the bond price, for given
government policy functions. This result allows me to establish the existence of a uniquemarket clearing price for each state of the economy and for given government policies inproposition 1. These results show we can find the bond price schedule face by the government
from condition (11).
For each s, I define investors’ aggregate net demand for bonds as

N D (s) ≡ α θh0 (s)

h

i

{z

|

Buyers’ demand

−α
|



θh1 (s)I − (1 − λ) B − α θ`1 (s) (1 − λ) ζB



}

|

{z

Sellers’ supply

(A.4)

}

(1 − λ) (1 − ζ ) B .
{z

Potential type h sellers}

I now show that it is decreasing and continuous in q (s), for all s ∈ S. The functions q̃1 (s)and q̃2 (s) are defined as in (A.2) and (A.3), respectively.
Lemma 3. For any s ∈ S, and given a government’s default policy function δ (x), investors’
aggregate net demand defined in (A.4) is continuous and decreasing in q (s). Moreover, if
q̃1 (s) < q̃2 (s) it is strictly decreasing for all q (s) ∈ R+ . And, if q̃1 (s) ≥ q̃2 (s) it is constantfor all q (s) ∈ [q̃2 (s), q̃1 (s)] and strictly decreasing for all q (s) in the complement of this set
in R+ .
Proof. First, notice that for any given s = (y, B, B 0 ), N D (s) is continuous since α (·) is
continuous by assumption and by proposition 2 we know that θh0 (s) and θ`1 (s) are also
continuous in q (s). In addition, notice in the case in whichq̃1 (s) ≡ u` +11
E 1 (s) + γ < uh +E 0 (s) − γ ≡ q̃2 (s),
1+r
1+r
5

from proposition 1 there is always at least one type of investors participating in secondary
markets, so from proposition 2 it follows that N D (s) is strictly decreasing in q (s). Thisis because α (·) is a strictly increasing function, θh0 is strictly decreasing in q (s) so buyers’
demand is strictly decreasing in q (s), and because θ`1 (s) and θh1 (s) are strictly increasing in
q (s) so then the negative of sellers’ supply is strictly decreasing. In the case in whichq̃1 (s) ≥ q̃2 (s),there is always at least one type of investor participating in secondary markets as long as
q (s) 6∈ [q̃2 (s) , q̃1 (s)], so from proposition 2 it follows that N D (s) is strictly decreasing in
q (s) ∈/ [q̃2 (s) , q̃1 (s)]. When, q (s) ∈ [q̃2 (s) , q̃1 (s)], from proposition 1 we know that there
are no investors participating in the secondary market. Therefore, N D (s) = 0 and constantfor the whole interval.

A.2

Market Clearing Price

I use lemma 3 to show that for each s and given a government’s default policy function
δ (y, B ), there is a unique price that is consistent with market clearing. Then, I characterize
the pricing schedule faced by the government. Throughout this subsection I denote B (s)and B 0 (s) the second and third components of s = (y, B, B 0 ), respectively.
For each s and any given price q, define the excess demand function for bonds in the
primary market as
h

n

o

i

ED (s; q ) ≡ N D (s; q ) − max B 0 (s), 0 − (1 − λ) B (s) ,(A.5)

with N D (s; q ) defined as in (A.4). Define q̃1 (s) and q̃2 (s) as in (A.2) and (A.3).
Proposition 1. For any policy function δ (x) and any s ∈ S such that B 0 (s) > 0, if q̃1 (s) <q̃2 (s) there is a unique price q (s) ∈ R+ consistent with
q (s)ED (s; q (s)) = 0.

(A.6)

Moreover, either q (s) > 0 and ED (s; q (s)) = 0, or q (s) = 0 and ED (s; q (s)) ≤ 0. In addition, when q̃1 (s) ≥ q̃2 (s) the result still holds except when B 0 (s) = (1 − λ)B (s), in whichcase any price within [q̃2 (s), q̃1 (s)], is consistent with q (s)ED (s; q (s)) = 0.
Proof. Consider first the case in which s is such that, N D (s; 0) ≤ B 0 (s) − (1 − λ) B (s). Then,since q ≥ 0 and from lemma 3 N D (s; q ) is decreasing in q, either N D (s; 0) = max {B 0 (s), 0}−(1 − λ) B (s) or there is no price such that N D (s, q ) = B 0 (s) − (1 − λ) B (s). Therefore, for
any q there is an excess supply of bonds in the primary market (i.e. ED (s; q ) ≤ 0). Thus,
the unique price consistent with (A.6) is q (s) = 0.
Next, consider the case in which s is such that N D (s; 0) > B 0 (s) − (1 − λ) B (s). Here wehave two cases. First, if
q̃1 (s) < q̃2 (s),from proposition 3 we know that N D (s; q ) is strictly decreasing in q, for all q ∈ R+ . Thenwe just need to increase the price until N D (s; q ) = B 0 (s) − (1 − λ) B (s). The second case,6

is the case in whichq̃1 (s) ≥ q̃2 (s) .
Now, if N D (s, 0) > B 0 (s) − (1 − λ) B (s) > 0, from proposition 1 we know that for any q ∈[0, q̃2 (s)] N D (s) is strictly decreasing and continuous, and also we know that N D (s, q̃2 (s)) =0, since above that price no type h investor with a = 0 is participating in secondary markets.
Thus, by the intermediate value theorem, there must exist a price between 0 and q̃2 (s) suchthat ED (s, q (s)) = 0. A similar argument applies if B 0 (s) − (1 − λ) B (s) < 0. In this casewe will find a price above q̃1 (s) such that ED (s, q (s)) = 0. The only case that is a little
more subtle is the case in which B 0 (s) = (1 − λ) B (s). Here, we have that government supply is zero. In addition, we know that for any q (s) ∈ [q̃2 (s) , q̃1 (s)], secondary markets shut
down, so N D (s, q (s)) = 0. In this case, ED (s, q (s)) = 0 for any q (s) ∈ [q̃2 (s) , q̃1 (s)]. This
multiplicity arises because the government is not trying to sell or buy bonds, and investors
have no incentives to participate in secondary markets.
The price schedule faced by a government, conditional on a given policy function δ (x),can be charaterized as in Corollary 1.
Corollary 1. The price schedule faced by a government conditional on a given policy function
δ (x) is given by(

q (s) ={p ∈ R+ : ED (s; p) = 0} if N D (s; 0) > B 0 (s) − (1 − λ) B (s).
0if N D (s; 0) ≤ B 0 (s) − (1 − λ) B (s)(A.7)

Proof. Directly follows from the previous results.
The price schedule in (A.7) replaces the standard no-arbitrage condition usually foundin the literature of sovereign default. Proposition 1 states that the price that clears theprimary market for sovereign bonds is unique, except for the knive edge case in which neither
the government nor the dealers participate in the primary market. To be more precise, thisuniqueness statement is conditional on a given government’s default policy function, investors
value functions, and future expected prices. However, the result highlights the parallelismof the pricing schedule to the standard no-arbitrage condition that maps future prices anda default policy function into current prices. In this sense, solving this model is not harderthan other models of sovereign default. Instead of having a closed form expression for the
price as in the standard no arbitrage condition, I need to find the price consistent with (A.6).

A.3

Amplification in two periods example

Proposition 2. Under the assumptions of section 3.2 the bond price elasticity with respect
to 1 − δ is larger than 1.
Proof. I proceed using the implicit function theorem on the market clearing condition forbonds defined by F = 0, with defined as:
L
M
H
1/2
F=
1/2 − 1/2 − 1/2 (1 − δ )
γ
Uh
U`
q
q
− 1−δ + 1+r
1−δ − 1+r

7

Then taking the derivative of F withrespect to (1 − δ ) I obtain:qq−H (1−δ
L (1−δ
∂F
M
)2
)2
= 
(1 − δ )−3/2
3/2 + 3/2 −
1/2∂ (1 − δ ) 2 − q + Uh
2γ
2 q − U`
1−δ

1+r

1+r

1−δ

Similarly taking the derivative of F with respect to q I obtain:11

H (1−δ )L (1−δ )
∂F
= 
−
3/2

3/2
∂q
2 − q + Uh
2 q − U`
1−δ

1+r

1+r

1−δ

Therefore by the implicit function theorem it is the case that:



∂F

−H

q
(1−δ )2q

U

2 − 1−δ + 1+hr∂q
∂ (1−δ )= − ∂F = −1
H (1−δ
∂ (1 − δ )
)
∂q



U

q
2 − 1−δ+ 1+hr3/2 + 2

3/2 − 2L

q(1−δ )2U`
q
1−δ − 1+r

3/2

+

1
L (1−δ
)U`q
1−δ − 1+rM
(1 − δ )−3/2
2γ 1/2
1
1
L (1−δ
H (1−δ
)
)

3/2



U

q
2 − 1−δ+ 1+hr3/2 − 2U`q1−δ − 1+rThis expression simplifies to :M
(1 − δ )−3/2
2γ 1/21
1
H (1−δL (1−δ
)
)

∂q
q
=
+
∂ (1 − δ )(1 − δ )Uq
2 − 1−δ+ 1+hr3/2 − 2U`q1−δ − 1+r

3/2

Defining
1
H (1−δ)1
L (1−δ
)

Z≡ 
− 
> 0,


Uh 3/2U` 3/2qq
2 − 1−δ + 1+r
2 1−δ − 1+r
and multiplying both sides by 1−δq we can get the expression in terms of the elasticity:M

1/2 (1 − δ )1 − δ ∂q
= 1 + 2γq ∂ (1 − δ )
Zq
−1/2
M
1/2 (1−δ )Since 2γ

Zq

> 0 it is the case that:1 − δ ∂q>1
q ∂ (1 − δ )Which concludes the proof.

8

−1/2

3/2

B

Solution Algorithm

It is well known that long-term debt models have convergence issues. To obtain convergence,
I follow the literature and solve an approximate model that adds preference shocks to thedefault debt issue choices of the government as in Dvorkin et al. (2021) and Gordon (2019).
Preference shocks are as small as needed to get convergence and do not significantly change
government optimal choices. I describe the approximate model and the solution algorithm
in appendix B.1. Appendix B.2 describes how simulations are computed.

B.1

The Approximate ModelI first show the equations of the approximate model that I solve. As in Dvorkin et al. (2021)
and Gordon (2019), the problem of the government consist on a discrete choice model withpreference shocks for default and repayment, as well as preference shocks for each choice ofdebt for next period B 0 . Conditional on repayment, the sovereign chooses a level of debt for
next period B 0 from a discrete set with Nb options B ≡ {B1 , B2 , ..., BNb }. The problem of
the government is given by
V (y, B, ε) = maxδ∈{0,1}n(1 − δ )V R (y, B, ε) + δV D (y, εD )ho

V (y, B, ε) = max
u(c) + εnB0 + βEy0 |y Eε0 V (y 0 , B 0 , ε0 )
0
R

i

B ∈B

h

s.t. : c = y − [λ + (1 − λ)z ] B + q (y, B, B 0 ) B 0 − (1 − λ)B
n

h

i

i
h

i o

V D (y, εD ) = u(h(y )) + εD + βEy0 |y Eε0 (1 − θ ) V D (y 0 , ε0D ) + θ V (y 0 , 0, ε0 ) ,where ε ≡ {ε1 , ε2 , ..., εNb , εD } is a vector with Nb + 1 specifying a utility value for defaulting,and for each choice of next period debt B 0 . It is assumed that ε is i.i.d over time and has
the following CDF




Nb
X







εnB0 − µ εD − µ − exp −,
F (ε) = exp −
exp −
σ
σ
n 0 =1B

where µ is the mean of the shocks and σ is the variance of the shocks. This is the Generalized
Extreme Value distribution pioneered by McFadden (1978). What this means, is that if thegovernment needs to choose a level of debt for the next period it would choose elementn∗ ∈ 1, ..., Nb if the utility derived from Bn0 ∗ is the largest among all choices.
For a given price schedule, I can determine the optimal bond supply from the governmentand default decisions. To compute the price of the bond we need to compute the net
demand of bonds from investors that dealers channelize to the primary market. To computeinvestors’ demand and supply of bonds we need to solve their respective problems. For eachstate x = (y, B ) at the beginning of the period and s = (x, B 0 ), which is the information9

needed by an investor27 , the value of an investor without bonds is1
E 0 0 [1 − δ (y 0 , B 0 , ε0 )] I 1 (s0 )I (s) = max{α (θ (f )) −q (s) − f + uh +1 + r y ,ε |yf ∈F̄
1 − α (θ (f ))+Ey0 ,ε0 |y [1 − δ (y 0 , B 0 , ε0 )] I 0 (s0 )}.
1+r


0



The value of holding a bond at the beginning of the period is,I 1 (s) = λ + (1 − λ) [z + ζJ` + (1 − ζ )Jh ] .
where the values Ji , i ∈ {`, h} are given by
Ey0 ,ε0 |y [1 − δ (y 0 , B 0 , ε0 )] I 1 (s0 )J` = max α (θ (f )) [q (s) − f ] + [1 − α (θ (f ))] u` +
1+r
f ∈F̄
(

"

#)

Ey0 ,ε0 |y [1 − δ (y 0 , B 0 , ε0 )] I 1 (s0 )Jh = max α (θ (f )) [q (s) − f ] + [1 − α (θ (f ))] uh +
1+r
f ∈F̄
(

"

#)

where s0 = (y 0 , B 0 , B 00 (y 0 , B 0 , ε0 )).
In each s, the optimal submarket choices for each type of investor are given byθ0 (s) = α0−1 



γ
1
0
0 0
1 0
−q (s) + uh + 1+
r Ey 0 ,ε0 |y [1 − δ (y , B , ε )] I (s )









 × I0 {active},γ
 × I0 {active}θ`1 (s) = α0−1 `1q (s) − u` − 1+r Ey0 ,ε0 |y [1 − δ (y 0 , B 0 , ε0 )] I 1 (s0 )γ
 × I0 {active},θh1 (s) = α0−1 h1q (s) − uh − 1+r Ey0 ,ε0 |y [1 − δ (y 0 , B 0 , ε0 )] I 1 (s0 )where Iai {active} is an indicator function that takes value equal one when that type ofinvestor is active in the secondary market. Investors are active in the secondary market
whenever the gains from trade can at least compensate the entry cost for dealers, γ. Finally,knowing the optimal submarket choices of investors, we can determine the fraction of eachtype of investor that meets a dealer in the secondary market and compute the excess demand
function in the primary market as
ED (s) ≡ α θh0 (s)


h

i

h

n





o

I − (1 − λ) B − max B 0 , 0 − (1 − λ) Bi−α θ`1 (s) (1 − λ) ζB − α θh1 (s) (1 − λ) (1 − ζ ) B.
I can pin down the unique price that is consistent with q (s)ED (s) = 0. These steps provide
me with an algorithm to solve the model. This algorithm begins with a guess of the priceschedule, updates the policy functions of the government, solves for investors optimal submarket choice as a function of price q (s) in each state s ∈ S, and finally finds the price q (s)27 Note that investors do not need to know the preference shock because they observe the choices of thegovernment and the preference shocks for the government are i.i.d. over time.

10

consistent with zero excess demand in primary markets, for each s ∈ S. Then, I used theresulting price schedule as a new guess and iterate until all equilibrium objects converge.

B.2

Model Simulation

I simulate the model over T = 500, 000 periods. Then, I burn the T1 = 1, 000 initial periods.
Find all episodes of length T = 69 periods where the 69th period is a default episode andnone of the previous 99 periods are default periods. I discard the first T0 = 30 periods andkeep 69 periods before default. Length T is chosen to be 69 because I use data on Greek
GDP from 1995Q1 and default happens 68 quarters later in 2012Q1. Since after re-gaining
access to international financial markets the government re-enters with B = 0, I choose to
discard T0 periods before the beginning of each replica of the economy so that I let the model
reach the ergodic set. Then, I compute the summary statistics for each of them. Finally, I
average over these episodes and report the averages of the moments.

C

Data Description

I collect the following time series for Greek economy.
National Accounts. Quarterly time series from 1995Q1 - 2012Q4 for consumption, exports, imports, and GDP are obtained from Eurostats. I use seasonal adjusted and calendar
adjusted chain-linked (2010) in million euros.
Debt and Investment Position. Data on net international investment position (as %of GDP) for the period 2003Q4 - 2012Q4 is obtained from Eurostat. Public and Publicly
Granted debt (as % of GDP) for the period 2000Q1 - 2012Q4 is obtained from the World
Development Indicators database from the World Bank.
Interest Rates and Spreads. Interest rates data is collected from Eurostats and Bloomberg.
Interest rate spreads is calculated as the difference in the annual interest rates between Greek
and German long term government yields in Eurostats. Long term debt yields are composedfrom central government bonds with residual maturity of around 10 years. Computing interest rate spreads using generic central government bonds from Greece and Germany collectedfrom bloomberg results in almost identical time series. Daily bid and ask prices are collectedfrom Bloomberg using generic central government bonds. I compute quarterly time series
using average of active days in each quarter. I use time series for bid and ask prices for 10
year bonds. Daily bid and ask prices for 5 year bonds have some missing values but quarterly
time series results in almost identical bid-ask spreads as for 10-year bonds.
Secondary Market Volumes and Turnover Rates. Information on secondary market
trade volumes is obtained from the electronic secondary securities market (HDAT) available
at the Bank of Greece website.28 Monthly traded volumes is available from January 2001 to
December 2012. Quarterly time series are calculated as the sum of monthly traded volumes.
28 https://www.bankofgreece.gr/Pages/en/Markets/HDAT/statistics.aspx

11

D

Details on the Calibration

D.1

Calculating utility from holding bonds

I calibrate preferences bonds uh and u` such that they satisfy to conditions: (i) that ex-ante
expected utility from holding the asset is zero for a given ζ, and (ii) such that uh − u` targets
the average bid-ask spread in the data. Condition (i) imposes the following restriction
0 = uh +"∞ XPr {δj = 0}
j =1

∞
X

1+r



"

#j

(1 − λ) (1 − ζ )Pr {δj = 0}=
(1 − λ) (1 − ζ )
1+r
j =0

#j

"

Pr {δj = 0}uh +(1 − λ) ζ (1 − α (θ`,j ))
1+r

∞
X

"

Pr {δj = 0}uh +(1 − λ) ζ (1 − α (θ`,j ))
1+r
j =1

#j




u` 

#j

u` .

Although I use the constraint above to calibrate the model, we can develop further intuition
assuming that the probability of default is more or less constant around the targeted average
default probability δ̄. In such case, the previous expression is approximately equal to∞X"

1 − δ̄
0 ≈(1 − λ) (1 − ζ )j =0 1 + r

=
=⇒

#j

∞
X

"

1 − δ̄uh +(1 − λ) ζ (1 − ᾱ)j =1 1 + r#ju`(1 − δ̄ ) (1 − λ) ζ (1 − ᾱ)u`(1 + r ) uh+
1 + r − (1 − δ̄ ) (1 − λ) (1 − ζ ) 1 + r − (1 − δ̄ ) (1 − λ) ζ (1 − ᾱ)(1 − δ̄ ) (1 − λ) ζ (1 − ᾱ)(1 + r )uh ≈ −u` ,1 + r − (1 − δ̄ ) (1 − λ) (1 − ζ )1 + r − (1 − δ̄ ) (1 − λ) ζ (1 − ᾱ)

where ᾱ ≡ α (θ`,j | Pr[δ = 1] = δ̄ ) is the probability of matching for a low type investor conditional on a default probability equal to δ̄, which would also be a constant if default risk isconstant.

D.2

Computing Turnover Rates

The turnover rate is 78% in HDAT in Greece is per quarter. This includes transactions
between dealers and investors as well as interdealer transactions. To calculate the turnoverrate in the model we have to compute all the transactions that happen in secondary markets
between dealers and investors and also interdealers transactions in primary markets. This
is not exactly what happens in reality as some dealers hold inventories and do not need to
trade with other dealers and some other trades occur through a long chain of dealer to dealer
transactions. We will assume that the number of transactions in the model approximates
the amount of transactions in the data. In the model, we can calculate the amounts oftransactions in both the primary and secondary market as follows. In the primary market,
whenever the government issues debt it is purchased by a dealer. Then, the number oftransactions in the primary market is given bynomax B 0 , 0 − (1 − λ) B.

12

Now, in the secondary markets, the following trades occur between a dealer and an investor:
















` − type investors to dealer : α θ`1 ζ (1 − λ) B
Potential h − type investors to dealer : α θh1 (1 − λ) (1 − ζ ) B
Dealer to old buyer : α θ0 (I − B )Dealer to new buyer : α θ0 λB.
Since we do not model the amounts of trades in the interdealers market this is the minimum
amount of transactions in secondary markets. The maximum amount of transactions adds tothese transactions the maximum amount of possible trades between dealers in the interdealersmarket, so we have to clean the turnover rate in the data to remove those trades. This is
the sum of the following trades.
noPrimary buyers to selling dealers : max B 0 , 0 − (1 − λ) B












Secondary buyers to selling dealers : α θ`1 ζ (1 − λ) H1 + α θ`1 (1 − λ) L1 α θh1 (1 − λ) (1 − ζ ) H1 .
Since primary market clears we can simplify this calculation using total purchases by investors,










Total investors’ purchases : α θh0 H0 + α θh0 λB = α θh0hi

I − (1 − λ) B .

Then, the maximum amount of transactions in secondary markets, given that interdealersmarket is competitive, is given by
h 







i













X max = 2 α θh0 H0 + α θh0 λB + α θ`1 ζ (1 − λ) H1 + α θ`1 (1 − λ) L1 + α θh1 (1 − λ) (1 − ζ ) H1 .
In reality the minimum amount of trades in secondary markets is given by




















X min = α θh0 H0 + α θh0 λB + α θ`1 ζ (1 − λ) H1 + α θ`1 (1 − λ) L1 + α θh1 (1 − λ) (1 − ζ ) H1 .
This happens when the same dealer is connecting both the investor selling and the investor
buying and just acting as a bridge. As mentioned before, the number of transactions inreality could be lower or higher than X SM as long chains of dealers would be require totransfer one bond from an investor to another one. Li and Schurhoff (2018) find that formunicipal bonds in United States, the average chain involves 1.5 dealers. So, we can compute
an intermediate amount of trades in secondary markets as
h 







i













X mean = 1.5 α θh0 H0 + α θh0 λB + α θ`1 ζ (1 − λ) H1 + α θ`1 (1 − λ) L1 + α θh1 (1 − λ) (1 − ζ ) H1 .
We will use X mean to compute the model’s implied turnover rate as
Turnover rate =13

X mean.
B

D.3

Alternative values for I¯This appendix shows that increasing the parameter I¯ does not significantly change the resultscompared to the baseline calibration. This is because the amount of potential demand I¯ issufficiently large in the baseline calibration and it does not bind in any significant way theamount of debti issued by the government in equilibrium. If course, reducing the parameterI¯ can affect the results as debt issuance might start being bound by the potencial world
demand for sovereign bonds. Table D.1 shows how the main moments change as we change¯ For larger values of I¯ a small re-calibration of the model would restore the baselineI.
moments.
Table D.1: The role of secondary market’s trade frictions and flows
Moments
Mean bond spread (%)
Std. dev. bond spread (%)
Debt to output (%)
Mean bid-ask spread (bps.)
Bonds turnover rate (%)

Baseline (I¯ = 5)
3.42
2.18
124
76
78

I¯ = 7.5
3.38
2.17
125
76
78

I¯ = 10
3.37
2.17
126
76
78

I¯ = 15
3.37
2.18
126
75
78

Note: The table shows the values for the targeted moments as we increase the parameter determining the¯
potential demand for sovereign bonds I.

14