The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.
Sovereign Debt and Credit Default Swaps
WP 23-05
Gaston Chaumont
University of Rochester
Grey Gordon
Federal Reserve Bank of Richmond
Bruno Sultanum
Federal Reserve Bank of Richmond
Elliot Tobin
Harvard Business School
Sovereign Debt and Credit Default Swaps*
Gaston Chaumont
Grey Gordon
University of Rochester
Federal Reserve Bank of Richmond
Bruno Sultanum
Elliot Tobin
Federal Reserve Bank of Richmond
Harvard Business School
March 14, 2023
Abstract
How do credit default swaps (CDS) affect sovereign debt markets? The answer depends crucially
on trading frictions, risk-sharing, arbitrage violations, and spillovers from secondary to primary markets. We propose a sovereign default model where investors trade bonds and CDS over the counter
via directed search. CDS affect bond prices through several channels. First, CDS act as a synthetic
bond. Second, CDS reduce bond-investing risks, allowing exposure to be unwound. Third, CDS availability increases trading profitability, which induces entry and reduces trading costs. Last, these direct
effects feedback into default decisions. Our novel identification strategy exploits confidential microdata
to quantify the extent of trading frictions and risk-sharing. The model generates realistic CDS-bond
basis deviations, bid/ask spreads, and CDS volumes and positions. Our baseline specification predicts
large effects of frictions generally but small spillovers from a naked CDS ban. These predictions hinge
crucially on the identified parameters.
Keywords: sovereign debt, CDS, directed search, over-the-counter
JEL codes: F34, G12
* We are thankful to our discussants, Juan Passadore and Juliana Salomao, and the Minneapolis Fed Sovereign Debt conference participants and
organizers Manuel Amador, Cristina Arellano, Javier Bianchi, Luigi Bocola, and Mark Wright. We also thank our previous discussant Pierre-Olivier
Weill and Vivian Yue for providing us with CDS-bond basis data. Thanks also go to seminar participants at the Richmond Fed Sovereign Debt
Workshop, the Central Bank of Colombia, University of Rochester, Universidad Adolfo Ibanez, Universidad Diego Portales, Banque de France,
the NASMES, the SED, the 5th SAFE conference on market micro-structure, the 9th Annual UTDT Economics Conference, and the PAELLA
Conference. Kyler Kirk and Tre’ McMillan provided excellent research assistance. This research was supported in part through computational
resources provided by the BigTex High Performance Computing Group at the Federal Reserve Bank of Dallas. The views expressed are those of
the authors and do not necessarily reflect those of the Federal Reserve Bank of Richmond or the Board of Governors.
1
Introduction
Credit default swaps (CDS) are financial derivatives created in the 1990s that provide insurance
against default risk. The volume of transactions on CDS linked to sovereign government bonds,
which we refer to as sovereign CDS, has steadily increased since their inception.1 However, very
little is known about the interactions between sovereign bond and sovereign CDS markets and,
specifically, how quantitatively relevant these interactions are. One possible reason for this lack of
knowledge is that both of these tightly-related assets are traded in opaque over-the-counter (OTC)
markets, where transactions occur bilaterally between market participants.
We propose a model of sovereign default where bonds and CDS trade over the counter, and
we leverage regulatory data to quantify the impact of the CDS market on economic outcomes of
Argentina. The model uses directed search to capture key liquidity properties of the data, such as
bid-ask spreads, dealer CDS positions, and CDS-bond basis deviations, the last of which measures
arbitrage opportunities available to CDS and bond traders. We obtain CDS positions for sovereign
debt via regulatory data from the Depository Trust and Clearing House Corporation (DTCC), and
large dealers’ exposure to sovereign default (CDS and bond holdings) from the FR Y-14Q regulatory filings as part of the Federal Reserve’s Capital Assessments and Stress Testing information
collection. We expand this data with bid and ask quotes for bonds and CDS from Bloomberg. This
data lets us determine how frictional bond and CDS markets are, how risk is shared among dealers
and investors, and ultimately how the sovereign is affected by CDS markets.
In the model, we divide market participants into a sovereign government, dealers and investors.
Dealers and investors have the same preferences, but investors have ex-ante heterogeneity in exposure to default risk, which is subject to shocks. There are three assets: a risk-free asset with
a perfectly elastic supply, a sovereign bond in positive supply, and CDS on the sovereign bond in
zero net supply. Dealers have access to competitive interdealer markets where they can trade bonds
and CDSs without frictions. Investors have to search for dealers to trade bonds and CDS. Search
is directed, and investors can choose to trade in one of a continuum of submarkets where dealers
charge different intermediation fees. There is free entry of dealers into each submarket, and, in
equilibrium, investors can increase their matching probability by paying higher fees to dealers.
Investors first choose a submarket in bond markets and, if matched, purchase as many bonds as
they want at interdealer prices. Then investor exposure potentially changes, and investors choose
CDS submarkets and desired quantities of CDS at interdealer prices.2 The risk-free asset market
1
According to the BIS Quarterly Review of June, 2018, the volume traded of sovereign CDS by 2007 was around
$1.6 trillion and representing 3.4% of the total CDS market and had more than duplicated its size to $3.3 trillion by
mid-2013, accounting for 13.3% of the CDS market.
2
While this timing of bonds first then CDS is potentially important, Section 5 shows it is not quantitatively important.
1
can be accessed freely by dealers and investors, which without loss of generality occurs after CDS
trading.
We embed this OTC market structure into a standard sovereign default model (Arellano, 2008).
The state at the beginning of the period is a country’s endowment and a stock of debt. The decision
to repay or default is made strategically comparing the two alternatives. In particular, the value of
repaying depends on the market value of bonds, an outcome of the OTC structure of the model.
If the government repays its debt, it chooses the amount of new bonds to be issued in order to
maximize the lifetime utility of the representative consumer in its country. We prove that the fully
liquid (zero trading frictions) version of the model nests the standard risk-neutral pricing model.
Trading frictions resulting from matching and portfolio restrictions determine equilibrium bond
prices through several mechanisms. Three of these are direct effects. First is a classical Walrasian
demand effect. For example, eliminating CDS induces substitution from CDS to bonds, changing
aggregate bond demand. This force would be present absent search frictions. Second is an intermediation effect. Portfolio restrictions change dealer profits and, through free entry, trading fees.
Different fees change the probability of matching and consequently aggregate demand. Third is an
entry effect. As investors’ incentives to trade change, so do the fees they are willing to pay. This
induces more or less entry by dealers, and this measure matters when dealers share risk. These
three direct effects are complemented by an indirect effect—the default risk effect. When one of
the direct effects changes bond prices, the sovereign’s borrowing and default decisions change as
well, resulting in a general equilibrium effect on bond prices.
How much these effects matter for the impact of any policy change depends crucially on trading
frictions. Our first quantitative contribution is in identifying these frictions using data and showing
the model delivers key empirical patterns. In the data, as in the model, we divide market participants into three categories: sovereign governments; large banks active in CDS markets (dealers);
and other market participants (investors). CDS buy and sell volumes and dealer bond and CDS
holdings identify exposure heterogeneity, both between dealers and investors and among investors.
The average bid-ask spreads of bonds and CDS identify trading fees. The elasticities of bid-ask
spreads to changes in default risk identify the elasticities of the matching technology in OTC markets. Identified in this way using Argentinean data, the model replicates key untargeted patterns,
including a positive correlation between CDS volume and risk and a large and positive CDS-bond
basis deviation that is increasing in risk.
This last result tends to be hard to obtain because it indicates CDS are expensive relative to
bonds, which normally can be arbitraged away by selling CDS protection and buying bonds (a
trade permitted by the benchmark’s portfolio restrictions). For instance, Oehmke and Zawadowski
(2015) prove in their corporate CDS model the basis must be negative (Proposition 4). We show
the model delivers a positive basis through higher gains from trade leading to higher matching
2
probabilities. For the identified elasticities, the higher matching probabilities boost aggregate bond
demand more than CDS demand. Section 3.5 discusses in detail how this works.
Our second quantitative contribution is determining the impact of trading frictions on the
sovereign bond market. We first exam the impact of trading frictions relative to the model’s perfectlyliquid version and find that the direct effect of frictional trading worsens prices on average but improves them at small debt levels. The indirect effect / default risk effect amplifies these movements,
resulting in a large, state-dependent impact on prices. We then turn to the bond market and allow
for unbounded short positions in bond holdings. Allowing bond shorting has a strong negative impact on bond prices due to a strong demand effect amplified by the default risk effect. In contrast,
closing the CDS market has only a minimal impact on bond prices. Finally, we analyze a policy
actively debated during the European debt crisis and implemented by the European Union in 2012:
a ban on “naked” CDS purchases—CDS purchases that exceed bond holdings and consequently
generate negative exposure. We find that banning naked CDS has a small, positive effect on bond
prices driven by the intermediation, demand, and default risk effects. We investigate how sensitive
these results are to parameter values, and show naked CDS bans and eliminating CDS can have
large effects, even when altering just a single parameter controlling matching elasticities. This result highlights the importance of our first contribution, showing results are not generically robust
but depend crucially on how large trading frictions are. The welfare effects of trading frictions
can be large, both for the sovereign and investors, and the default risk effect / general equilibrium
feedback from prices to default rates proves essential for assessing them.
Relation to the Literature. Our paper contributes to two strands of the literature. First, it contributes to the sovereign default literature that followed the seminal work of Eaton and Gersovitz (1981) and the quantitative literature arising after the influential work of Arellano (2008) and
Aguiar and Gopinath (2007). Our contribution is to bring into consideration how the CDS market
affects sovereign bond markets and consequently a sovereign’s ability to issue debt. The closest
paper to ours is Salomao (2017)—to our knowledge the first paper in this literature incorporating
CDS. Her work investigates how CDS affect debt restructuring outcomes and the corresponding
implications for government decisions. She finds that CDS can generate uncertainty on the recovery value of defaulted bonds and such uncertainty make investors be more aggressive in debt
restructuring negotiations. Our work is complementary, highlighting the risk-sharing role of CDS
while taking into account the frictional nature of bond and CDS markets. In this respect, our paper is also closely related to the emergent literature on illiquidity in sovereign debt markets with
random search (Passadore and Xu, 2022) and directed search (Chaumont, 2022). We extend those
analyses by incorporating CDS, identifying trading frictions in a novel way, and assessing the
impact of the naked CDS ban implemented during the European sovereign debt crisis.
3
Second, our paper contributes to the large finance literature that followed Duffie et al. (2005)
and studies search frictions in OTC markets. Our model is closer to those in Lagos and Rocheteau
(2009) and Lester, Rocheteau, and Weill (2015) since we allow for assets holdings to be traded in
perfectly divisible quantities using directed search. Most of this literature focuses on outcomes of a
single asset market, but some recent work investigates interactions between multiple assets. This is
the case for recent work by Oehmke and Zawadowski (2015) and even more closely related work
by Sambalaibat (2022).3 Oehmke and Zawadowski (2015) characterize the interactions between
corporate bonds and CDS and propose a theory where corporate CDS are not redundant because
they are cheaper to trade. This assumption does not necessarily carry over to the sovereign CDS
market, and we show how to identify the costs from the data. Additionally, CDS are not redundant
in our model for any positive trading cost because there is a risk of not matching with dealers.
Sambalaibat (2022) uses a theoretical model to study the interactions between bonds and CDS
in OTC markets. In her model, heterogenous investors, some of whom would like to take short
positions in bonds, trade through random search. The main finding is that allowing for CDS trade
improves bond prices by reducing the bond illiquidity discount. Many of the mechanisms in her
model are also operative in ours. For instance, allowing CDS in her model expands the set of feasible trades and attracts more investors into the market. This is embedded in our demand effect,
where investors can match with higher probability, and in our entry effect, where increased investor
activity induces more dealers to actively trade. However, there are also important differences. Contrary to her main finding, the introduction of CDS in our benchmark calibration has a small negative
demand and total effect on bond prices. In other calibrations, introducing CDS has a large positive
effect, or a large negative effect. The disagreement with her main result might be attributable to
investors inability to obtain more than one unit per individual in her model, preventing individual
investor demand from playing a larger role. But there are also effects in our model not present
in hers. For example, because of directed search, increases in gains from trade result in a higher
probability of matching due to investors choosing higher fee submarkets.4 This ends up being an
important quantitative channel for generating a positive CDS bond-basis. One important qualitative difference between our models is our inclusion of general equilibrium feedback (the default
risk effect). This amplifies the indirect effects and can change the welfare conclusions. Finally, our
model is quantitative, matching key data well. We show many of the results, both qualitatively and
quantitively, depend on trading frictions that our calibration strategy lets us identify from the data.
The paper is organized as follows. We present our model in Section 2. In Section 3 we calibrate
the model and discuss the results. Section 4 studies the effects of different regulations and perform
3
Additionally, Sambalaibat (2018) examines empirically the effects of the European naked CDS ban finding the
permanent ban decreased bond market liquidity.
4
She does, however, consider an extension with endogenous search intensity.
4
a welfare analysis. Section 5 performs robustness exercises, and Section 6 concludes.
2
The model
Our model is comprised of an OTC block and a sovereign block. The OTC block takes the issuance
of bonds and default probability as given, and determines the market clearing bond and CDS prices.
The sovereign block takes the price schedule for the bond as a function of the bond issuance, and
determines the bond issuance and default probability.
2.1
The OTC block
We begin with the OTC block since it is the novel component of our sovereign debt model.
2.1.1
Agents, preferences, and endowments
At any given moment, there are two types of agents in action: a finite measure I of investors and
an infinite measure of dealers. To ensure that prior histories do not affect current outcomes, we
assume investors and dealers permanently disappear from the market after closing their trades and
consuming (and are replaced with fresh ones).
Investors and dealers have quasi-linear utility functions that value consumption g this period
and g 0 next period as g +βEδ ui (g 0 ) and g +βEδ ud (g 0 ), respectively. (In the calibration, investor and
dealer utility will be the same up to a constant.) These preferences enable portfolio decisions to be
swayed by risk while simultaneously keeping endowments from influencing anything other than
the risk-free asset, which is not relevant for our purposes. Thus, we normalize dealer and investor
endowments to zero in the current period. In the next period, the endowment is potentially correlated with default, paying out ω in repayment and 0 in default. Hence, investors have exogenous
exposure to default in the amount ω.5 Investors will also have endogenous exposure equal to their
bonds position less any net CDS protection. Exogenous exposure is stochastic, evolving from ω e
before bonds are traded to ω after according to ω = (1 − ρω )µω + ρω ω e + σω εω with εω ∼ N (0, 1).
We assume the unconditional distribution (across investors) of ω e and ω is the same, which requires
ωe ∼ N (µω , σω2 /(1 − ρ2ω )).
Heterogeneity in ω e and ω serves two purposes. First is that previous financial investments,
from which our model abstracts, have ongoing implications. Second is that investors are heterogeneous in default-cost incidence. E.g., investors with local-currency debt and USD-denominated
assets may gain from a default if it produces rampant inflation. Or, investors may be hedged across
multiple countries, reducing idiosyncratic risk. The role of the within-period exposure shock εω
makes CDS useful for hedging previous bond choices, capturing its use as a tool for marginally
5
Lemma 1 in Appendix A shows this formulation of the exogenous exposure is equivalent, due to quasi-linearity,
to one with payments 0 in no-default states and −ω in default states.
5
increasing or decreasing exposure, which we discuss later on.
2.1.2
Financial markets and technology
There are three assets: a risk-free asset with a perfectly elastic supply, a sovereign bond b in fixed
supply B 0 > 0, and CDS contracts c in zero net supply. To simplify the language, we refer to the
sovereign bond as just the bond when not ambiguous. One unit of the risk-free asset a pays one
unit of consumption next period and costs qf . The supply of the risk-free asset is perfectly elastic.
That is, investors and dealers can buy or sell any quantity of the risk-free asset at the price qf .
One unit of the sovereign bond (CDS) pays one unit of consumption next period in states
where the sovereign repays (defaults) and zero in states where it defaults (repays).6 As there will
be essentially only two states of uncertainty for investors and dealers, the bond is like an Arrow
security paying out in repayment and the CDS an Arrow security paying out in default. We use
δ = 1 to denote the state of the world in which the bond defaults, and δ = 0 to denote the state
of the world in which it does not default. The default probability is a sufficient statistic for dealers
and investors, and we denote it δ̄. We allow for a technological constraint on shorting bonds in the
form of b ≥ b, where b ≤ 0 and b = −∞ means there is no constraint. Similarly, we allow for
a constraint on endogenous exposure x ≡ b − c ≥ 0 where x ≤ 0 and x = −∞ means there is
no constraint. In a naked CDS ban, x = 0, preventing an agent from benefiting financially from
default. We assume that constraints have to hold with probability one at the end of the period and
that x ≤ b ≤ 0.7 Agents may take any short or long position in CDS as long as it satisfies the
previous constraints on endogenous exposure (i.e., c is chosen from R).
For investors to trade bonds or CDS, they must match with dealers in frictional markets. Specifically, if they wish to purchase bonds in the amount b, they choose a submarket characterized by
dealer fee fb ∈ R+ . If matched, they pay an inter-dealer unit price q for a total of qb plus the
fee fb . In choosing fb , they take as given the market tightness θb (fb ) in that submarket, which is
the measure d of dealers active in submarket fb relative to the measure of investors n active in
that submarket. The constant returns-to-scale matching technology Mb (n, d) determines investors’
matching probability αb (θb (fb ))) ≡ Mb (1, θb (fb )) = Mb (n, d)/n.8 Likewise, to purchase or sell
CDS c, they must pay fc plus the inter-dealer cost pc if they match, which occurs with probability
αc (θc (fc )).
Active dealers trade bonds or CDS with investors. To do so, they must pay an entry cost to
6
We abstract from counterparty risk. In the data, CDS contracts pay an exogenous coupon when there is no default.
The CDS contract here is identical to a CDS contract with a coupon κ paired with a short-position in the risk-free asset
equal to κ. Since the risk-free asset is liquid, setting the coupon to zero is a normalization.
7
The restriction x ≤ b is so that investors who matched in in bond markets and, e.g., took a position of b = b do
not have to match with probability one in CDS markets in order to satisfy the exposure constraint.
8
We assume Mb (0, ·) = Mb (·, 0) = 0.
6
enter the respective market. Active dealers in the bond market have to pay an entry cost γb > 0,
while active dealers in the CDS market have to pay an entry cost γc > 0. After entering either
market, they can purchase any desired amount of bonds b and CDS c at inter-dealer prices in a
frictionless inter-dealer market. An active dealer in the bond market chooses a submarket fb to
visit, matching at rate ρb (θb (fb )) ≡ Mb (1/θb (fb ), 1) = Mb (n, d)/d.9 Similarly, an active dealer in
the CDS market chooses a submarket fc and matches at rate ρc (θc (fc )). A useful property of the
matching technology is that αi (θ) = ρi (θ)θ for i = b, c.
2.1.3
Timing
To define the model timing, we break the current period into three sequential sub-periods, s1 , s2 , s3 .
In s1 , dealers decide to become active in bonds or CDS. Investors decide how many bonds they
wish to purchase and the submarket they wish to enter.10 Bond-market matching realizations occur
at the end of s1 . At the beginning of s2 , the exposure shock εω is realized. Then investors decide
how much CDS they wish to purchase and the submarket they wish to enter. CDS-market matching realizations occur at the end of s2 . In s3 , investors choose their risk-free bond position. Dealers
choose their bonds, CDS, and risk-free positions, and settle all their outstanding obligations (delivering b or c to investors as promised) simultaneously. At the beginning of the next period, the
default shock is realized, payments are settled, and consumption occurs.11
2.1.4
The dealer’s problem
For an active dealer, the demand for risk-free asset, sovereign bonds and CDS is independent of
which markets she intermediates. This is because dealers have access to a perfectly competitive
inter-dealer market. An active dealer’s portfolio problem is
π
s.t.
= max −qf a − qb − pc + βEδ ud (a + δc + (1 − δ)b)
a,b,c
(1)
b ≥ b, b − c ≥ x
In general, the optimal policies will not be unique because there is a redundant asset.
The value from becoming an active bond dealer is given by the value of being active, π, minus
the entry cost to become active, γb , plus the expected benefits from trading with an investor in their
9
The trading rate ρ may be greater than unity, which can be interpreted as dealers executing more than one bilateral
transaction in a given period. The more transactions dealers make (in expectation) in a given submarket, the higher
expected profits are. Thus, more dealers enter such submarkets to satisfy the zero expected profit condition that follows
from free entry of dealers into each submarket. In submarkets where transaction fees are relatively low, the zero profit
condition may imply that each dealer entering the submarket makes more than one transaction, in expectation, in order
to cover the fixed entry cost.
10
If they do not wish to trade, they can choose a zero-fee submarket where trade will occur with zero probability.
11
Because the bond market opening before the CDS market is potentially important in driving the results, Section
5 explores a version of the model where investors access the CDS market first. We find similar quantitative results.
7
preferred submarket. That is,
Πb = π − γb + max ρb (θb (fb ))fb .
fb
Similarly, the value from becoming an active CDS dealer is
Πc = π − γc + max ρc (θc (fc ))fc .
fc
Becoming an active bond dealer and CDS dealer yields the following value
Πb,c = π − γb − γc + max ρb (θb (fb ))fb + max ρc (θc (fc ))fc .
fb
fc
Finally, an inactive dealer’s value is
Π0 = max −qf a + βEδ ud (a).
a
Without loss of generality, we assume ud , β, and qf are such that Π0 = 0.
We assume free-entry of dealers. Therefore, in equilibrium Πb , Πc , Πb,c ≤ Π0 = 0. Given this,
it is immediate that
Πb,c = Πb + Πc − π ≤ 0,
with strict inequality whenever π > 0. As we prove in the appendix, π attains a minimum of
Π0 = 0 only at risk-neutral prices.12 Consequently, entering only one of the bond or CDS markets
is always optimal, and strictly so except for one specific set of prices (which will generally not clear
markets). Therefore, we will look for an equilibrium where dealers enter bond or CDS markets,
but not both.
For use in the investor problem, we briefly characterize active submarkets, that is, submarkets
with θ > 0. In an active bond submarket fb , one must have the choice of fb attain the maximum Πb
and, because of free-entry, have Πb = Π0 = 0; the situation is identical for active CDS submarkets.
Defining net entry costs as γ̃b ≡ γb − π and γ̃c ≡ γc − π, this requires
ρb (θb )fb = γ̃b
and
ρc (θc )fc = γ̃c ,
or, equivalently,
fb =
γ̃b θb
αb (θb )
and fc =
γ̃c θc
.
αc (θc )
We will use this mapping from positive θ to fees in the investor problem.
12
See Lemma 4. The “only at” requires x < 0. If x = 0, a range of prices generates π = Π0 = 0.
8
(2)
2.1.5
The investor’s problem
We solve the investor’s problem by backward induction starting from the market for risk-free assets
in sub-period s3 . In sub-period s3 , an investor with bond and CDS holdings b and c solves
V (ω, b, c; s3 ) ≡ max −qf a + βEδ [ui (a + (1 − δ)(b + ω) + δc)] .
a
(3)
Let the optimal choice be denoted ai (ω, b, c). Here, the cost of acquiring bonds and CDS does
not appear because we have exploited quasi-linear utility to treat them as sunk. The first order
condition of this problem is
qf = βEδ [u0i (a + (1 − δ)(b + ω) + δc)] ,
(4)
which is the Euler equation for risk-free assets. Under the assumed preferences: (i) any risk-free
endowment in the next period would simply shift the optimal choice of a by a commensurate
amount, but result in the same next-period consumption; and, (ii) any endowment in the current
period would just scale current consumption.
In sub-period s2 , after observing ω the investor has to choose a submarket to enter and a demand
for CDS in case of matching with a dealer. Using (2), which provides a map between the dealers’
fees and the tightness of the market, we can write the investor’s problem as choosing the tightness
directly. Thus, an investor with bond holdings b solves
V (ω, b; s2 ) ≡ max − γ̃c θc + αc (θc ) [V (ω, b, c; s3 ) − pc] + [1 − αc (θc )]V (ω, b, 0; s3 )
θc ≥0,c
(5)
subject to b − c ≥ x.
Note the fee, given implicitly by γ̃c θc /α(θc ), is paid conditional on matching, resulting in an expected fee of γ̃c θc . The first order conditions of this problem are
p ≤ Vc (ω, b, c; s3 ), with equality if b − c > x, and
γ̃c = αc0 (θc ) [V (ω, b, c; s3 ) − pc − V (ω, b, 0; s3 )] ,
(6)
(7)
where Vc (ω, b, c; s3 ) denotes the partial derivative of V (ω, b, c; s3 ) with respect to c. The optimal
choice of c conditional on matching satisfies the conventional Euler equation for CDS. Let an
optimal CDS choice and optimal market tightness be denoted ci (ω, b) and θc (ω, b), respectively.
In sub-period s1 , the investor with exposure ω e has to choose a submarket to enter and a demand
for sovereign bonds upon meeting a dealer. Using equation (2) again, we can write the investor’s
problem as choosing the tightness directly. Thus, an investor solves
V (ω e ; s1 ) ≡ max − γ̃b θb + Eω|ωe αb (θb ) [V (ω, b; s2 ) − qb] + [1 − αb (θb )]V (ω, 0; s2 ) ,
θb ≥0,b
subject to b ≥ b.
9
(8)
The first order conditions of this problem are
q ≥ Eω|ωe Vb (ω, b; s2 ), with equality if b > b, and
γ̃b = αb0 (θb )Eω|ωe [V (ω, b; s2 ) − qb − V (ω, 0; s2 )] ,
(9)
(10)
where Vb (ω, b; s2 ) denotes the partial derivative of V (ω, b; s2 ) with respect to b. Let an optimal
bond choice and optimal market tightness be denoted bi (ω e ) and θb (ω e ), respectively.
2.1.6
Market clearing
There is a perfectly elastic supply of the risk-free asset at the price qf , the market clearing price.
We next provide market clearing conditions for the sovereign bond and CDS markets.
Let us introduce some notation. Define as db (ω e ) the measure of dealers active in the bond market in sub-period s1 trading with investors with exposure ω e , dc (ω, b) the measure of dealers active
in the CDS market serving investors with b bond holdings and exposure shock ω in sub-period s2 ,
R
R
and D = d¯ + e db (ω e )dFω +
dc (ω, b)dFω,b is the total measure of dealers. Fω represent the
ω
ω,b
distribution of exposure shocks and Fω,b is the endogenous joint distribution of investors’ realized
exposure shocks and bond holdings. In s1 , denote nb (ω e ) the mass of investors with exposure ω e .
In sub-period s2 , investors who matched with a dealer have bond holdings bi (ω e ) while those in¯ is
vestors who did not match have bond holdings equal to 0. The exogenous mass of dealers, d,
always active in the interdealer market.
The bond market clears if
0
Z
B =
Mb (nb (ω e ), db (ω e ))bi (ω e )dFω + Dbd .
(11)
ωe
The CDS market clears if
Z
0=
Mc (nb (ω, b), dc (ω, b)) ci (ω, b)dFω,b + Dcd .
(12)
ω,b
We assume that the following regularity conditions hold:
Assumption 1. ud is unbounded above.
Assumption 2. At least some short-selling is allowed, b < 0, at least some financial exposure is
allowed, x < 0, and there is a strictly positive measure of exogenous dealers, d¯ > 0.
Assumption 3. α0 (0) = ∞, α0 (θ) > 0, and α00 (θ) < 0 for all θ > 0.
Assumption 1 guarantees that dealer utility from trading in interdealer markets π goes infinite
when consumption next period goes infinite. This ensures that if they can make infinite profits,
an infinite amount of dealers enter. Allowing some short-selling in assumption 2 ensures that investors generally have some gains from trade. If there is literally no short-selling, then one can
have a situation where investors want to have negative bonds but cannot, meaning they begin with
10
their optimal level of bonds (zero) and have literally zero gains from trade. Zero gains from trade
consequently results in no entry, and markets do not clear in a natural way.13 However, that lack of
gains from trade is okay as long as there are always some dealers, d¯ > 0. Assumption 3, which is
satisfied by the matching function of section 3, ensures that θ is interior and behaves in predictable
ways as net entry costs go to zero.
To see how these assumptions work to make aggregate demand move naturally, reconsider the
optimal choice of θb given by
γ̃b = γb − π = α0 (θb )(V (b; s2 ) − qb − V (0; s2 ))
The gains from trade are (V (b; s2 ) − qb − V (0; s2 )). If this is strictly positive, as generically it is
by virtue of assumption 2, and γ̃b > 0, then assumption 3 guarantees that (1) there is an optimal,
interior θb ∈ R++ and (2) as γ̃b ↓ 0, θb ↑ ∞ for any q. Consequently, the aggregate bond demand
grows infinite, in absolute value, as γ̃b ↓ 0. By the strict quasi-convexity of π (established in
lemma 4 in the appendix), γ̃b = 0 at two points, one with high q (q̄) and one with low q (q). As one
approaches either of these extremes, aggregate demand explodes, and so equilibrium is in (q, q).
2.1.7
Definition of equilibrium in OTC markets
We define equilibrium in OTC markets given δ̄ and B 0 as follows:
Definition 1. A family {V, bi , ci , ai , θb , θc , π, bd , cd , q, p} is a symmetric equilibrium if satisfy equations (1)-(12).
2.2
The sovereign block
We now describe the sovereign block of the model, which endogenizes the supply of bonds, B 0 ,
and default decisions, δ. For any given default probability and bond supply, equilibrium in the OTC
markets determines a bond price q.14
2.2.1
Agents, preferences, and endowments
There is a sovereign government who has a stochastic, Markov “potential” output stream Y . If the
sovereign does not default and is not in autarky, this potential output stream is actual output. If the
sovereign does default or is in autarky, this output stream is reduced to h(Y ) ≤ Y . The sovereign
P
values stochastic consumption streams {Ct } according to E t βgt u(Ct ).
13
They might still clear, but it would be through exploiting indifference conditions at θ = 0.
This part of the model is completely standard following Arellano (2008), and readers familiar with that model
can probably skip to (13).
14
11
2.2.2
Financial markets
For tractability, we assume that bonds mature in one period. With one-period bonds, investors hold
bonds and CDS contracts at most for one period. Because of this, the distribution of bond holdings
is re-started every period. Thus, the distribution of investor types and bond holdings is not part of
the aggregate state of the economy, which greatly simplifies the solution of the model.15
At the beginning of each period, the sovereign has some amount of existing debt obligations B.
It then chooses to honor those obligations, δ = 0, or default on them δ = 1. If it defaults, then the
sovereign enters autarky, i.e., is unable to save or borrow, from which it escapes with probability
φ. In autarky, we say the sovereign’s debt is zero.
If the sovereign is not in autarky and does not default, then it chooses an amount of debt B 0 to
issue, taking the price schedule q(Y, B 0 ) as given.
2.2.3
Timing
The timing of the sovereign block is as follows. Shocks determining the level of potential output
Y and whether the sovereign leaves autarky (if applicable) are realized. The sovereign makes its
default decision (if not in autarky). If the sovereign was not in autarky and repays maturing debt,
the sovereign can issue new debt B 0 .
2.2.4
Government’s problem
At the beginning of each period, the government has outstanding level of debt that it needs to repay,
B, and observes the new realization of the endowment, Y . Since bonds mature in one period and
the distribution of bond holdings is re-started every period, the pair (Y, B) is the relevant state of
the economy. After observing the state of the economy, the government decides whether to repay
the outstanding level of debt or to default.
The optimal decision is determined by weighting the costs and benefits of repaying the outstanding level of debt. The benefit of default is debt service costs can instead be used to boost current period consumption. The costs are lost output and a temporary exclusion from international
credit markets. The temporary exclusion from international credit markets reduces the ability of
the government to use credit as a source for consumption smoothing. The length of the exclusion is
captured by an exogenous probability of regaining access to credit markets, φ ∈ (0, 1). When the
government re-gains access to credit markets, it starts next period with no debt. The output cost is
an endowment loss while the government is in default and it is given by the function h(y) ≤ y, for
15
Assuming long-term debt would require tracking distributions of bond and CDS holdings or assuming that investors who leave the economy after one-period somehow can re-allocate all bonds and CDS positions to new investors
without being subject to trading frictions in OTC markets. This is an interesting extension with even more effects to
consider, but it is worth understanding the many mechanisms present in this simpler formulation, first.
12
all y.
The recursive formulation of government’s problem is then given by
W (Y, B) = max {δW d (Y ) + (1 − δ)W r (Y, B)},
δ∈{0,1}
W (Y ) = u(h(Y )) + βg E{φW (Y 0 , 0) + (1 − φ)W d (Y 0 )},
d
W r (Y, B) = max
u(C) + βg E{W (Y 0 , B 0 )},
0
(13)
C,B ≥0
s.t. : C = Y + q(Y, B 0 )B 0 − B
where W is the option value of repaying the debt, W d is the value of defaulting, and W r is the
value of repaying. Whenever the government decides to repay its debt, it is allowed to choose the
new debt issuances, B 0 , taking as given the price scheduled that it faces in the market, q(Y, B 0 ). Let
the optimal default policy be denoted δ(Y, B) and an optimal bond policy be denoted B 0 (Y, B).
The frictions in secondary markets for bonds and CDS play an important role in the default
decision of the government. They enter the problem of the government by affecting the market
price of newly issued bonds, q, and thus directly affecting the value of repaying debt, W r .
2.2.5
Definition of equilibrium in the sovereign block
We define partial equilibrium in the sovereign block given q as follows:
Definition 2. A partial equilibrium in the sovereign block given a price schedule q is a family
{W, W r , W d , B 0 , δ} that is a solution to the sovereign’s problem.
2.3
Combining the model blocks
To combine the two blocks, we need to impose consistency between the price schedule arising from
the OTC block, and the bond issuance and default probability optimally chosen by the government.
We say the price schedule q(Y, B 0 ) is consistent with OTC equilibrium if, for every Y, B 0 > 0, there
exists an equilibrium in the OTC markets, given the default probability δ̄ = Eδ(Y, B 0 ) and debt
issuance B 0 , that results in price q(Y, B 0 ). We are now ready to state equilibrium.
Definition 3. An equilibrium is a set of functions {q, W, W r , W d , B 0 , δ} such that {W, W r , W d , B 0 , δ}
solves the sovereign’s problem taking q as given and q is consistent with OTC equilibrium.
We characterize equilibrium in the case of σω = 0 to build intuition and provide some additional insights on the equilibrium outcomes. As the focus of our paper is quantitative, this analysis
is provided in Appendix A for interested readers.
3
Calibration and benchmark results
The model is calibrated at a quarterly frequency to match Argentina.
13
3.1
Functional Forms
The output cost of default is
h(y) = y − max{0, d0 y + d1 y 2 }
Utility function curvature is the same for the sovereign, dealers, and investors:
c1−σ − 1
= ud (c) + κ,
1−σ
where κ is chosen to deliver Π0 = 0. Log output follows a Gaussian AR(1) process, log Y =
u(c) = ui (c) =
ρY log Y−1 + σY Y . The matching function is given by
n−ξi
n−ξi + d−ξi
for ξi ∈ (0, 1) and i ∈ {b, c}. This results in a matching probability of
1
.
αi (θ) =
1 + θ−ξi
for i ∈ {b, c}. The parameter ξi controls the matching probability elasticity and can vary across
Mi (n, d) = n
bond and CDS markets.
3.2
Measurement
To bring the model to the data, a few key concepts are needed. First is that our CDS payments
fc are upfront but can equivalently be quoted in terms of a running spread. A running spread is
an endogenous coupon payment in the case of non-default such that the expected, discounted, net
present value of a CDS contract is zero. Second is that our bond payments fb are also upfront but
can be quoted in terms of a Z-spread, which is just the internal rate of return less the risk-free rate.
Our measure of the CDS-bond basis breakdown follows the literature by looking at the running
CDS spread less the Z-spread of sovereign debt. Appendix C describes these measures in more
detail. The reason the CDS-bond basis is expected to hold is because, for small default rates, both
spreads should approximately equal the expected default rate.16
The bid/ask spreads for bonds in our data are quoted in yield to maturity (YTM), which is
equivalent to being quoted in Z-spreads. The bid/ask spreads for CDS are likewise quoted in running spreads. We define a bid/ask spread in the model by noting the following. First, CDS dealers’
net profit is ρ(θ(fc ))fc . To be willing to transact with at least some positive probability, this must
be positive. As any bid less than pc will never be transacted (fc ≤ 0 ⇒ ρ = 0), we think of the
“bid” price as being pc. On the other hand, fc > 0 will sometimes transact and sometimes not.
16
One peculiarity of the conventional measure is that even when q +p = qf , so no arbitrage holds exactly, there can
still be a measured deviation in the CDS bond basis due to nonlinearities. In our model, we will have the deviations
due both to this nonlinearity and the lack of arbitrage in transacted investor-dealer prices.
14
Hence, we say dealers ask for pc + fc , in which case if that is met, they will transact for sure. (As
the fc are heterogeneous, there will also be heterogeneity here, but we will aggregate to a single
number.) Consequently, a bid/ask upfront spread per unit of notional CDS as (pc + fc − pc)/c or
fc /c, which can be volume weighted to derive an aggregate number. However, for a bid/ask spread
in terms of the running spread, we take the volume-weighted price (pc + fc )/c and interdealer
price p, convert those both to running spreads, and take their difference. Similarly for bonds, we
get the volume-weighted average bond price (qb + fb )/b and the interdealer price q, convert both to
Z-spreads, and difference them to get a spread in terms of YTM. In measuring the CDS-bond basis
deviation, we use the average transacted prices inclusive of fees, convert CDS to running spreads
and bonds to Z-spreads, and then difference them.
A final key measurement issue is that our bonds and CDS are both short-term contracts, while
in the data they are five-year contracts. For bonds, we handle this in the standard way by focusing
on the debt service targets rather than debt stocks. For CDS, we do something similar, reducing
our position and volume measures to one quarter’s worth (5%) of a five-year contract.
3.3
Identification strategy
Table 1 provide the parameters that we set exogenously with some rationale. Most of these values
are standard, but some deserve further explanation. As discussed at the end of Section 2.1.1, exogenous exposure in part captures the impact of previous financial investments. To capture legacy
exposure to CDS and bonds, which our model assumes are short-term but in the data are typically
5-year contracts, we set the persistence ρω = 0.95 to give a 5-year half-life. To ensure market
clearing, we allow a small measure of bond shorting b = −0.01, which ensures gains from trade
exist generically. Likewise the exogenous measure d¯ of dealers is set to 0.001 (which will be tiny
relative to the estimated measure of investors) to ensure even if there is no entry (due to no gains
from trade) the bond market can still clear.
This leaves ten parameters: seven for the OTC block and three for the sovereign block. In the
sovereign block we need to determine the sovereign’s discount factor and the two default cost
parameters. We pin these down using standard moments in the literature in the liquid version of
the model, where investors and dealers can trade bonds without frictions. Specifically, we target
the debt-service output ratio, the standard deviation of spreads (defined as the yield-to-maturity
over a risk-free rate) and mean of spreads. We find (β, d0 , d1 ) = (0.849, −0.119, 0.128), and the
corresponding moments are denoted in Table 3 with an asterisk. Calibrating these parameters separately prevents the estimation from distorting OTC frictions to improve model fit on these standard
dimensions.
In the OTC block we need to determine the bond and CDS entry costs, the matching function
elasticities for bonds and CDS, the mean and variance of the exposure shocks to investors (expo15
Table 1: Exogenously fixed parameter values with explanations
Parameter
Sovereign
σ
ρY
σY
φ
Dealers
β
σ
d¯
Investors
β
σ
ρω
Markets
b
r
qf
Description
Value
Risk aversion
GDP persistence
GDP innovation std.
Autarky escape prob.
Discount factor
Risk aversion
Exogenous measure
Discount factor
Risk aversion
Exposure persistence
Bond shorting limit
CDS coupon
Risk-free price
2
0.949
0.027
0.0385
Reason
Standard value
Chatterjee and Eyigungor (2012)
Chatterjee and Eyigungor (2012)
Chatterjee and Eyigungor (2012)
0.99 Standard value
2 Standard value
0.001 Market clearing regularity
0.99 Standard value
2 Standard value
0.95 Five-year half life
-0.001 Market clearing regularity
0.0 Normalization
0.99 Standard value
sure shocks), and the measure of investors. To identify the first four, we exploit the model’s tight
relationship between entry costs, matching elasticities, and bid/ask spreads. To see this relationship, consider the bid/ask spread for bonds quoted in upfront payments, fb /b, is related to entry
costs γb by the free entry condition (2) in a direct way:
γb − π θb
fb
=
.
b
b αb (θb )
A similar relationship holds for CDS. We use the entry cost parameters γb and γc to match bid/ask
spreads for bonds and CDS, respectively.
Combining the above with the submarket optimality condition (10), we obtain
Eω|ωe [V (ω, b; s2 ) − qb − V (ω, 0; s2 )]
fb
αb0 (θb )θb 1 γb − π
=
=
(θ
)
,
α
,θ
b
b
b
b
αb (θb ) b α0 (θb )
b
where f,x (x) denotes the elasticity of f with respect to x evaluated at x. The elasticity of the
matching probability controls how much changes in gains from trade (the expectation term) translate into increases in bid/ask spreads (the lefthand side). And since default risk movements drive
changes in the gains from trade, the slope in a regression of bid/ask bond spreads on default risk,
which can be measured using YTM, identifies the elasticity of αb and similarly for CDS and αc .
In the data, there is a tight relationship between bid/ask spreads and YTM for both bonds and
CDS, as Figure 1 shows. These elasticities are crucial for some of the model’s predictions, and the
16
model’s tight link between bid/ask spreads and risk is a novel way to identify these key parameters.
Figure 1: Bid/ask bond and CDS spreads against yield-to-maturity for ARG
Spread as Yield Changes in Argentina
Correlation: .3871
Correlation: .8937
0
CDS Spread (in Basis Points)
10
20
30
40
Spread between Bond YTM (in Basis Points)
0
5
10
15
20
25
Spread as Yield Changes in Argentina
400
600
800
1000
1200
Average Yield to Maturity (in Basis Points)
1400
400
Data frequency is daily with dates between Jan 20, 2017 and May 05, 2019
600
800
1000
1200
Average Yield to Maturity (in Basis Points)
1400
Data frequency is daily with dates between Jan 20, 2017 and May 05, 2019
The remaining parameters are the measure of investors and the mean and standard deviation of
exogenous exposure ω (and ω e ). Since exposure shocks ω are orthogonal to default risk, the firstorder effect of increases in the variance of ω is to increase the desire for risk reallocation across
investors. This allows the standard deviation σω to be cleanly identified by aggregate CDS volume.
Due to risk-sharing, increases in the mean level of exposure µω passthrough both to investors and
dealers, and this is reflected (for investors who match in CDS markets) in a lower CDS position for
investors and, by market clearing, a higher CDS position for dealers.
The measure of investors I is the last parameter. Note that the total amount of exposure in the
economy—how many resources stand to be lost in the case of a default—is the bond supply B plus
Iµω . For a given level of µω > 0, increasing the measure I of investors increases total risk in the
economy due to ω, but also decreases how much exposure needs to be allocated per agent, which
is (B + Iµω )/(I + D)—at least ignoring the endogenous response of the dealer measure D. So
as I goes infinite, the amount of exposure borne by each agent tends to decrease (with B/(I + D)
shrinking) and with it the amount of debt. Conversely, for I tiny, investors have little capacity to
absorb debt and most of it is bought by dealers. So the exposure of dealers and bond holdings of
dealers identifies the measure of investors.
3.4
Model fit
Our identification strategy and estimated fit for the OTC block are summarized in Table 2. The
model reproduces the targeted moments with one exception. Specifically the bid/ask spread for
CDS is too high on average. We cannot exactly pin down the average level of both bid/ask spreads,
in part because of strong theoretical linkages between the bonds and CDS market (these can be
seen in Proposition 5 in the appendix). However, the fit is quite good overall, and the key matching
17
elasticity identification strategy works as expected.
Table 2: Targeted moments from the calibration
Moment
Model
Data
Bid-ask spread for bonds mean (%)
Bid-ask spread for CDS mean (%)
Agg. dealer CDS position / mean GDP (%)
Agg. dealer net exposure / mean GDP (%)
Agg. dealer bond position / mean GDP (%)
Agg. dealer-investor volume / mean GDP (%)
Reg. coef. of YTM on bid-ask bond spreads
Reg. coef. of YTM on bid-ask CDS spreads
0.0641
0.2333
0.0006
0.0165
0.0171
0.1951
0.0052
0.0205
0.0629
0.0984
0.0010
0.0170
0.0180
0.1945
0.0042
0.0204
Parameter
Value
Bond entry cost
CDS entry cost
Investor exposure mean
Investor measure
0.0893
1.8232
0.1106
3.4309
Investor exposure s.d.
Matching function elasticity bonds
Matching function elasticity CDS
0.0174
0.0901
0.3925
The entry costs for bonds and CDS turn out to be quite different, with the CDS entry cost about
twenty times larger than for bonds. Because bonds are an inferior asset to CDS, only the latter of
which can offer insurance against default, the model demands that CDS be more expensive than
bonds to rationalize B clearing the market with a small average CDS position of dealers. The
estimated elasticity parameters also appear considerably different. However, the matching probabilities αb and αc are not constant elasticity, and so one cannot directly map from ξi to αi ,θi . One
important lesson from our model is that liquidity cannot simply be read off of differences in bid/ask
spreads (or volume)—how those bid/ask spreads change in risk can be equally or more important.
In other words, both the means and slopes in Figure 1 are necessary to properly characterize trading
frictions.
The model’s fit for some untargeted moments is reported in Table 3. The correlations between
volumes and risk all have the correct sign, though the magnitudes are overstated. The correlation
between the dealers’ CDS position and risk has the wrong sign, which is natural because as risk
increases, risk-sharing dictates that dealers should take on more risk, reflected in selling protection.
The data’s positive correlation holds in the whole sample, as can be seen in Figure 2, but evidently
weakens substantially on the subsample having higher levels of risk. Additionally, over a wide
swath of risk levels, the mean position of dealers stays quite flat, reflected in the small volatility of
the dealers’ position (small relative to volume, e.g.), which the model captures. Interestingly, the
model has a large positive mean and standard deviation for CDS-bond basis deviations, like in the
data. In fact, we will show the model’s CDS-bond basis deviations are increasing in risk, as is the
case for Argentina.
3.5
Benchmark results
The model generates three key empirical measures of trading frictions and risk-sharing, which we
depict in Figure 4. The first one is that, as yields and default probabilities increase, dealers tend
18
Table 3: Untargeted moments from the calibration
Moment
Model
Data
Bond spread mean (%)∗
Bond spread std. (%)∗
Debt service to output ratio (%)∗
Bid-ask spread for bonds std. (%)
Bid-ask spread for CDS std. (%)
CDS-bond basis deviation mean (%)
CDS-bond basis deviation std. (%)
Agg. dealer buy volume / mean GDP (%)
Agg. dealer sell volume / mean GDP (%)
Std. of agg. dealer CDS position / mean GDP (%)
Default probability (%) (full sample)
Corr. of log IDP and agg. dealer CDS / mean GDP
Corr. of log IDP and agg. dealer-investor volume / mean GDP
Corr. of log IDP and agg. dealer buy volume / mean GDP
Corr. of log IDP and agg. dealer sell volume / mean GDP
Corr. of YTM spreads and bond bid-ask spreads
Corr. of YTM spreads and CDS bid-ask spreads
Reg. cons. of YTM on bid-ask bond spreads
Pred. bid-ask bond spreads at YTM target
Reg. cons. of YTM on bid-ask CDS spreads
Pred. bid-ask CDS spreads at YTM target
7.4446
3.8296
5.5530
0.0367
0.1004
1.3014
1.6148
0.0978
0.0973
0.0312
1.0808
-0.6104
0.9619
0.7826
0.9664
0.5453
0.7840
0.0003
0.0428
0.0008
0.1683
8.1500
4.4300
5.5300
0.0266
0.0554
6.4800
7.1100
0.0975
0.0970
0.0015
0.5100
0.0900
0.0900
0.0900
0.3870
0.8940
0.0344
0.0690
-0.0383
0.1280
INTERNAL FR/OFFICIAL USE // FRSONLY
Note: the moments marked with ∗ are targeted in a first stage calibration using the liquid
version of the model; correlations are for CDS position and dealer volume are based on
The data used in this project is from DTCC TIW and POS CDS confidential data, and public
quarterly values.
Bloomberg data from 2010 Q1 to 2019 Q4 across 50 countries. The DTCC derived data includes
either 25 international and 5 U.S. based BHCs or 5 select U.S. BHCs also active in the Sovereign
CDS market (J.P. Morgan Chase & Co, Goldman Sachs, Wells Fargo, Citigroup, and Morgan
Stanley).
This scatter plot shows the Dealer’s net CDS position as a percentage of Argentina’s GDP
Figure 2:
Net CDS position of dealers versus yield-to-maturity for ARG
against their 1 year implied default probability (log) during the beginning of 2010 through the
end of 2019.
Note: data is daily, and correlation is for daily observations (which is not identical to the correlation in table 3).
19
Figure 3: CDS-bond basis deviations in the data
CDS bond basis deviations in basis points
0
10000
20000
30000
CDS bond basis deviations for Argentina
2000m1
2005m1
2010m1
Time
2015m1
2020m1
Data from Gilchrist, Wei, Yue, and Zakrajsek (2022).
to remain more or less neutral in terms of protection (the data counterpart of this model result is
Figure 2). At moderate to high levels of risk, their position in the model is tightly clustered around
zero, like the data. Hence it is not the case that dealers are providing large amounts of insurance.
The second key feature is that the CDS-bond basis breaks down when default risk is very high,
exploding positive. Figure 3 shows the data’s CDS-bond basis, which is usually close to zero but
explodes up in each of Argentina’s high default risk epsiodes, 2001, 2009, 2014, and 2020. The
final fact is that, as yields and default probabilities increase, bid/ask spreads for both bonds and
CDS increase, reflecting increased gains from trade and the correspondingly larger fees.
Figure 5 sheds light on why the model generates these patterns. It plots for a fixed debt supply
some key OTC variables as the expected default rate varies. Note that only investors with low
(in fact, negative) exogenous exposure ω are substantially active in the bond market (the more
exposed investors are allowed to take a small short position). As default risk increases, they reduce
their risk by reducing individual bond demand. At the same time, the gains from trade increase,
meaning investors are willing to pay higher fees to access the market and consequently match
at higher rates. Because of the model’s tight connection between fees and bid/ask spreads, this
generates an increase in bid/ask spreads (not pictured). It also generates an increase in matching
probabilities, which means aggregate demand is decoupled from individual demand.
The dealers’ small and stable CDS position is reflected in how investor demand for CDS and
bonds hinges on the exogenous exposure level. Dealers, who have zero exogenous exposure, look
like the moderate ω investor type, demanding almost no bonds and having little CDS demand. This
behavior of dealers is targeted and so must hold in the simulation on average because dealers have
a small CDS position on average, which gives buy and sell volume roughly equal—so investors
are selling and buying protection from each other.
20
Figure 4: Simulated key variables
The most challenging pattern to understand is the CDS-bond basis deviation. It necessarily
begins close to zero when the default rate is close to zero: In the limit where the default rate is zero,
bond and CDS both become risk-free assets with a price of qf and 0, respectively, implying the CDS
running spread and bond YTM spreads are both equal to zero. But as default rates increase, it first
rises (as the running spread on CDS rises faster than Z-spread for bonds) but subsequently falls
sharply. Why?
The CDS-bond basis deviation is really a measure of how cheap bonds are relative to CDS.
The more demand there is for bonds compared to demand for synthetic bonds (−c) via CDS, the
greater the CDS-bond basis deviation will be. An increase in the basis as default risk increases
from low levels reflects that aggregate demand for bonds is increasing faster/decreasing slower
than aggregate demand for synthetic bonds. Here, individual demand for bonds b is falling since
risk is going up. Aggregate demand is falling due to this intensive margin but rising due to the
R
extensive margin: the probability of matching is increasing. So aggregate demand, I αb (θb )bdF ,
necessarily falls by less in percentage terms (and could in principle increase) than the fall in b. In
contrast, as the gains from matching goes up in CDS markets, the matching probability increases
both for those selling protection and those buying. So these extensive margin forces partially offset.
Hence, the extensive margin effect means aggregate bond demand can increase faster/fall slower
than aggregate synthetic bond demand in CDS markets. This makes bonds cheap relative to CDS,
21
Figure 5: Investor behavior in the benchmark, varying default rates
Note: plotted for a bond supply of 0.03; “Unmatched b” (“Matched b”) means investors
who did not (did) match with a dealer in the bond market; this graph applies to both
the benchmark and the case with liquid sovereign policies but frictional OTC markets
since the bond issuance is fixed and default risk is on the horizontal axis.
22
resulting in a positive and increasing CDS-bond basis at low default rates.
This extensive margin effect is necessarily limited, however. Once the matching probability
levels off—as strongly influenced by its elasticity—additional risk shows up as a decrease in aggregate demand for bonds. And that is where the short sale constraint on bonds really matters. In
the CDS market, increases in risk are borne by all types of investors, reflected in the individual CDS
protection choices varying. This is how a planner would allocate risk—in fact, for investors who
match in CDS markets, markets are complete and risk is fully shared among them and dealers.17
This efficient risk-sharing means CDS risk premia and running spreads increase at a comparatively
slow rate. In contrast, in the bond market only the less-exposed investors are active because of the
short-sale constraint on bonds. To bear all the risk themselves is inefficient, resulting in a greater
compensation for risk reflected in a Z-spread that rises quickly in default risk (at higher levels of
risk).
But why don’t low-exposed bond investors arbitrage away the positive deviations by buying
cheap bonds and buying protection in CDS markets? The reason is that frictions make this trade
risky and so not pure arbitrage. If they buy bonds, there’s two types of imperfectly insurable risk.
First, they could fail to match in CDS markets, in which case they would be stuck with bonds
and excess risk. They could mitigate this risk by paying larger fees, but that is expensive and
eats into the profits of the “arbitrage” trade. Second, their exposure could change after purchasing
bonds, amplifying the risk associated with not matching in CDS markets. Given these disincentives,
the low-exposed investors are happy to let the CDS-bond basis deviate upwards. The negative
bond basis is much harder to arbitrage for anyone—it requires selling expensive bonds and selling
protection in CDS markets. But because of the short-sale constraint, it is very hard to do that. It also
comes with the aforementioned risks of a suboptimal exposure level arising from not matching.
The documented patterns between bid/ask spreads, volume, dealer CDS positioning, and CDSbond basis breakdowns all show up when considering default events, as is done in Figure 6. As
output declines, default risk and spreads increase. With larger default risk, the consumption gap
between matched and unmatched investors is larger and gains from trade increase, inducing investors to pay higher intermediation fees to achieve their optimal level of exposure to default risk.
Larger intermediation fees are mapped into larger bid-ask spreads in the bond and CDS markets.
The increases in risk and reduction in debt both generate a larger deviation of the CDS-bond basis
(the first we discussed, the latter is evident in Figure 8). The greater desire to trade also results in
more dealers entering the market just before default (as larger θb , θc choices result in more dealers), but this effect is very small. Dealers in the aggregate sell a little more CDS protection, but the
17
As discussed in Section 2.1.2, there are effectively only two states of uncertainty (repayment and default) with
the CDS being an Arrow security paying in the default state. The risk-free bond then spans the other state (in proper
combination with the CDS).
23
absolute value is small relative to debt.
Figure 6: Default events
4
Counterfactuals
In this section we analyze a series of counterfactuals. Firstly, we assess the quantitative importance
of trading frictions on bond prices and the response of the government by comparing our baseline
model to an alternative version of the model in which bonds and CDS are liquid and can be traded in
competitive markets—that is, dealers face zero entry cost (Proposition 8 in Appendix A). Secondly,
we study the equilibrium responses to policy changes that modify the constraints of trade in CDS
and bonds. We consider the following policies: allowing bond shorting; eliminating trading in
CDS; and banning naked CDS. Finally we consider the welfare gains or losses associated with
these policies.
Table 4 reports key simulation statistics for the various cases. These simulations combine the
effects on prices and the optimal debt issuance response. We can isolate the effects of the changes
at the same debt issuance by looking at the average price schedules. That is, EY [q(Y, B 0 )] for
differing levels of B 0 . These, plotted as differences from the benchmark average price schedule
and smoothed, are displayed in Figure 7. (Unsmoothed values are given in Appendix D.) Even
these changes have multiple effects, including not only direct effects on investors but feedback
24
Table 4: Model comparison
Statistic
Bond spread mean (%)
Bond spread std. (%)
Debt service to output ratio (%)
Bid-ask spread for bonds mean (%)
Bid-ask spread for bonds std. (%)
Bid-ask spread for CDS mean (%)
Bid-ask spread for CDS std. (%)
CDS-bond basis deviation mean (%)
CDS-bond basis deviation std. (%)
Aggregate dealer
CDS position (%)
dealer net exposure (%)
dealer bond position (%)
dealer-investor volume (%)
dealer buy volume (%)
dealer sell volume (%)
Std. of agg. dealer CDS position (%)
Default probability (%) (full sample)
Correlation of
log IDP and agg. dealer CDS
log IDP and agg. dealer-investor volume
log IDP and agg. dealer buy volume
log IDP and agg. dealer sell volume
YTM spreads and bond bid-ask spreads
YTM spreads and CDS bid-ask spreads
Welfare gain
Sovereign (CEV ×104 )
Investor (agg., money metric ×104 )
(1)
Benchmark
(2)
No Naked CDS
(3)
No CDS
(4)
Short Bonds
(5)
Liquid
(6)
Liq. pol., OTC
7.44
3.83
5.55
0.06
0.04
0.23
0.10
1.30
1.61
7.45
3.83
5.55
0.06
0.04
0.98
0.48
2.98
2.90
7.45
3.83
5.55
0.06
0.04
-
7.36
3.64
5.09
0.02
0.02
0.20
0.13
-0.63
0.38
7.65
4.18
5.70
-
8.72
4.29
5.70
0.08
0.05
0.26
0.10
1.20
1.63
0.00
0.02
0.02
0.20
0.10
0.10
0.03
1.08
-0.02
0.02
0.00
0.02
0.00
0.02
0.00
1.08
0.01
0.01
1.08
0.05
0.02
0.07
0.15
0.10
0.05
0.04
0.89
1.17
0.00
0.02
0.02
0.21
0.11
0.10
0.03
1.17
-0.61
0.96
0.78
0.97
0.55
0.78
-0.93
0.83
0.39
0.83
0.54
0.97
0.55
-
-0.25
0.88
0.38
0.86
-0.11
-0.09
-
-0.56
0.96
0.75
0.94
0.71
0.68
-
0.31
-0.24
0.22
-0.25
-41.69
24.27
-4.91
11.03
-
Note: all CDS position and volume measures are deflated by mean GDP; welfare measures are relative to the benchmark and compare
average utilities along the simulated path; CEV stands for consumption equivalent variation; the “Liq. pol., OTC” case uses the policy
functions from the liquid version of the model with benchmark OTC frictions; welfare for the “Liq. pol., OTC” case is not reported because
prices and policies are inconsistent.
25
from default decisions into prices, and we will decompose these forces.
Figure 7: Price schedule differences under alternative policies
Note: bond schedules are averaged across GDP using the invariant distribution and
then smoothed; the appendix reports the non-smoothed values.
4.1
Comparing the liquid and frictional markets
We assess the quantitative importance of frictional markets by taking the limit as entry costs γb , γc
go to zero, recovering a version of the Arellano (2008) model. The results for a few key variables
are displayed in Table 4 in columns (1) and (5). OTC frictions have two types of effects on bond
prices. There is a direct effect, which changes bond prices as investor and dealer demand changes,
and an indirect effect, which is how q changes as the sovereign reoptimizes default and debt issuance. To delineate these, we consider an intermediate case that uses the sovereign policies from
the liquid model with pricing from the benchmark model. The results for this experiment are displayed in the column (6). The direct effect of moving from a liquid to frictional environment is
captured in moving from (5) to (6), while the indirect effect is found in moving from (6) to the
benchmark (1).
The direct effect of frictional trading is an adverse impact on prices, reflected in the 1pp increase in average bond spreads from 7.65% to 8.72% (Table 4). Looking beyond averages and to
specific debt levels in Figure 7, the indirect effect (which is the negative of the “Liq. pol., OTC”
curve) improves prices at debt levels below 0.05 and worsens them at higher debt levels. Essentially, whenever the direct effect is negative, the sovereign’s value of repayment decreases (which
26
follows from a simple budget constraint argument).18 This drives up default rates and creates a
negative indirect effect. So the direct and indirect effect covary, with the indirect effect amplifying
the direct effect. Since the direct effect of moving from the liquid case to frictional OTC markets
is positive when debt levels are low, the indirect effect is also positive when debt levels are low.
Generally one can expect the total effect to give the sign of both the direct and indirect effect
conditional on a debt level.
Investors with positive (negative) exposure have less (more) demand for bonds than a riskneutral agent. So the distribution of exogenous exposure ω is key for determining whether the
frictional model implies better or worse prices for the sovereign than the risk-neutral case. Based
on Figure 5 where the less (in fact, negatively) exposed agents were the only bond purchasers, one
would expect better pricing than the risk-neutral case. But that figure was constructed with a low
amount of debt (0.03). Figure 8 is analogous to Figure 5 but varies the bond supply while holding
expected default rates fixed. There, one can see that at higher levels of debt, other investors also
trade the bond. Because the marginal investor shifts from less-exposed to more-exposed agents as
debt increases, one should expect that at low (high) levels of debt OTC frictions improve (worsen)
the bond price. And this is exactly what happens, as can be deduced from Figure 7. The “Liq. pol.,
OTC” curve lies above the “Liquid” curve for debt levels below 0.035, implying trading frictions
increase prices when debt is small. The increase in average spreads reflects the average debt level
is closer to 0.056 (since debt service to GDP is targeted to be 5.6% and GDP is roughly 1), which
is in the higher-debt region where OTC frictions lower prices.
The mechanism through which the OTC frictions change prices is key for understanding the
results of the other counterfactuals: Frictions change who the marginal investor is. In the liquid
case, it is as if agents of all types can trade freely, and in the end everyone bears an infinitessimal amount of risk, and all agents price the bond. The OTC frictions in the benchmark have an
asymmetric effect on risk-taking, making it harder to buy protection against default than to sell
protection: Investors with positive exogenous exposure would like to short the bonds, which is not
allowed, or buy CDS, which is more costly in terms of fees (γc γb ) than accessing the bond market. As a result, agents with negative exposure end up as the marginal investors at low debt levels
(as seen in Figures 5 and 8), pushing the bond price up relative to risk-neutral. However, investors
are also risk-averse and their capacity to absorb bonds is limited. As the total amount of debt issued
increases, these negatively-exposed active bond traders are supplemented by more-exposed bond
traders who become active, changing the marginal investor.
We saw already that—conditional on debt—the indirect effect amplifies the direct effect. The
18
Given any feasible debt issuance and default policy, a larger q means that policy is still feasible but results in
higher consumption; additionally, other choices may become feasible as well, resulting in a further boost from option
value. Conversely, a smaller q reduces consumption in every state and can make some consumption plans infeasible.
27
Figure 8: Investor behavior in the benchmark
Note: plotted for a default rate of 8.7%; “Unmatched b” (“Matched b”) means investors
who did not (did) match with a dealer in the bond market; this graph applies to both
the benchmark and the case with liquid sovereign policies but frictional OTC markets
since the bond issuance is fixed and default risk is on the horizontal axis.
28
indirect effect averaged over the simulated path, which can be seen in moving from column (6)
to column (1) in Table 4, has a more subtle interpretation. Faced with higher spreads conditional
on typical debt levels, the sovereign reduces debt issuance (resulting in slightly lower average
debt service costs), which brings down equilibrium spreads to a level that’s even lower than in
the liquid case. This behavior is driven by the sovereign’s Euler equation and impatience. Loosely
speaking, the sovereign will always borrow up to the point that the effective bond price equals its
discount factor.19 When prices improve, all else equal, this incentivizes borrowing that drives the
price back down. When prices worsen, the reverse happens. In this respect, whatever a policy does,
the sovereign will tend to undo in the simulation, which is the main reason why in columns (1) to
(5) of Table 4, the spreads are remarkably similar but the debt service can vary substantially.20
4.2
The role of bond shorting
We turn our attention now to examine the effects of allowing investors and dealers to take negative
bond positions (i.e., to short sovereign bonds), which Figure 7 shows has a large adverse impact
on bond prices.
Viewed from a classic Walrasian perspective, allowing bond-shorting should always hurt bond
prices, as it does in this case. The reason is that bond demand must equal bond supply. By suddenly
allowing agents to take short positions, this necessarily reduces bond demand (weakly) at any given
price. We label this canonical Walrasian force as the demand effect.
While the demand effect on prices plays an important role in our model, there are three other
forces at work: an entry effect given by the dealers who entered or exit the market; an intermediation effect, which arises from changes in dealers’ trading profits π which are are passed on
to investors in the form of intermediation fees and matching probabilities; and a default risk effect given by the change in default risk due to the sovereign’s response to pricing. The latter is
the indirect effect already discussed, while the demand, entry, and intermediation effects are all
components of the direct effect.
We quantify these components by allowing only one channel to operate at a time. We first maintain the benchmark policy environment but change EY 0 |Y [δ(Y 0 , B 0 )] to its value in the new policy
environment (bond shorting in this case). The incremental change, from q0 (Y, B 0 ) to q def (Y, B 0 ) is
our default risk effect. We then hold dealer profit π and the dealer measure D fixed but otherwise
solve for a “general equilibrium” (with misspecified π, D) allowing dealer and investor b, c (and
θb , θc for investors) to change. The incremental change q demand (Y, B 0 ) − q def (Y, B 0 ) is our demand
19
This statement ignores differentiability issues and supposes that consumption growth is approximately zero. In
that case, the Euler equation approximately gives d(q(Y, B 0 )B 0 )/dB 0 = β. The effective price is not just q but also
reflects how much prices move against them when issuing more debt.
20
Note column (6) does not have optimal policies, and so the sovereign is off the Euler equation.
29
effect. We then allow the measure of dealers to change and recompute equilibrium (with misspecified π). Our entry effect is the change q entry (Y, B 0 ) − q demand (Y, B 0 ). Last, we allow π to change,
resulting in the new equilibrium prices q1 (Y, B 0 ). The difference q1 (Y, B 0 ) − q entry (Y, B 0 ) is our
intermediation effect. By construction, the individual effects sum to the total effect, q1 (Y, B 0 ) −
q0 (Y, B 0 ).
Figure 9: Bond-shorting decomposition
Note: bond schedules are averaged across GDP using the 40th-60th percentile conditional distribution and then smoothed; the
appendix reports the non-smoothed values.
The decomposition in Figure 9 reveals a large negative demand effect, consistent with the
Walrasian predictions. The magnitude of the response, however, lies in the large disparity between
γb and γc . To rationalize the data, γb is an order of magnitude smaller than γc . Although trading
bonds is significantly less costly in the benchmark calibration, for many investors bonds are not a
great substitute for CDS. Not only can they not be shorted, but there’s also the risk that exogenous
exposure ω e changes. Once bond-shorting is allowed, investors who wanted to short—but not at the
“exorbitant” fees implied by γc —now find it attractive to short by paying the smaller γb -implied
fees.
The large demand effect is amplified by a large default risk effect / indirect effect: Since the
combined direct effects push prices lower, that drives up default rates which further depresses
prices. In contrast, both the entry and intermediation effect are essentially zero. The intermediation
effect must in fact be zero in this experiment. The reason is that dealers can attain any consumption
allocation they want using a risk-free asset and CDS irrespective of whether they have access to
bond-shorting. Consequently, π and the net entry fees are not directly affected in this counterfactual.
That the entry effect is close to zero is a numerical result. Bond shorting does change trading behavior, submarket tightnesses, and equilibrium fees, which implies the measure of dealers
30
changes. However, the measure of dealers is small relative to the measure of investors (contrast
Tables 2 and 5); and when dealers enter they do not bring any exogenous exposure ω with them
nor do they generally have much demand for endogenous exposure, as dictated by the calibration.
Consequently, the entry effect is small, which will also be true in the other counterfactuals.
4.3
Closing the CDS market
CDS are a relatively modern financial innovation that are widely used by dealers and investors. We
now consider the consequences of shutting down the CDS market.
Despite their evident usefulness, the benchmark model predicts almost no spillover from the
CDS market to the bond market in terms of prices (Figure 7). This is not a generic result, and we
will show in Section 5 that some parameterizations of the model predict large spillovers. But to
understand why they are estimated to be small here, we decompose the almost zero total effect into
its four components in Figure 10.
Figure 10: No CDS decomposition
Note: bond schedules are averaged across GDP using the 40th-60th percentile conditional distribution and then smoothed; the
appendix reports the non-smoothed values.
The figure reveals that the positive default risk effect—arising from larger q overall—and a
negative intermediation effect offset one another. The negative intermediation effect arises from
the ban on CDS making some allocations infeasible for dealers, which reduces their profits π and
drives up net entry costs. Higher net entry costs reduce risk-sharing and result in increased risk
premia / lower q. As these two forces mostly offset, the total effect looks almost identical to the
demand effect.
The demand effect is signed as theory would predict: By closing an opportunity to take positions in synthetic bonds, the demand curve for bonds shifts out and prices rise. However, the
magnitude is much smaller than in a qualitatively similar bond-shorting-ban counterfactual (the re31
verse of allowing bond shorting). The demand effect is weaker here partly because the CDS market
is estimated to be much more frictional the bond market. Technically investors can take short positions in synthetic bonds. But, fees are larger (γc γb ) and matching probabilities lower (Figures
5 and 8) than for bonds. So allowing bond-shorting has a much larger demand effect than eliminating CDS as the bond market is estimated to be much less frictional. Additionally, the volume
of CDS traded is fairly small relative to debt service in our calibration. So when that market gets
shut down, the demand boost from CDS-protection-selling investors substituting into bond-buying
is only a small percentage of overall bond demand.
4.4
A ban on naked CDS
Motivated by the regulatory change implemented in the European Union in 2011-2012, we now use
the model to investigate the consequences of a naked CDS ban—defined as a ban on the purchase
of CDS protection in excess of the amount of bond exposure. As with the elimination of CDS, the
benchmark calibration predicts there is almost zero response in terms of prices (Figure 10), but this
is likewise not a generic result. We will show in Section 5 that a naked CDS ban can significantly
improve prices, worsen prices, or leave them unchanged, depending on the calibration.
Figure 11: Naked CDS ban decomposition
Note: bond schedules are averaged across GDP using the 40th-60th percentile conditional distribution and then smoothed; the
appendix reports the non-smoothed values.
To understand why the total effect is small here, we apply our decomposition strategy to create
Figure 11. The demand effect begins positive but transitions to a negative value. From theory, we
expect the demand effect to be positive for the following reason. If we think about a unified asset
that is just endogenous exposure (i.e., b − c but excluding ω), then the total supply of exposure is
the bond while the total demand is bond demand plus net demand for synthetic bonds. By eliminating the ability to have negative endogenous exposure, the demand for exposure increases which
32
should drive up the exposure price, q. The model captures some of that, but there are additional
complicating factors in the model. One pure demand force that cuts in the opposite direction arises
from precautionary behavior. Investors are more willing to buy bonds if, after the realization of ω,
they can undo some of that exposure. Banning naked CDS reduces the ability to reallocate risk in
this way, reducing demand for the bond ex ante.
Like with all the counterfactuals, the entry effect is small and the default risk effect moves in
line with the total effect for the same reasons as before. One should generally expect the intermediation effect to be negative because the portfolio choice of the dealers is being restricted, which
lowers π else equal and raises net entry costs else equal. However, it is positive here for an interesting reason. As we show in Figure 15 in the appendix, the naked CDS ban effectively segments
bond and CDS markets. Only the less exposed investors trade bonds, and only the more exposed
trade CDS (buying protection from dealers). The consequence of this can be seen in the substantial
increase in the CDS-bond basis deviation in Table 4, which indicates bonds are cheaper relative
to synthetic bonds (c < 0) than in the benchmark. Dealers benefit as they can obtain the exposure
they want by selling CDS protection to the very exposed investors, increasing their profits. This
behavior is perhaps most clearly seen in Table 5. The naked CDS ban has dealers drop almost all
bond holdings but increase exposure by selling protection to highly exposed investors.
Table 5: Dealer behavior along the simulated path
Statistic
Measure of dealers
Individual dealer bond
Individual dealer CDS
Individual dealer exposure
Individual dealer profit
Individual dealer buy vol
Individual dealer sell vol
4.5
Benchmark
No Naked CDS
No CDS
Short Bonds
0.0011
0.1452
-0.0048
0.1500
0.0005
0.9004
0.9058
0.0011
0.0187
-0.1572
0.1759
0.0008
0.0000
0.1640
0.0011
0.0831
0.0011
0.0820
0.0002
0.0000
0.0000
0.0014
0.5055
0.3364
0.1691
0.0004
0.6902
0.3539
Welfare
In this section we consider the welfare impact of each of the counterfactual policies.
To measure welfare for the sovereign, we use the standard consumption equivalent variation
(CEV) measure, which scale the consumption policy in the benchmark up or down until the value
associated with that policy equals the value under the new regime. Dealers’ welfare is always zero
by free entry. To measure welfare for investors, we use a money metric. Specifically, given indirect
utility U1 from a new regime and U0 from the benchmark, the welfare measure is simply U1 − U0
times the measure of investors I. Because of quasi-linear utility, this gain is effectively measured
33
in terms of the consumption good; and, since GDP is close to one, the money metric gain can be
thought of as a share of GDP. Similarly, because GDP is close to one and the sovereign’s consumption roughly equals GDP, the CEV measure is also in terms of a share of GDP (approximately).
The measures therefore are roughly comparable.
Figure 12 depicts the changes in welfare relative to the benchmark for different levels of debt
issuance. Consistent with the previous findings that the ban on naked CDS and eliminating the
CDS market had little effect on prices, we find virtually no impact of such policies on welfare.
Since the effects are small, Figure 16 in the appendix plots only those two policies, which reveals
that the sovereign benefits from them while investors end up worse off.
Figure 12: Welfare analysis
Note: the top panels and the bottom right panel are functions of (Y, B) and have been averaged using the invariant
distribution and then smoothed; the panel labeled “Conditional” is a function of B 0 , δ̄ and has had a numerical average
taken across a grid of default rates; all series have been smoothed, the appendix reports the non-smoothed values.
We observe, however, that introducing sovereign bond shorting or perfect liquidity has a significant welfare effect on investors and sovereign governments. With sovereign bond shorting, both
the sovereign government and investors welfare decrease. The sovereign loses about 1-3 bps of
GDP (top left panel), while investors lose around 30 bps of GDP (top right panel). That investors
lose is actually a general equilibrium effect induced by higher default rates (bottom right panel).
In partial equilibrium with the default rate fixed, welfare for investors is higher by 20bps-40bps
34
(bottom left panel). So it is welfare-improving for both the sovereign and investors for investors
to limit their ability to short. This perhaps counterintuitive result requires the combination of the
OTC and sovereign debt model blocks, as our model does.
Eliminating OTC frictions entirely has a significant positive effect on the sovereign government
and investors. The sovereign gains about 1.5-2 bps of GDP while investors can gain close to 100
bps of GDP (top right panel) and even more in partial equilibrium (bottom left panel).
5
Robustness
This section examines the robustness of the quantitative results to alternative parameterizations and
an alternative timing.
5.1
Naked CDS and CDS ban large effects, bond shorting small effects
A ban on naked CDS trading or a total ban on CDS trading has a positive, but very small, impact
on prices and welfare. This is not a generic result coming from the model but rather a result of the
calibration. To illustrate this point, we consider variation in a single parameter, ξb , which changes
the elasticity of bond matching probabilities. Figure 13 shows the bond price effects when we
increase it from the benchmark’s value of 0.09 to 0.7. With a more elastic bond supply, a naked
CDS ban or eliminating CDS has a large negative impact on the bond price schedule. Similarly,
the effect of bond shorting goes from a large negative effect in the benchmark parameterization
to zero. Clearly then the matching elasticity parameters ξb and ξc are key, and we identified these
carefully by using how bid/ask spreads varied in default risk (see Section 3.3).
5.2
Alternative parameterizations
We now consider more general variation of the key matching and risk-sharing parameters identified in Section 3.3 and reassess the counterfactuals’ impacts on bond prices. Table 6 provides a
summary of the decompositions from (an unsmoothed version of) Figure 7 for a number of parameter specifications. The maximum change in q relative to the benchmark appears in the first of the
five numeric columns, while the minimum change in q appears in the last five. A finding that holds
in virtually every parameter specification is that allowing for bond shorting reduces bond prices,
often by a substantial amount but sometimes only weakly. As we showed in the decomposition
exercise of Section 4.2, bond-shorting is special in that every effect works the same way: demand,
intermediation, entry, and default risk are all weakly negative. So it makes sense that this finding
is robust.
The effects of naked CDS bans and eliminating CDS are usually close to zero, but can be large
and negative for some matching elasticities and sometimes large and positive. That these results are
35
Figure 13: Price schedule differences for an elasticity parameter of ξb = 0.7
Note: bond schedules are averaged across GDP using the invariant distribution and then smoothed; the appendix reports
the non-smoothed values.
Table 6: Robustness of counterfactuals to various parameters
Maximum ∆q × 100
Benchmark
ξb = 0.7
ξb = 0.7, ξc = 0.1
ξb = 0.1, ξc = 0.7
ξb = 0.4, ξc = 0.4
γb = 4, γc = 1/4
γb = 1, γc = 1
γb = 1/4, γc = 4
µω = 0.2
µω = 0.0
µω = −0.2
σω = 0.01
σω = 0.03
σω = 0.05
I = 0.1
I=1
I = 10
Liquid
Short No CDS No Naked
4.18
13.98
5.18
4.24
8.65
4.49
4.28
4.40
5.43
0.82
0.00
4.70
1.28
0.00
24.83
8.44
0.46
0.00
0.00
0.07
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.10
0.00
0.00
0.00
0.00
0.27
0.06
0.12
0.14
0.07
0.07
0.02
1.08
2.65
0.00
0.01
0.28
0.09
0.00
0.00
0.00
0.00
0.26
0.04
0.10
0.11
0.77
1.97
0.01
1.15
1.40
0.00
0.01
0.31
Minimum ∆q × 100
Liq. pol., OTC
Liquid
Short
No CDS
No Naked
Liq. pol., OTC
2.31
7.80
2.37
2.32
4.68
2.37
2.32
2.36
2.37
0.00
0.00
2.37
0.00
0.00
12.28
3.48
0.00
-1.00
0.00
0.00
-0.97
0.00
-0.33
-0.95
-0.78
0.00
-3.14
-7.40
0.00
-3.81
-9.39
0.00
-0.01
-2.98
-3.77
-0.00
-0.00
-3.74
-1.18
-3.35
-3.69
-3.55
-3.88
-3.65
-3.48
-2.71
-7.71
-13.87
-0.15
-2.68
-4.85
-0.03
-4.82
-12.03
-0.00
-2.23
-0.34
-0.02
-0.09
-0.03
-0.03
-0.03
-0.07
0.00
0.00
-1.87
-0.52
0.00
-0.03
-4.82
-11.91
-0.00
-2.23
-0.36
-0.03
-0.10
-0.03
-0.03
-0.07
-0.08
0.00
-1.42
-1.87
-0.54
0.00
-0.66
0.00
0.00
-0.62
0.00
-0.12
-0.61
-0.50
0.00
-2.37
-4.05
0.00
-2.37
-4.69
0.00
0.00
-2.37
Note: maximums and minimums are taken after averaging across output but using unsmoothed values.
sensitive reflects the conflicting forces in the decomposition exercise. Eliminating OTC frictions
does not necessarily improve prices, like in the benchmark, but can sometimes reduce them by a
large amount, particularly when there are at least some agents with negative exogenous exposure.
The key there is who the marginal investor is. That is why the average exposure µω and the measure
of investors I play such a crucial role for that counterfactual.
36
5.3
Reverse Bond-CDS timing
In our benchmark model, agents first trade bonds and then, after the exposure shocks εω are realized, they trade in the CDS market. In order to assess the importance of this timing assumption,
in this section we reverse the timing.21 With the reverse timing assumption, a naked CDS ban effectively eliminates CDS: investors in the first period do not have access to bonds yet, and so they
cannot take out positive amounts of CDS. There can still be some CDS trade, but it is exclusively
between dealers buying protection and investors selling protection (dealer sell volume is zero).
Figure 14: Price schedule differences under alternative policies with reverse Bond/CDS timing
Note: bond schedules are averaged across GDP using the invariant distribution and then smoothed; the appendix reports
the non-smoothed values.
Figure 14 depicts the average difference in bond prices across debt issuance for different policies relative to the benchmark under the reverse timing convention. When comparing the impact
of the policies with the reverse timing (Figure 14) and without it (Figure 7), one can see that prices
move more for the no CDS and no-naked CDS cases with the reverse timing. However, the general
message is still the same—such policies have a small impact. This is confirmed by the welfare
analysis in Figure 17 in Appendix D. The analysis of the impact of bond shorting and market liquidity also reveals that the reverse timing does not impact the qualitative outcomes for prices. The
welfare results are also robust, with liquidity tending to improve welfare for both the sovereign and
investors and bond shorting tending to decrease welfare for both. The magnitudes of the effects on
prices and welfare are also similar.
21
We thank our discussants for emphasizing this issue.
37
There are two main reasons for the results to be similar across timing conventions. The first
one is that, in our calibration, the CDS market is very frictional relative to the bond market. In
particular, the probability of matching in bond markets is much higher than in CDS markets (see,
e.g., Figure 8), and increasing that probability through higher fees is expensive. This implies that,
in the benchmark timing, investors do not rely on matching in CDS markets later because it is a
low probability event (and costly to do). So individual investor bond demand is mostly unchanged
when the probability of subsequent access to the CDS market goes from a small positive value (in
the benchmark timing) to zero (in the reverse timing). And, in the reverse timing, the measure of
investors who access the CDS market is small irrespective of investor type. This makes the pool of
potential bond investors in the reverse timing similar to the pool in the benchmark timing.
The second reason for similar reverse timing results is that the information premium associated
with buying bonds after the exogenous exposure shock, as opposed to before, is small. The variance
of the exposure shock σω2 is small relative to the variance of overall exposure σω2 /(1 − ρ2ω ) (recall
ρω = 0.95). This makes ex-post exogenous exposure ω highly predictable using the ex-ante state
ω e . As a result, bond demand before and after the information is revealed are roughly the same,
and so the information premium embedded in bond prices is small.
6
Conclusions
The market for sovereign debt CDS is relatively new and not well understood. At the same time,
policy is being implemented regulating it. To understand the role of CDS, it is essential to incorporate and understand issues of liquidity, risk-sharing, violations of arbitrage, and relationships
between primary and secondary markets. In this paper, we have proposed a model that addresses
many of these issues in an attempt to understand and quantify the market as it is and as it could
be. We show how to identify the model’s trading frictions, and show the model reproduces key
risk-sharing and trading-friction patterns in the data. The model’s counterfactuals indicate that
CDS regulation can have substantial effects on sovereign debt markets. However, our identification strategy forces us to conclude that eliminating CDS or implementing a naked CDS bans has
few spillover effects to sovereign bonds. In contrast, liquidity can help or hurt depending on debt
levels while bond shorting almost always hurts prices. While there is much more to study and understand in these markets, our data-disciplined model provides novel insights into existing policy
proposals, both for those that have been enacted and those that may be in the future.
38
References
Aguiar, M. and G. Gopinath (2007). Emerging market business cycles: The cycle is the trend.
Journal of political Economy 115(1), 69–102.
Arellano, C. (2008). Default risk and income fluctuations in emerging economies. American
economic review 98(3), 690–712.
Chatterjee, S. and B. Eyigungor (2012). Maturity, indebtedness, and default risk. American Economic Review 102(6), 2674–2699.
Chaumont, G. (2022).
manuscript.
Sovereign debt, default risk, and the liquidity of government bonds.
Duffie, D., N. Gârleanu, and L. H. Pedersen (2005). Over-the-counter markets. Econometrica 73(6), 1815–1847.
Eaton, J. and M. Gersovitz (1981). Debt with potential repudiation: Theoretical and empirical
analysis. The Review of Economic Studies 48(2), 289–309.
Gilchrist, S., B. Wei, V. Z. Yue, and E. Zakrajšek (2022). Sovereign risk and financial risk. Journal
of International Economics 136, 103603. NBER International Seminar on Macroeconomics
2021.
Lagos, R. and G. Rocheteau (2009). Liquidity in asset markets with search frictions. Econometrica 77(2), 403–426.
Lester, B., G. Rocheteau, and P.-O. Weill (2015). Competing for order flow in otc markets. Journal
of Money, Credit and Banking 47(S2), 77–126.
Oehmke, M. and A. Zawadowski (2015). Synthetic or real? the equilibrium effects of credit default
swaps on bond markets. The Review of Financial Studies 28(12), 3303–3337.
Passadore, J. and Y. Xu (2022). Illiquidity in sovereign debt markets. Journal of International
Economics 137, 103618.
Salomao, J. (2017). Sovereign debt renegotiation and credit default swaps. Journal of Monetary
Economics 90, 50–63.
Sambalaibat, B. (2018). Naked cds bans and the bond market: Empirical evidence. Available at
SSRN 3438856.
Sambalaibat, B. (2022). A theory of liquidity spillover between bond and cds markets. The Review
of Financial Studies 35(5), 2525–2569.
39
Appendix for
“Sovereign Debt and Credit Default Swaps ”
Gaston Chaumont, Grey Gordon, Bruno Sultanum and Elliot Tobin
A
Theoretical results
It is worth highlighting here that, because of quasi-linearity, having ω paying in no-default states
is equivalent to having payments 0 in no-default states and −ω in default states for all the model’s
predictions except risk-free borrowing, which is not of interest.
Lemma 1. Consider an alternative formulation for the exogenous exposure with payment 0 in nodefault states and −ω in default states. Then the investors choice of θb , b, θc and c are the same as
in the original model.
Proof of lemma 1. To see this note that
V (ω, b, c; s3 ) = max −qf a + βEδ [ui (a + (1 − δ)(b + ω) + δc)]
a
= max −qf a + βEδ [ui (a + ω + (1 − δ)b + δ(c − ω)]
a
= max −qf ã + βEδ [ui (ã + (1 − δ)b + δ(c − ω)] +qf ω
ã
{z
}
|
value function of alternative problem
≡ Ṽ (ω, b, c; s3 ) + qf ω,
where we performed the substitution ã = a + ω, and defined Ṽ (ω, b, c; s3 ) as the value function of
the alternative problem with payments 0 in no-default states and −ω in default states. We can see
that if we replace V (ω, b, c; s3 ) = Ṽ (ω, b, c; s3 ) + qf ω in problem (5), we obtain the same problem
up to a constant qf ω. Therefore, we obtain the same choices of θc and c in the alternative model and
that V (ω, b; s2 ) = Ṽ (ω, b; s2 ) + qf ω. Similarly, if we replace V (ω, b; s2 ) = Ṽ (ω, b; s2 ) + qf ω in
problem (8), we obtain the same problem up to a constant qf E[ω|ω e ]; resulting in the same choice
of θb and b in the alternative model and V (ω e ; s1 ) = Ṽ (ω e ; s1 ) + qf E[ω|ω e ].
Now we turn our attention to the CDS-bond basis, which is an no-arbitrage condition that
implies bonds, CDS, and the risk-free asset must satisfy a certain relationship. Here, we must make
distinction between measurement and theory. In measurement, the CDS-bond basis is measured as
the CDS running spread minus the Z-spread of the bond. Theoretically, that does not have to be
zero, even when no-arbitrage holds. (It must hold approximately though for small default rates).
We say the CDS-bond basis holds if the actual no arbitrage relationship, q + p = qf , holds. With
one caveat, we can show the basis holds for inter-dealer prices:
Proposition 1. In any OTC equilibrium, q + p ≤ qf . If bd > b, then q + p = qf , i.e., CDS-bond
basis holds for inter-dealer prices.
Proof of proposition 1. The dealer can always purchase one unit of bonds and one unit of CDS
at a cost q + p and sell one unit of the risk-free asset at a cost qf . The resulting next period
consumption allocation is unaltered. Consequently, for an optimal portfolio choice to exist, profits
must be bounded requiring q + p − qf ≥ 0.
1
Additionally, if bd > b, then the dealer can implement the reverse trade, implying q+p−qf ≤ 0.
Consequently, q + p = qf .
We can characterize the equilibrium allocations further by focusing on the limiting case in
which there is no uncertainty or heterogeneity over the value of ω, i.e., σω = 0. The remaining
characterizations assume that dealer’s optimal portfolio choice is not constrained and the CDSbond basis holds for inter-dealer prices. We also assume Inada conditions for ud and ui .
The first results shows that, when the CDS-bond basis holds for inter-dealer prices, there is no
point in entering the bond market and then entering the CDS market. The reason is any desired
consumption allocation can be achieved with the risk-free asset and bonds or the risk-free asset
and CDS, so there’s no point in paying fees twice. In other words, the only reason why investors
hold both bonds and CDS is because there is uncertainty about the optimal exposure to default risk,
and, once the uncertainty is resolved, investors trade CDS to adjust their exposure.
Proposition 2. Suppose investors are unconstrained at their optimal choices. When the CDS-bond
basis holds, bi is such that θc (bi ) = ci (bi ) = 0.
Proposition 3. Suppose investors are unconstrained at their optimal choices.
The CDS-bond basis must hold if α(θc (bi )) = 1. In general, investors bound the CDS-bond
basis
α(θc (bi ))(q + p − qf ) = (1 − α(θc (bi )))(β(1 − δ̄)u0 (a(bi , 0) + bi ) − q).
Below, we combine the proof of proposition 2 and 3.
Proof of proposition 2 and proposition 3. Let q ∗ ≡ β(1 − δ̄) and p∗ ≡ β δ̄ be the risk-neutral bond
and CDS prices.
The first order condition (FOC) for risk-free assets solves
qf = q ∗ u0 (a(b, c) + b + w) + p∗ u0 (a(b, c) + c).
For use below, we note that the implicit function theorem gives
−p∗ u00 (a(b, c) + c)
<0
ac (b, c) = ∗ 00
q u (a(b, c) + b + w) + p∗ u00 (a(b, c) + c)
When the naked CDS constraint doesn’t bind, p = Vc = p∗ u0 (a(b, c) + c) and this is
qf = q ∗ u0 (a(b, c) + b + w) + p,
giving
qf − p
= u0 (a(b, c) + b + w),
q∗
which holds when matched with CDS.
When the short-sale constraint doesn’t bind bi > b, the FOC is satisfied
q = Vb (b; s2 )
2
By the envelope theorem, this is
q = α(θc (b))Vb (b, c; s3 ) + (1 − α(θc (b)))Vb (b, 0; s3 )
= α(θc (b))q ∗ u0 (a(b, c) + b + w) + (1 − α(θc (b)))q ∗ u0 (a(b, 0) + b + w)
qf − p
= α(θc (b))q ∗ ∗ + (1 − α(θc (b)))q ∗ u0 (a(b, 0) + b + w)
q
= α(θc (b))(qf − p) + (1 − α(θc (b)))q ∗ u0 (a(b, 0) + b + w)
Then we can write
(1 − α(θc (b)))q + α(θc (b))(q + p − qf ) = (1 − α(θc (b)))q ∗ u0 (a(b, 0) + b + w)
or
α(θc (b))(q + p − qf ) = (1 − α(θc (b)))(q ∗ u0 (a(b, 0) + b + w) − q)
The LHS is the expected gain from entering the CDS market and arbitraging, and the RHS is the
expected cost, which is the failure to meet and hence have suboptimal consumption.
Note the following:
• Whenever α(θc ) = 1, the CDS-bond basis must hold.
Now consider when the CDS-bond basis holds. Then if α(θc ) < 1,
q = q ∗ u0 (a(b, 0) + b + w)
We already established that when matched in CDS markets,
qf − p
= u0 (a(b, c) + b + w),
q∗
When CDS-bond basis holds, this then gives
q
u0 (a(b, 0) + b + w) = ∗ = u0 (a(b, c) + b + w),
q
which implies a(b, 0) = a(b, c). We noted above that ac < 0, so this implies c = 0. At c = 0,
θc = 0 since there is no gain from trade.
Therefore,
• whenever the CDS-bond basis holds and α(θc ) 6= 1, then c = 0 and θc = 0 = α(θc ).
Now we must rule out α(θc ) = 1. Suppose α(θc ) = 1. Then the CDS-bond basis holds. A
choice of bi portfolio with ci (bi ) 6= 0 results in consumption of a(bi , ci (bi )) + bi + w = u0−1 (q/q ∗ )
in repayment and a(bi , ci (bi )) + ci (bi ) = u0−1 (p/p∗ ) in default. Consider now c̃i (b̃i ) = 0 = θc (b̃i )
by choosing b̃i such to satisfy
a(b̃i , c̃i (b̃i )) + b̃i + w = a(b̃i , 0) + b̃i + w = u0−1 (q/q ∗ )
Then because consumption is the same in repayment for (b̃i , c̃i ) and (bi , ci ), the FOC for assets
ensures that consumption is the same in default. Consequently, the two portfolios result in the
same lottery over consumption. Now consider the price of the portfolios as viewed from an investor
already matched in the bond market. The cost of (b̃i , c̃i ) is just q b̃i + qf a(b̃i , 0). The cost of (bi , ci )
3
is qbi + γ̃c θc (bi ) + pci + qf a(bi , ci ). Comparing the cost, we find
qbi + γ̃c θc (bi ) + pci + qf a(bi , ci )
= (q − qf )bi + γ̃c θc (bi ) + pci + qf (a(bi , ci ) + bi )
= (q − qf )bi + γ̃c θc (bi ) + pci + qf (a(b̃i , c̃i ) + b̃i )
= (q − qf )(bi + ai − ai ) + p(ci + ai − ai ) + γ̃c θc (bi ) + qf (a(b̃i , 0) + b̃i )
= (q − qf )(b̃i + ãi − ai ) + p(c̃i + ãi − ai ) + γ̃c θc (bi ) + qf (a(b̃i , 0) + b̃i )
= (q − qf )(b̃i + ãi − ai ) + p(c̃i + ãi − ai ) + γ̃c θc (bi ) + qf (a(b̃i , 0) + b̃i )
= (q − qf )(b̃i + ãi ) + p(c̃i + ãi ) − ai (q + p − qf ) + γ̃c θc (bi ) + qf (a(b̃i , 0) + b̃i )
= (q − qf )(b̃i + ãi ) + p(c̃i + ãi ) + γ̃c θc (bi ) + qf (a(b̃i , 0) + b̃i )
= q(b̃i + ãi ) + p(c̃i + ãi ) + γ̃c θc (bi )
= q(b̃i + ãi ) + p(ãi ) + γ̃c θc (bi )
= q b̃i + qf ãi + γ̃c θc (bi )
So we see the cost is strictly greater with θc > 0, as it must be to have α(θc ) = 1 provided that
γ̃c > 0. Even with γ̃c = 0, it is still weakly better to have θc = 0.
Almost as a corollary of the preceding result, we next show that consumption allocations are
the same conditional on matching in the CDS or bond markets.
Proposition 4. Let investors be unconstrained. When the CDS-bond basis holds, consumption
allocations are the same if matched in the CDS or the bond market. Moreover,
−ci (0) = bi .
Proof of proposition 4. Let gδ denote consumption in state δ.
Consider an investor who matches in the bond market. Since allocations are unconstrained, q
pins down q = q ∗ u0 (g0 ), and qf = q ∗ u0 (g0 ) + p∗ u0 (g1 ) = q + p∗ u0 (g1 ) so p∗ u0 (g1 ) = qf − q, with
q ∗ ≡ β(1 − δ̄) and p∗ ≡ β δ̄. Because of the CDS-bond basis, that’s the same as p∗ u0 (g1 ) = p,
so consumption allocation for g1 is the same as if matching in CDS. The converse, for an investor
who matches in the CDS market, follows an analogous argument and results in the same g0 as if
matched in bonds. So the allocations are the same.
So we have that the allocation g1 is the same if trading in bonds and CDS markets, and the
same happens for g0 . So the allocation for the case with δ = 1 is a(bi , 0) + 0 = a(0, ci (0)) + ci (0))
and for the case with δ = 0 the allocation is a(bi , 0) + bi + w = a(0, ci (0)) + w =⇒ a(bi , 0) + bi =
a(0, ci (0)). Combining these two equations we have that a(0, ci (0))+ci (0))+bi = a(0, ci (0)) =⇒
bi = −ci (0).
The prices of the portfolios must also be the same. So current and next period consumption
allocations are the same.
Since the consumption allocations are the same conditional on matching, the value of matching must be as well. This leads to a tight relationship between tightness in the two markets that
is independent of the gains from trade (except through submarket choices). In particular, when
choosing to match in the bond market, investors weigh the possibility that they fail to gain access
through bonds and then can gain access through CDS. Increasing the fee paid in bonds increases
4
the probability of matching, which reduces the need to match in CDS markets in sub-period two.
How optimal that is depends on the matching probability elasticity.
Proposition 5. Suppose investors are unconstrained at their optimal choices. When the CDS-bond
basis holds and θ choices are interior,
α0 (θb )
γb − π
= 0
(1 + (α,θ − 1)α(θc (0))),
γc − π
α (θc (0))
where α,θ ≤ 1 is the elasticity of α evaluated at θc (0). When γb = γc , θ choices are interior, and
α is strictly concave,
θb < θc (0).
Lemma 2. For a weakly increasing and weakly (strictly) concave function f : R+ → R having
f (0) = 0, the elasticity of f is weakly (strictly) less than one.
Rx
Proof of lemma 2. Consider strict concavity. Since f (0) = 0, f (x) = 0 f 0 (t)dt. By concavity, f 0
Rx
0 (x)
< 1, which says the elasticity of
is decreasing, and so f (x) > 0 f 0 (x)dt = xf 0 (x). Hence, xff (x)
f is strictly less than one. The case of weak concavity obviously follows.
Proof of proposition 5. An investor matching in the bond market gets V (bi ; s2 ) − qbi . And an investor matching in the CDS market, get V (0, ci (0); s3 ) − pci (0). Proposition 4 shows those alternative matches result in the same consumption bundles. Hence, define V ∗ ≡ V (0, ci (0); s3 ) −
pci (0) = V (bi ; s2 ) − qbi . Also, define the utility of being unmatched in either market as V =
V (0, 0; s3 ).
The optimal choice of θc satisfies
γ̃c = α0 (θc )(V ∗ − V ).
The optimal choice of θb satisfies
γ̃b = α0 (θb )(V ∗ − V (0; s2 ))
= α0 (θb )(V ∗ − [α(θc (0))V ∗ + (1 − α(θc (0)))V − γ̃c θc (0)])
= α0 (θb )((1 − α(θc (0)))(V ∗ − V ) + γ̃c θc (0))
Replacing V ∗ − V with γ̃c /α0 (θc (0)), we have
0
γ̃b = α (θb ) (1 − α(θc (0)))
γ̃c
+ γ̃c θc (0))
α0 (θc (0))
γ̃b
α0 (θb )
= 0
{1 − α(θc (0)) + θc (0)α0 (θc (0))}
γ̃c
α (θc (0))
α0 (θb )
θc (0)α0 (θc (0))
= 0
1 − α(θc (0)) 1 −
α (θc (0))
α(θc )
0
α (θb )
= 0
[1 + α(θc (0))(α,θ − 1)] .
α (θc (0))
The observations for γb = γc follow from the LHS being equal to 1, which imposes requirements on α0 (θb )/α0 (θc (0)) being greater, less than, or equal to one depending on α,θ . Then those
α0 (θb )/α0 (θc (0)) requirements are mapped to θb orderings of θc noting that α0−1 is a decreasing
function for α00 ≤ 0.
⇔
5
For the claim that α,θ ≤ 1, note that the second order condition for θc (0) requires that α00 (θc ) ≤
0. Then concavity implies the elasticity is less than one.
If α is strictly concave, then the elasticity is strictly less than one, by Lemma 2.
The result says investors pay lower fees in the bond market and face a lower market tightness
than in the CDS market ceteris paribus. This is because investors can achieve exactly the same
exposure and consumption allocation by trading in the CDS market or the bond market. At the
time they are choosing the submarket to purchase bonds, investors know that if they fail they can
still achieve the desired exposure by trading in the CDS market. However, investors who are trading
at the CDS market are those that failed to trade bonds, and thus they are willing to pay relatively
larger fees to trade because there are no further opportunities to do so after the CDS market closes.
The following proposition shows that the equilibrium q must be less than the risk-free price
when there is no exogenous exposure, µω = 0. The proof works by first establishing that at riskneutral prices, investors and dealers want no exposure—they are risk-neutral at the margin, and if
µω = 0, then risk-neutral prices result in no trade. Without any trade, markets can’t clear because
no one is willing to hold bonds. So, the bond price must fall to induce trade and holding of bonds
in equilibrium.
Proposition 6. In equilibrium with µω = 0, q is strictly less than the risk-neutral price.
Lemma 3. If µω = 0, then when q and p are given by the risk-neutral prices (in partial equilibrium), investors and dealers hold zero bonds and CDS, and market tightnesses are zero.
Proof of lemma 3. By definition, at risk-neutral prices, the CDS bond basis holds, since β(1 − δ̄) +
β δ̄ = β = qf . Let q ∗ denote the risk-neutral price of the bond, which is β(1 − δ̄).
Let us guess the solution to the investor’s problem has a = u0−1 (1) and bi = ci (bi ) = ci (0) =
θb = θc (bi ) = θc (0) = 0 is optimal. To confirm this, note that q = Vb is satisfied:
q = β(1 − δ̄)u0 (a + b) = q ∗ u0 (u0−1 (1)) = q ∗ .
Likewise, p = Vc holds
p = β δ̄u0 (a + c)
= β δ̄u0 (a) + βu0 (a) − βu0 (a)
= −β(1 − δ̄)u0 (a) + βu0 (a)
= −β(1 − δ̄)u0 (u0−1 (1)) + βu0 (u0−1 (1))
= −β(1 − δ̄) + β
= −q ∗ + qf ,
which holds from the risk-neutral CDS-bond basis. Then, with b = c = 0 optimal, there are no
gains from trade. Therefore θb = θc = 0 is optimal.
For the dealer’s problem, it is likewise trivial to show that bd = cd = 0 is optimal.
Proof of proposition 6. Suppose the CDS-bond basis holds. From Lemma 3, at q = q ∗ , investors
and dealers both demand zero exposure. Moreover, an increase in q causes both dealer and investor’s to hold fewer bonds. Therefore, to induce the bonds to be held in equilibrium, one must
have q < q ∗ to incentivize strictly positive bond demand.
6
This reasoning also means that investors and dealers both want to have exposure, which immediately implies investor bond positions are positive and the short-selling constraint is not binding.
It also implies that investors’ CDS positions are negative, implying dealers own CDS protection
from market clearing. But dealers also want exposure, which means they must own bonds too,
implying the short-sale constraint cannot bind for them either.
Proposition 7. In equilibrium with µω = 0, the short-selling constraint on bonds is not binding.
Proof of proposition 7. Note that at q given by the risk-neutral price, Lemma 3 gives optimal bond
demand as zero for dealers and investors. In equilibrium, q must therefore be lower to induce
someone to hold the bonds. Lower q thus increases demand for bonds by both dealers and investors.
Therefore, in equilibrium, both dealers and investors with a match in the bond market demand a
strictly positive amount of bonds and the short-selling constraint does not bind.
The remaining results hinge on dealer entry. However, entry hinges not only on the matching
function and entry costs, but also on gains from trade both from the investor and dealer perspective.
Proposition 8 uses that q and q converge to the risk-neutral prices as entry costs shrink to zero,
meaning the equilibrium price must converge to the risk-neutral one.
Proposition 8. Assume 1 and 2. If the CDS-bond basis holds (such as when b = −∞ or when
µω = 0) and x < 0, the model limit as γb goes to zero for fixed γb /γc has q given by the riskneutral price.
Lemma 4. Assume 1 and 2. If the CDS-bond basis holds and x < 0, then π is strictly quasi-convex,
attains a minimum of zero when q is the risk-neutral price, and goes to ∞ as q ↓ 0 or q ↑ ∞.
Proof of lemma 4. Let the risk-neutral prices for bonds and CDS be denoted q ∗ = β δ̄ and p∗ =
β(1 − δ̄), and note q ∗ + p∗ = qf . Then the dealer profit problem can be written
π = max −qf a − qb − pc + q ∗ u(a + b) + p∗ u(a + c)
a,b,c
b−b≤0
c + x − b ≤ 0.
a+b≥0
a+c≥0
(θb )
(θx )
Because the CDS-bond basis holds, and using the envelope theorem, the total derivative of π with
respect to q is
dπ
∂π ∂p ∂π
=
−
dq
∂q
∂q ∂p
= −b + c.
Totally differentiating again,
d2 π
d(c − b)
=
.
2
dq
dq
The first order conditions (FOCs) of the problem are
−q + q ∗ u0 (a + b) = −θb − θx
−p + p∗ u0 (a + c) = θx
−qf + q ∗ u0 (a + b) + p∗ u0 (a + c) = 0
7
(where we ignore the last two feasibility constraints because of an Inada condition). Summing the
first two of these, and using q + p = qf , one has 0 = −θb . Consequently, the short-constraint on
bonds cannot bind for dealers when the CDS-bond basis holds.
Solving for a + c and a + b from the second and first FOC, and then differencing, gives
q − θx
p + θx
0−1
0−1
−u
.
(14)
c−b=u
p∗
q∗
Now consider if θx > 0. In that case, c − b = −x and so d(c − b)/dq = 0 in that case. If θx = 0,
then totally differentiating gives
1
dp/p∗
1
dq/q ∗
d(c − b)
=
−
dq
dq
dq
u00 ◦ u−1 p
u00 ◦ u−1 q
p∗
=−
q∗
1
u00 ◦ u−1
p
p∗
1
1
1
−
p∗ u00 ◦ u−1 q q ∗
q∗
>0
So,
d(c − b)
d2 π
=
≥ 0.
2
dq
dq
This proves π is necessarily quasi-convex, but not strictly quasi-convex because there could be
a flat spot (where dπ/dq = 0 for a range of values).
To establish strict quasi-convexity of π, we will show that dπ/dq = 0 at a single point, which
is risk-neutral pricing. This will then give risk-neutral pricing as the minimizer of dealer profits.
= −b + c = −x > 0. So the function is strictly
With x < 0, the case of θx > 0 means dπ
dq
increasing when the constraint is binding, and so not flat. With θx = 0, dπ
= −b + c is only zero if
dq
> 0 for θx = 0, that can occur only at one point, which is where c = b.
−b + c = 0. Since d(c−b)
dq
For θx = 0 and c = b, (14) requires
q
qf − q
q
p
= ∗ ⇔
= ∗ ⇔ q = q∗.
∗
∗
p
q
qf − q
q
That is, only at risk neutral prices does the sovereign demand zero exposure. Since this is where
dπ/dq = 0 and the function is strictly quasi-convex, this point must be the profit minimizer.
For q less than risk-neutral, consider the portfolio a = 1 and c = 0 with b such that qb is a
positive constant. This plan is feasible. And as q goes to 0, b goes infinite, as does u(a + b). So, if
u is unbounded above, π goes infinite.
Similarly, if there is no exposure constraint (x = −∞), then a similar argument with a = 1, b =
0 and pc constant gives π infinite as q goes to qf , implying p goes to 0.
When there is an exposure constraint that binds for q ∈ (q ∗ , qf ), the exposure constraint also
binds at all higher q. And in this region profit increases linearly at a constant rate −x > 0 since we
already established dπ
= −b + c = −x > 0. So as q goes infinite, so does π.
dq
Proof of proposition 8. In any equilibrium, the net entry costs must be non-negative, γ̃b , γ̃c ≥ 0.
As γb and γc go to zero, this requires that π goes to zero.
By lemma 4, profits π are strictly quasi-convex and attain a minimum of zero at the risk-neutral
8
prices. Each level of π = min{γb , γc } is associated with two values of q, say q ≤ q corresponding
to min{γ̃b , γ̃c } = 0. The equilibrium q must be in [q, q]. Both of these bounds move continuously
towards the risk-neutral price.
B
Data Description
We construct a database containing information about bond and CDS prices, bid/ask spreads for
bonds and CDS, CDS and bond holdings of dealers, and standard aggregate macro variables for
Argentina. In this appendix, we describe the data sources and some additional details.
B.1
Data Sources
We obtain a given bank’s CDS position on a sovereign’s CDS via regulatory data from the Depository Trust and Clearing House Corporation (DTCC). The Dodd-Frank Wall Street Reform and
Consumer Protection Act (2010) requires real-time reporting of all swap contracts to a registered
swap data repository (SDR), which the DTCC operates in the CDS market. The Dodd-Frank Act
also requires SDRs to make all reported data available to appropriate prudential regulators.22 As
a prudential regulator, the Federal Reserve has access to the transactions and positions involving
individual parties, counter-parties, or reference entities that are regulated by the Federal Reserve.
The DTCC data contains every US-regulated CDS trade. We drop any trades where the reference entity is not a country’s government. This means that in addition to dropping any CDS trades
where the reference entity is a non-governmental organization, we also remove CDS trades with
city or state level reference entities.
In addition to CDS Positions, we calculate the quarter-end net sovereign bond exposure (CDS
and bond holdings) of U.S.-headquartered banks classified as dealers in the DTCC database. We
obtain this information from the banks’ FR Y-14Q regulatory filings as part of the Federal Reserve’s Capital Assessments and Stress Testing information collection.
Our definition of dealers in the data consist of CDS dealers as classified by the DTCC and
banks for whom we have data in Y14Q. Ultimately, this gives us a list of five banks. In our sample,
every dealer actively trades CDS in every quarter.
We obtain bond prices and CDS prices from Bloomberg.
B.2
CDS Positions
We create a CDS position for each bank at each point in time. We observe the initial position for
each bank as of January 1st, 2010. Every time a bank buys (sells) protection in the entity during
the month, its position increases (decreases). We also subtract any expiring contracts from dealer’s
position in that reference entity. Thus, we assume a dealer d has CDS position at time t relative to
the end of the previous period (t − 1) as follows:
P ositiondt = P ositiond(t−1) + CDSBoughtdt − CDSSolddt − ExpiredContractsdt .
After calculating the position for each bank, we aggregate the positions of dealers. The volume
measures likewise reflect aggregate volumes.
22
See Sections 727 and 728 of The Dodd-Frank Wall Street Reform and Consumer Protection Act.
9
B.3
Bond and exposure position of dealers
In Y14Q, we have the notional quarter-end Sovereign Debt Securities and CDS net exposure to
Argentinean sovereign debt as reported on the Securities Main and Hedging schedule and Trading
Sovereign schedule. Combining this bonds-less-net-CDS protection information with the quarterend net position on CDS we get from DTCC allows us to infer the quarter-end bond position of
dealers.
B.4
CDS and Bond Prices
CDS and bond price data was collected from Bloomberg by downloading data for generic 5-year
CDS and bond. The 5 year CDS was chosen because it is the most commonly traded CDS contract.
We also collected generic 5-year bond yield data from Bloomberg.
C
CDS-bond basis deviation measurement
To measure the CDS-bond basis deviations, we use the Z-spread approach, consistent with our data
that comes from Gilchrist et al. (2022).23 This compares the CDS running spread with the usual
spread from sovereign debt. We now describe the measurement of these in the model and the data.
The running spread is the endogenous coupon scds amount paid at predetermined intervals such
that—assuming a constant Poisson arrival rate λ for default and some recovery rate in the case of
default—the net present value of the CDS contract is zero. In the model, default intensity λ is given
implicitly by the solution to
λ
=
F
(1 − e−(ρ+λ)/4 )
|{z}
ρ+λ
“Fixed leg” value
{z
}
|
“Floating leg” value
R 1/4
0
e−ρτ λe−λτ dτ
where e−ρ = (1 + r∗ )−4 gives the discount rate ρ and F is the upfront payment per unit of notional.
(The IDP associated with λ is 1 − e−λ .) The running spread is then the s that solves
λ
−(ρ+λ)/4
(1 − e−(ρ+λ)/4 ) = se
| {z }
ρ+λ
Expected coupon value
⇔s=
λ
(e(ρ+λ)/4 − 1)
ρ+λ
The annualized spread is scds = 4s. Small default rates and discount rates imply scds ≈ 1 − (1 −
E[d])4 , i.e., the running spread is approximately the default rate. However, it should be kept in
mind this is an approximation. In particular, as λ goes large, scds can become very large.
The Z-spread is the usual spread in sovereign debt. Specifically, it is the constant Z such that
the net present value of the bond, discounted by 1 + r∗ + Z, is zero. Annualized then, this spread
is
sbond = (1/q)4 − (1 + r∗ )4 .
For small default rates, sbond ≈ (1 + r∗ )4 E[d]4 .
23
We thank the authors for providing us with their CDS-bond basis deviations data.
10
The CDS bond basis deviation is defined as
scds − sbond .
Consequently, for small default rates and small risk-free rates, the deviations should be roughly
4r∗ times the annual default probability, or just a few basis points. In our model, the deviations do
not occur simply because the approximations fail to hold but also because the fees investors pay in
CDS and bond markets break the no-arbitrage relationship.
D
Additional quantitative results
This section provides additional quantitative results.
D.1
Naked CDS ban OTC equilibrium behavior
Figure 15 is identical to Figure 8 but with a naked CDS ban.
D.2
More welfare analysis
Figure 16 reports welfare just for the naked CDS and CDS bans. Figure 17 gives the welfare
breakdown when the timing convention is reversed.
D.3
Unsmoothed versions of figures
The unsmoothed versions of Figures 7, 13, and 14 are in Figures 18, 19, and 20, respectively.
Unsmoothed versions of Figures 9, 10, and 11 are in the top, middle, and bottom panel of Figure
21, respectively. The unsmoothed version of Figure 12 appears in Figure 22.
11
Figure 15: Investor behavior with a naked CDS ban
Note: plotted for a default rate of 8.7%; “Unmatched b” (“Matched b”) means investors
who did not (did) match with a dealer in the bond market.
12
Figure 16: Welfare analysis (just naked CDS and no CDS cases)
Note: panels labeled “Realized” have been averaged using the invariant distribution and then smoothed; the panel
labeled “Conditional” has had a numerical average taken across a grid of default rates; all series have been smoothed,
the appendix reports the non-smoothed values.
13
Figure 17: Welfare with reverse Bond/CDS timing
Note: panels labeled “averaged across output” have been averaged using the invariant distribution and then smoothed;
panels labeled “averaged across default rates” have had a numerical average taken across a grid of default rates; the
appendix reports the non-smoothed values.
14
Figure 18: Price schedule differences under alternative policies (unsmoothed)
Note: bond schedules are averaged across GDP using the invariant distribution.
Figure 19: Price schedule differences for an elasticity parameter of ξb = 0.7 (unsmoothed)
Note: bond schedules are averaged across GDP using the invariant distribution.
15
Figure 20: Price schedule differences under reversed timing (unsmoothed)
Note: bond schedules are averaged across GDP using the invariant distribution.
16
Figure 21: Decompositions (unsmoothed)
Note: bond schedules are averaged across GDP using the 40th-60th percentile conditional distribution.
17
Figure 22: Welfare evaluations (unsmoothed)
Note: the top panels and the bottom right panel are functions of (Y, B) and have been averaged using the invariant
distribution and then smoothed; the panel labeled “Conditional” is a function of B 0 , δ̄ and has had a numerical average
taken across a grid of default rates.
18
@FedFRASER