Are CDS Auctions Biased and Inefficient?∗
Songzi Du†
Haoxiang Zhu‡
Simon Fraser University
MIT Sloan and NBER
February 9, 2015
Abstract
We study the design of CDS auctions, which determine the payments by CDS sellers
to CDS buyers following the defaults of bonds. Through a simple model, we find that
the current two-stage design of CDS auctions leads to biased prices. First, dealers may
manipulate the first-stage quotes to profit from their CDS positions. Second, various
restrictions imposed on both stages of CDS auctions prevent certain investors from
participating in the price-discovery process. The resulting allocations of bonds are also
inefficient. As a remedy, we propose a double auction design that delivers more efficient
price discovery and allocations.
Keywords: credit default swaps, credit event auctions, price bias, manipulation, allocative efficiency, double auction
JEL Classifications: G12, G14, D44
∗
First draft: November 2010. An earlier version of this paper was distributed under the title “Are CDS
Auctions Biased?” and is available on our websites. For helpful comments, we are grateful to Jeremy Bulow,
Mikhail Chernov (discussant), Darrell Duffie, Vincent Fardeau (discussant), Jacob Goldfield, Alexander Gorbenko (discussant), Yesol Huh, Jean Helwege, Ron Kaniel, Ilan Kremer, Yair Livne, Igor Makarov, Andrey
Malenko, Lisa Pollack, Michael Ostrovsky, Rajdeep Singh (discussant), Ken Singleton, Rangarajan Sundaram, Andy Skrzypacz, Bob Wilson, Anastasia Zakolyukina, Adam Zawadowski (discussant), and Ruiling
Zeng, as well as seminar participants at Stanford University, the University of South Carolina, the ECBBank of England Workshop on Asset Pricing, Econometric Society summer meeting, Financial Intermediary
Research Society annual meeting, SIAM, NBER Summer Institute Asset Pricing meeting, European Finance
Association annual meeting, Society for the Advancement of Economic Theory meeting, and Western Finance Association annual meeting. We are also grateful to a number of senior executives at ISDA for their
comments and insights. Paper URL: http://ssrn.com/abstract=1804610.
†
Simon Fraser University, Department of Economics, 8888 University Drive, Burnaby, B.C. Canada, V5A
1S6. songzid@sfu.ca.
‡
MIT Sloan School of Management and NBER, 100 Main Street E62-623, Cambridge, MA 02142.
zhuh@mit.edu.
1
Introduction
Credit default swaps (CDS) are default insurance contracts between buyers of protection
(“CDS buyer”) and sellers of protection (“CDS seller”), written against the default of firms
or countries. Since the financial crisis, CDS have been one of the financial innovations that
received the most policy attention and regulatory actions.1 As of June 2014, global CDS
markets have a notional outstanding of $19.5 trillion and a gross market value of $635 billion.2
This paper studies the design of CDS auctions, a unique and unusual mechanism that
determines the post-default recovery value for the purpose of settling CDS. Since the recovery
value is a fundamental parameter for the pricing, trading and clearing of CDS contracts,
achieving a fair, unbiased auction price is crucial for the proper functioning of CDS markets.
Our primary objective is to evaluate, from a theoretical and market-design perspective,
whether the current format of CDS auctions can discover efficient recovery values and achieve
efficient allocations of defaulted bonds. We show that it cannot. As a remedy, we propose a
double auction design that delivers better price discovery and more efficient bond allocations.
The current CDS auction mechanism was initially used in 2005 and hardwired in 2009
as the standard method used for settling CDS contracts after default (ISDA 2009). From
2005 to December 2014, CDS auctions have settled 113 defaults of firms (such as Fannie
Mae, Lehman Brothers, and General Motors) and sovereign countries (such as Greece and
Argentina). For many firms, separate CDS auctions are held for senior and subordinate
debt. As explained in Section 2, the current mechanism consists of two stages. The first
stage of the auction solicits market orders, called “physical settlement requests,” to buy or
sell defaulted bonds. The net market order is called the “open interest.” Simultaneously,
dealers quote prices on the bonds, and a price cap or floor is calculated from these quotes.
The second stage is a standard uniform-price auction, subject to the price cap or floor. If
the final auction price is p∗ per $1 of face value, the default payment by CDS sellers to CDS
buyers is 1 − p∗ .
To the best of our knowledge, Chernov, Gorbenko, and Makarov (2013) (CGM) provide
the only other theoretical model of CDS auctions. Their model is an extension of the
1
For example, the Dodd-Frank Act of United States has mandated and later implemented mandatory
central clearing of standard over-the-counter derivatives, including CDS. In its Financial Markets Infrastructure Regulation, the European Commission states: “Derivatives play an important role in the economy but
are associated with certain risks. The crisis has highlighted that these risks are not sufficiently mitigated
in the over-the-counter (OTC) part of the market, especially as regards credit default swaps (CDS).” See
http://ec.europa.eu/internal market/financial-markets/derivatives/index en.htm. In 2012, European regulators also made a controversial move of banning “naked” sovereign CDS in the EU.
2
See the semiannual OTC derivatives statistics, Bank for International Settlements, December 2014.
1
strategic bidding models of Wilson (1979) and Back and Zender (1993). CGM have an
important insight that participants may manipulate the final price in the second stage of
the auction in order to profit from their CDS positions. Such manipulation is possible and
effective because the second stage of the auction is one-sided, namely only buy orders or
only sell orders are allowed. Moreover, CGM’s model assumes that certain investors are
exogenously constrained such that they cannot buy the defaulted bonds. Those who can
buy the bonds would bid strategically. Combining these features, CGM conclude that the
final auction price can be either higher or lower than the bond fundamental value.
Empirically, CGM find that in 26 CDS auctions on U.S. firms from 2006 to 2011, CDS
auction prices tend to be lower than secondary market bond prices before and after auction
dates. This V-shaped price pattern is confirmed by Gupta and Sundaram (2012) (GS),
who also conduct a structural estimation of CDS auctions. In earlier empirical papers with
smaller samples, Helwege, Maurer, Sarkar, and Wang (2009) find that CDS auction prices
and bond prices are close to each other, and Coudert and Gex (2010) provide a detailed
discussion on the performance of a few large CDS auctions. To the best of our knowledge,
these four studies are the only ones in the small literature on CDS auctions.
While CGM’s model takes a first step in analyzing the mechanism of CDS auctions, it is
limited in several respects. First, in their model the fundamental bond value is commonly
known after default, that is, CDS auctions provide no additional price discovery. This
assumption is hardly realistic, since default is an important information event, and the
primary function of an auction in general is to provide price discovery. Second, CGM do
not model dealers’ incentive to manipulate the price cap or floor in the first stage of the
auction. Because dealers’ quotes determine the permitted range of the final auction price,
understanding dealers’ incentives is an essential step to better understand the CDS auction
design. Lastly, CGM’s model does not address allocative efficiency in CDS auctions. In
reality, efficient allocation of bonds in CDS auctions is valuable for investors, as trading
bonds in CDS auctions incurs zero bid-ask spread, whereas doing so in secondary markets is
much more costly.3
Our analysis provides novel insights on these dimensions not addressed by CGM’s model.
We show that CDS auction prices are biased for two reasons. First, dealers may strategically
manipulate the price cap or floor in order to profit from their existing CDS positions. Second,
various restrictions imposed on the auctions prevent certain investors from participating in
3
For example, Feldhutter, Hotchkiss, and Karakas (2013) find that the round-trip bid-ask spread of
defaulted bonds are about 1.5% of market value near default dates. The Amihud illiquidity measure also
roughly doubles in the week of default, suggesting a higher price impact costs of trading large quantities.
2
the price-discovery process. These two channels are orthogonal to those in CGM’s model
(second-stage manipulation and restrictions on buying bonds). Moreover, we show that
CDS auctions deliver inefficient allocations of defaulted bonds. Finally, we propose a double
auction design that achieves better price discovery and more efficient allocations.
Our analysis is built on a simple model of divisible auctions, which works roughly as
follows. A continuum of traders have high or low values for owning the defaulted bonds.
The proportion of high-value traders is unobservable and serves as a state variable. Traders
also have heterogeneous CDS positions. A trader’s utility is the payout from his CDS
positions plus the profit of buying or selling bonds in the auction, less a quadratic cost of
holding defaulted bonds. Traders select the optimal physical requests in the first stage and
the optimal demand schedules in the second stage. Moreover, a finite number of infinitesimal
dealers submit first-stage quotes, which determine the second-stage price cap or floor. This
infinitesimal-trader model greatly reduces technical complexity in illustrating our insights.4
Optimal strategies in the two stages are solved in a subgame-perfect equilibrium.
The first, manipulation channel for price bias is straightforward. A dealer’s benefit of
manipulating quotes in the first stage is the more favorable payout from CDS positions, and
a major cost is reputation. Without loss of generality let us suppose that dealers are CDS
buyers, who benefit from a low final auction price. If dealers’ CDS positions are sufficiently
large, dealers will quote prices below their best estimate of bond value in the first stage,
leading to downward biased prices. Conversely, if dealers are CDS sellers with sufficiently
large positions, they quote prices above their best estimate of bond value, leading to upward
biased prices. We further show that dealers’ ability to manipulate the quotes is stronger
if they are “in the money” on their CDS positions, namely if they tend to be CDS buyers
(sellers) when the post-auction bond value is low (high).
The manipulation incentive in CDS auctions is similar to the manipulation incentive in
survey-based financial benchmarks, such as the London Interbank Offered Rate (LIBOR).
Since LIBOR manipulation is already an established fact,5 the current CDS auction design
causes the same concern. Although the incentive of manipulation does not necessarily imply
4
Realism aside, we believe the only major missed opportunity of our model is that an infinitesimal trader
cannot unilaterally affect the second-stage price. But as discussed above, this strategic incentive is already
modeled by Chernov, Gorbenko, and Makarov (2013). Moreover, the large-trader model of Chernov, Gorbenko, and Makarov (2013) does not handle price discovery, first-stage manipulation, or allocative efficiency.
Therefore, to make a more interesting contribution to the literature, we opt for a infinitesimal-trader model
rather than a large-trader model.
5
See, for example, Market Participants Group on Reference Rate Reform (2014) and Official Sector
Steering Group (2014) for the institutional details and suggested reforms on reference rates such as LIBOR.
3
its occurrence, proper designs of auction mechanisms should not give dealers disproportionately large discretion in affecting the final price, such as by setting a price cap or floor.
The second, information channel for price bias is slightly more involved. For simplicity,
suppose for now that the price cap or floor is dropped altogether in the second stage, so
price manipulation does not occur. Without loss of generality, consider a CDS buyer who
has a high value for owning the defaulted bonds. (For example, this trader may specialize
in distressed investment.) This high-value trader wishes to buy defaulted bonds, but the
current design of the CDS auction stipulates that only CDS sellers can submit physical
requests to buy (i.e. market orders to buy) in the first stage. Thus, the demands to buy
bonds from high-value CDS buyers are suppressed completely in the first stage of the auction.
If the open interest is to buy, the auction protocol also stipulates that only orders in the
opposite direction, i.e. sell limit orders, are allowed in the second stage. Thus, the demands
of high-value CDS buyers to buy bonds are suppressed in the second stage of the auction as
well. Consequently, by completely missing the information from high-value CDS buyers, the
auction final price is downward biased if the open interest is to buy.6
This argument applies symmetrically if the open interest is to sell, but in this case it is
the low-value CDS sellers who are prevented from participating in either stage of the auction.
Thus, the auction final price is upward biased if the open interest is to sell.
The combination of the manipulation channel and the information channel leads to either
upward or downward biased prices, depending on dealers’ CDS positions. Biased prices also
lead to inefficient allocations of bonds in CDS auctions, even if the price cap or floor is set
“correctly.” While investors can trade the bonds in the secondary markets, doing so incurs
higher transaction costs because CDS auctions allow trading at zero bid-ask spread.
Our model generates a number of novel predictions regarding quoting behavior, price
biases, and post-auction trading activity. For example, in the first stage, dealers who are
CDS buyers quote relatively low prices, whereas dealers who are CDS sellers quote relatively
high prices. As a result, the final auction price is downward biased if dealers are net CDS
buyers, and is upward biased if dealers are net CDS sellers. Moreover, if the open interest
is to buy, low-value CDS traders get too much allocation in the auctions and will sell bonds
after the auctions. If the open interest is to sell, high-value CDS traders get too little
allocation in the auctions and will buy bonds after the auctions. These predictions are new
to the literature.
6
We can show that other types of traders—including high-value CDS sellers, low-value CDS buyers, and
low-value CDS sellers—either fully or partially participate in either stage of the auction.
4
A major challenge in testing these predictions is that it requires data on CDS positions by
identity or transaction data in the secondary bond markets by identity. These types of data
are highly confidential and only available to regulatory agencies. That said, our predictions
of price biases are generally consistent with the existing literature. For instance, CGM and
GS find that if the open interest is to sell, which is the case for the majority of CDS auctions,
the final auction prices are downward biased (relative to secondary market prices days before
and after the auctions). On the other hand, Vause (2011) documents that dealers are net
CDS buyers in 2010 and 2011, with hundreds of billions of dollars in net notional amounts.
The combination of these two patterns is consistent with our prediction that if dealers are
net CDS buyers, then CDS auctions tend to underprice the bonds.
Our analysis suggests that a double auction is a better design to reduce price biases and
allocative inefficiency. A double auction is by no means unusual or exotic; everyday open
auctions and close auctions on stock exchanges are double auctions. Under a double auction
design, limit orders in the second stage can be submitted in both directions, buy and sell,
regardless of the open interest. Two-way orders allow broader investor participation in the
price-discovery process. We also argue that the price caps or floors should be dropped to
mitigate the potential manipulation incentives of dealers. Indeed, we show that in our model
with infinitesimal traders, the double auction delivers both the competitive price and the
efficient allocation. If the number of auction participants is finite, say n, a double auction
still delivers the efficient price, with allocative inefficiency of order O(1/n).
The double auction design is yet another dimension in which we contribute to the extant
literature. For instance, CGM argue that the alternative allocation rule proposed by Kremer
and Nyborg (2004) would mitigate strategic bidding incentives and reduce underpricing in
CDS auctions; they additionally suggest that the price cap should depend on the firststage open interest in a fairly complex fashion. Separately, GS examine Vickery auction
and discriminatory auction as alternative formats in the second stage, holding the firststage strategies fixed. Under CGM’s proposal and GS’s proposal, however, certain investors
are still left out of the price-discovery process, and modification to the second stage alone
does not neutralize dealers’ first-stage manipulation incentives. We thus expect that those
modifications do not correct price biases caused by the manipulation channel and information
channel we have identified.
5
2
The Institutional Arrangements of CDS Auctions
This section provides an overview of CDS auctions. Detailed descriptions of the auction
mechanism are also provided by Creditex and Markit (2010).
A CDS auction consists of two stages. In the first stage, the participating dealers7
submit “physical settlement requests” on behalf of themselves and their clients. These
physical settlement requests indicate if they want to buy or sell the defaulted bonds as well
as the quantities of bonds they want to buy or sell. Importantly, only market participants
with nonzero CDS positions are allowed to submit physical settlement requests, and these
requests must be in the opposite direction of, and not exceeding, their net CDS positions.
For example, suppose that bank A has bought CDS protection on $100 million notional of
General Motors bonds. Because bank A will deliver defaulted bonds in physical settlement,
the bank can only submit a physical sell request with a notional between 0 and $100 million.
Similarly, a fund that has sold CDS on $100 million notional of GM bonds is only allowed
to submit a physical buy request with a notional between 0 and $100 million.8 Participants
who submit physical requests are obliged to transact at the final price, which is determined
in the second stage of the auction and is thus unknown in the first stage. The net of total
buy physical request and total sell physical request is called the “open interest.”
Also, in the first stage, but separately from the physical settlement requests, each dealer
submits a two-way quote, that is, a bid and an offer. The quotation size (say $5 million)
and the maximum spread (say $0.02 per $1 face value of bonds) are predetermined in each
auction. Bids and offers that cross each other are eliminated. The average of the best halves
of remaining bids and offers becomes the “initial market midpoint” (IMM), which serves as
a benchmark for the second stage of the auction. A penalty called the “adjustment amount”
is imposed on dealers whose quotes are off-market (i.e., too far from other dealers’ quotes).
The first stage of the auction concludes with the simultaneous publications of (i) the
initial market midpoint, (ii) the size and direction of the open interest, and (iii) adjustment
amounts, if any.
Figure 1 plots the first-stage quotes (left-hand panel) and physical settlement requests
7
For example, between 2006 and 2010, participating dealers in CDS auctions include ABN Amro, Bank
of America Merrill Lynch, Barclays, Bear Stearns, BNP Paribas, Citigroup, Commerzbank, Credit Suisse,
Deutsche Bank, Dresdner, Goldman Sachs, HSBC, ING Bank, JP Morgan Chase, Lehman Brothers, Merrill
Lynch, Mitsubishi UFJ, Mizuho, Morgan Stanley, Nomura, Royal Bank of Scotland, Société Générale, and
UBS.
8
To the best of our knowledge, there are no formal external verifications that one’s physical settlement
request is consistent with one’s net CDS position.
6
(right-hand panel) of the Lehman Brothers auction in October 2008. The bid-ask spread
quoted by dealers was fixed at 2 per 100 face value, and the initial market midpoint was
9.75. One dealer whose bid and ask were on the same side of the IMM paid an adjustment
amount. Of the 14 participating dealers, 11 submitted physical sell requests and 3 submitted
physical buy requests. The open interest to sell was about $4.92 billion.
Figure 1: Lehman Brothers CDS Auction, First Stage
13
First−Stage Quotes
12
Points Per 100 Notional
Physical Settlement Requests
1.5
Sell
Buy
1
11
Billion USD
0.5
10
9
8
5
Bid
Ask
IMM
−1
−1.5
BNP
BofA
Citi
CS
DB
GS
HSBC
ML
MS
RBS
UBS
Barclays
Dresdner
JPM
6
−0.5
BofA
Barclays
BNP
Citi
CS
DB
Dresdner
GS
HSBC
JPM
ML
MS
RBS
UBS
7
0
In the second stage of the auction, all dealers and market participants—including those
without any CDS position—can submit limit orders to match the open interest. Nondealers
must submit orders through dealers, and there is no restriction regarding the size of limit
orders one can submit. If the first-stage open interest is to sell, then bidders must submit
limit orders to buy. If the open interest is to buy, then bidders must submit limit orders to
sell. Thus, the second stage is a one-sided market. The final price, p∗ , is determined as in
a usual uniform-price auction, subject to a price cap or floor. Specifically, for an open sell
interest, the final price is set at
p∗ = min (M + ∆, pb ) ,
(1)
where M is the initial market midpoint, ∆ is a pre-determined “spread” that is usually $0.01
or $0.02 per $1 face value, and pb is the limit price of the last limit buy order that is matched.
If needed, limit orders with price p∗ are rationed pro-rata. Symmetrically, for an open buy
7
interest, the final price is set at
p∗ = max (M − ∆, ps ) ,
(2)
where ps is the limit price of the last limit sell order that is matched, with pro-rata allocation
at p∗ if needed. If the open interest is zero, then the final price is set at the IMM. The
announcement of the final price p∗ concludes the auction.
After the auction, bond buyers and sellers that are matched in the auction trade the
bonds at the price of p∗ ; this is called “physical settlement.” In addition, CDS sellers pay
CDS buyers 1 − p∗ per unit notional of their CDS contract; this is called “cash settlement.”
Figure 2 plots the aggregate limit order schedule in the second stage of the Lehman
auction. For any given price p, the aggregate limit order at p is the sum of all limit orders
to buy at p or above. The sum of all submitted limit orders was over $130 billion, with limit
prices ranging from 10.75 (the price cap) to 0.125 per 100 face value. The final auction price
was 8.625. CDS sellers thus pay CDS buyers 91.375 per 100 notional of CDS contract.
Figure 2: Lehman Brothers CDS Auction, Second Stage
Price (per 100 Notional)
12
10
8
Aggregate Limit Orders
Open Interest
Final Price
6
4
2
0
3
0
20
40
60
80
Quantity (Billion USD)
100
120
140
A Model of CDS Auctions
In this section we describe the baseline model and solve the optimal strategies in both stages
of the auction. We also derive the associated prices and allocations. To isolate the effects of
8
various restrictions imposed on the auctions, price caps or floors are not considered in this
section. Rather, an extended model with price caps and floors is solved in Section 4.
There is a unit mass of infinitesimal traders on [0, 1]. This infinitesimal-trader setting
greatly reduces technical complexity at little cost of economics. Each trader i ∈ [0, 1] has
a CDS position Qi , where {Qi } are independent and uniformly distributed on [−Q, Q], for
Q > 0. Each trader i also has a private value vi for holding the defaulted bonds. The private
value is a reduced form for heterogeneous information or heterogenous inventory among the
traders.
There are two states for the defaulted bonds: high (H) and low (L), with equal ex ante
probability. In the high state, a fraction m > 1/2 of the traders have value vH , and the rest
have value vL . In the low state, a fraction m of the traders have value vL , and the rest have
value vH . Value vi is independent of CDS position Qi .
Throughout the paper we impose the following parameter restriction:
Assumption 1.
vH − vL ≤ λQ.
(3)
This condition generates an interior solution and simplifies the analysis. The qualitative
results of this paper does not hinge upon this parameter condition.
In this section we study the following two-stage auction game:
1. In stage 1, each trader i ∈ [0, 1] submits a physical settlement request ri that satisfies
ri · Qi ≤ 0 and |ri | ≤ |Qi |.
Let
Z
R≡
ri di
(4)
i
be the open interest in the first stage of the auction.
2. (a) If R < 0, then in stage 2 each trader i ∈ [0, 1] submits a demand schedule xi :
[0, 1] → [0, ∞) to buy bonds.
(b) If R > 0, then in stage 2 each trader i ∈ [0, 1] submits a supply schedule xi :
[0, 1] → (−∞, 0] to sell bonds.
The final auction price p∗ is defined by
Z
ri + xi (p∗ ) di = 0.
i
9
(5)
The utility of trader i is:
λ
Ui = (1 − p∗ )Qi + (vi − p∗ )(ri + xi (p∗ )) − (ri + xi (p∗ ))2 .
2
(6)
The first and second terms of Ui are, respectively, the payout from CDS settlement and the
profits of trading the defaulted bonds. The last term of Ui , with some λ > 0, is interpreted as
the inventory cost of creating the bond position ri + xi in CDS auctions. This inventory cost
implies that traders do not wish to accumulate unlimited positions. In this regard it plays
a similar role as risk aversion, although it is not risk aversion. This kind of linear-quadratic
utility is also used by Vives (2011), Rostek and Weretka (2012), and Du and Zhu (2014),
among others.
Given the first-stage open interest, the second stage strategy is straightforward. Since
traders are infinitesimal, each trader takes the price p∗ in the second stage as given and
∗
.
wants to get as close as possible to his optimal bond allocation, vi −p
λ
Lemma 1. In any equilibrium, each trader i submits in the second stage the demand/supply
schedule:

max −r + vi −p , 0 , if R < 0;
i
λ
(7)
xi (p) =
min −r + vi −p , 0 , if R > 0.
i
λ
3.1
The competitive equilibrium benchmark
As a benchmark, we consider the competitive equilibrium, in which each trader submits an
unconstrained demand schedule, taking the price as given. The first-order condition of Ui
yields the competitive equilibrium strategy:
xci =
vi − p
.
λ
(8)
R
Given the market clearing condition i xci di = 0, the competitive equilibrium prices in the
high state and low state are, respectively,
pcH = mvH + (1 − m)vL ,
(9)
pcL = (1 − m)vH + mvL .
(10)
The allocations in the competitive equilibrium are efficient, so trader i’s efficient allocation
v −pc
v −pc
is i λ H in the high state and i λ L in the low state.
10
3.2
Equilibrium of the two-stage auction
We now characterize the equilibrium in the two-stage auction. Intuition of the equilibrium
are discussed immediately after the results.
Proposition 1. There is an equilibrium with the following strategies and properties:
1. In the first stage, every trader i submits:



min((vH − mp∗B − (1 − m)p∗S )/λ, −Qi ), if vi = vH , Qi < 0


ri (vi , Qi ) = max((vL − (1 − m)p∗B − mp∗S )/λ, −Qi ), if vi = vL , Qi > 0 ,



0,
otherwise
(11)
where
p∗B = xvL + (1 − x)vH ,
(12)
p∗S = xvH + (1 − x)vL ,
(13)
and where x is a function of m and
appendix.
vH −vL
λQ
that is spelled out in Equation (23) in the
2. In the second stage, every trader i submits the demand schedule in Equation (7).
3. In the high state, the open interest is to buy, and the final auction price is p∗B . In the
low state, the open interest is to sell, the final auction price is p∗S .
−1
−1
−vL
−vL
4. If m ≥ 8 12 − vHλQ
, p∗B ≥ p∗S . If m < 8 12 − vHλQ
, p∗B < p∗S .
Corollary 1. As Q → ∞, the final prices in the equilibrium of Proposition 1 converge to
m/2
1−m
vH +
vL ,
1 − m/2
1 − m/2
1−m
m/2
=
vH +
vL .
1 − m/2
1 − m/2
p∗B,∞ =
(14)
p∗S,∞
(15)
if m ≥ 2/3, and to
2m2 − m
2(1 − m)2
v
+
vL ,
H
4m2 − 5m + 2
4m2 − 5m + 2
2(1 − m)2
2m2 − m
=
v
+
vL .
H
4m2 − 5m + 2
4m2 − 5m + 2
p∗B,∞ =
(16)
p∗S,∞
(17)
11
if m < 2/3.
Proposition 2 (Price biases). In Proposition 1, the equilibrium price p∗S in the low state
is higher than the competitive price pcL ; and the equilibrium price p∗B in the high state is
lower than the competitive price pcH . The equilibrium allocation of bonds in Proposition 1 is
inefficient.
3.3
Intuition of the equilibrium
The intuition of the equilibria is best understood by asking: which types of traders are
constrained in the two stages?
Figure 3 plots the first-stage physical requests and second-stage allocations in equilibrium.
−vL −1
) , which corresponds to the three left-hand
We first discuss the case of m ≥ 8(12 − vHλQ
−vL −1
) , which corresponds to the three
subplots, and then turn to the case m < 8(12 − vHλQ
right-hand subplots. The prices biases are illustrated in Figure 4. The kink, where p∗B and
−vL −1
p∗S intercept, corresponds to the threshold of m, 8(12 − vHλQ
) .
−vL −1
) . The top-left subplot shows that high-value CDS buyers
The case of m ≥ 8(12− vHλQ
(vi = vH , Qi > 0, solid red line) and low-value CDS sellers (vi = vL , Qi < 0, dashed blue line)
are completely prevented from participating in the first stage. This is because ri must have
opposite sign from Qi . Moreover, if R > 0, the middle-left subplot shows that all high-value
traders are prevented from participating in the second stage. This is because only sell orders
are allowed for R > 0 but high-value traders only wish to buy. Putting these two stages
together, we see that if R > 0, high-value CDS buyers’ trading interests are suppressed
completely throughout the auction. We also see that other types of traders can at least
partially participate in either stage. Consequently, too few high-value traders participate in
the price-discovery process, leading to a downward biased price.
Similarly, if R < 0, then the top-left and bottom-left subplots reveal that low-value
CDS sellers are completely prevented from participating in both stages. (Other types can
participate in at least one stage.) The resulting final auction price is upward biased.
The price biases are the most transparent in Corollary 1, where we have taken the limit
of Q → ∞. In the limit the distribution of Qi becomes diffuse, and the cutoff point 8(12 −
vH −vL −1
) converges to 2/3. Thus, with probability 1, the max(·) or min(·) operator in (11)
λQ
no longer binds. This implies that high-value CDS sellers and low-value CDS buyers fully
express their trading interest in the first stage. If R > 0 (which happens in the high state),
12
low-value CDS sellers also fully express their trading interest in the second stage (see the
middle-left subplot of Figure 3). The only traders that are constrained in both stages are
the high-value CDS buyers. Therefore, the effective ratio of high-value traders to low-value
m/2
, lower than the unbiased ratio of m/(1 − m). This explains why in the limit,
traders is 1−m
the final price in the high state is
m/2
1−m
vH +
vL ,
1 − m/2
1 − m/2
lower than the competitive price of mvH + (1 − m)vL . The intuition for the upward price
bias if R < 0 is symmetric.
−vL −1
The case of m < 8(12 − vHλQ
) . From the three right-hand subplots of Figure 3, we
see that the first-stage strategies and second-stage allocations are similar to the case of
−vL −1
m ≥ 8(12 − vHλQ
) , with one important difference. Since m is relatively close to 1/2,
the buy-versus-sell trading interest is more balanced. For instance, in the high state, the
open interest is to buy, but it is small in magnitude. Indeed, as m → 1/2, R → 0. Thus,
the effective presence of high-value traders in the second stage through open interest is very
small. But because only low-value CDS sellers participate substantially in the second stage
(as can be seen in the middle-right subplot of Figure 3), they dominate the price-discovery
process. The resulting price is downward biased. Similarly, in the low state, there is a very
small open sell interest, and high-value CDS buyers dominate the price discovery in the
second stage, resulting in an upward biased price.
The price bias in this case is also the most transparent in Corollary 1, where we take the
limit Q → ∞. As m → 1/2, the final auction price converges to vL in the high state, but
converges to vH in the low state. These biases are large and extreme.
Inefficient allocations. It is therefore not a surprise that the equilibrium allocations of
bonds are inefficient. Figure 5 depicts the final allocations. In the high state, allocations
to low-value traders are uniformly too high. High-value traders either buy too much (if Qi
is sufficiently negative) or too little (if Qi is positive or mildly negative). In the low state,
allocations to high-value traders are uniformly too low. Low-value traders either sell too
much (if Qi is sufficiently positive) or too little (if Qi is negative or mildly positive).
13
Figure 3: First-stage physical requests and second-stage allocations. Parameters: vH = 1,
vL = 0, λ = 1, Q = 2.
Physical settlement requests (m=0.6)
ri
Physical settlement requests (m=0.9)
ri
0.6
0.2
0.4
0.1
-2
0.2
1
-1
2
Qi
-2
1
-1
2
Qi
-0.2
-0.1
-0.4
-0.2
-0.6
High-value trader
Low-value trader
High-value trader
Second stage allocations, open interest to buy (m=0.9)
xi*
-2
1
-1
Low-value trader
Second stage allocations, open interest to buy (m=0.6)
xi*
2
Qi
-2
1
-1
2
Qi
-0.05
-0.2
-0.10
-0.4
-0.15
-0.6
-0.20
-0.8
High-value trader
Low-value trader
High-value trader
Second stage allocations, open interest to sell (m=0.9)
xi*
Low-value trader
Second stage allocations, open interest to sell (m=0.6)
xi*
0.8
0.20
0.6
0.15
0.4
0.10
0.2
-2
-1
High-value trader
0.05
1
2
Qi
-2
Low-value trader
-1
High-value trader
14
1
2
Low-value trader
Qi
Figure 4: Final auction prices (parameters are the same as in Figure 3)
Second stage prices
Price
1.0
High state p*B
0.8
High state pcH
0.6
Low state p*S
Low state pcL
0.4
0.2
0.6
0.7
0.8
0.9
1.0
m
Figure 5: Total allocations in two stages (parameters are the same as in Figure 3).
Total allocations, high state (m=0.9)
Total allocations, high state (m=0.6)
0.6
0.2
-2
1
-1
2
0.4
Qi
0.2
-0.2
-2
-0.4
1
-1
2
Qi
-0.2
-0.6
-0.4
-0.8
-0.6
Equilibrium allocation, high-value trader
Equilibrium allocation, low-value trader
Efficient allocation, high-value trader
Efficient allocation, low-value trader
Total allocations, low state (m=0.6)
Total allocations, low state (m=0.9)
0.6
0.8
0.4
0.6
0.2
0.4
-2
0.2
-2
-0.2
1
-1
1
-1
2
Qi
-0.2
-0.4
-0.6
15
2
Qi
4
Imposing a Price Cap or Price Floor
In Section 3 we have shown that the two-stage CDS auctions without price cap or floor lead
to biased prices and inefficient allocations. In this section we consider the effect of adding a
price cap or a price floor on the second-stage orders, taking the cap or floor as given. We
assume that the price cap and floor are known by the traders in the first stage. In Section 5
we study the dealers’ strategic incentives in quoting prices that eventually determine the
price cap or floor.
Denote by p the price cap, which applies if R < 0, and denote by p the price floor, which
applies if R > 0. We impose the following assumption in this section.
Assumption 2. The price cap and floors are symmetric:
p − vL = vH − p.
(18)
They also satisfy
p < p∗S
and
p > p∗B ,
(19)
where p∗S and p∗B are the equilibrium prices from Proposition 1.
Condition (18) keeps the model symmetric and hence tractable. Condition (19) makes
the price cap and floor binding constraints; if the price cap and floor do not satisfy (19), the
equilibria would be unchanged from Proposition 1.
By Proposition 2, Assumption 2 nests an important special case: p = pcL and p = pcH .
That is, the cap and floor could be the competitive prices.
Proposition 3. Under Assumption 2, there exists an equilibrium in which every trader i
submits in the first stage:



min((vH − mp − (1 − m)p)/λ, −Qi )


ri (vi , Qi ) = max((vL − (1 − m)p − mp)/λ, −Qi )



0
vi = vH , Qi < 0
vi = vL , Qi > 0 ,
(20)
otherwise
and submits (7) in the second stage.
The equilibrium price is equal to the price cap p given an open interest to sell and is equal
to the price floor p given an open interest to buy. For either direction of the open interest,
there is strict pro-rata rationing of the second-stage orders at the equilibrium price. The
equilibrium allocation with price cap or floor is inefficient.
16
Figure 6: Total allocations in two stages with price cap or floor
In the high state the price floor is set at pcH , and in the low state the price cap is set at pcL .
Other parameters are the same as in Figure 3.
Total allocations, high state (m=0.9)
Total allocations, high state (m=0.6)
0.4
-2
1
-1
2
Qi
0.2
-0.2
-2
-0.4
1
-1
2
Qi
-0.2
-0.6
-0.4
-0.8
-0.6
Equilibrium allocation, high-value trader
Equilibrium allocation, low-value trader
Efficient allocation, high-value trader
Efficient allocation, low-value trader
Total allocations, low state (m=0.9)
Total allocations, low state (m=0.6)
0.6
0.8
0.4
0.6
0.2
0.4
-2
1
-1
2
Qi
0.2
-0.2
-2
-1
1
2
Qi
-0.4
Figure 6 depicts the final allocations with a price cap or floor for the equilibrium described
in Proposition 3. In the high state the price floor is set at pcH , and in the low state the price
cap is set at pcL . This parameter guarantees that the prices are “correct.” But as shown in
Proposition 3, the final allocations are still inefficient. The direction of allocative inefficiency
is qualitatively similar to that in Proposition 1 and is caused by the pro-rata rationing in
the second stage. In the high state, low-value traders receive too much allocation (they do
not sell enough). In the low state, high-value traders receive too little allocation (they do
not buy enough). This allocative inefficiency generally hurts investors because reallocating
the defaulted bonds in the secondary markets incurs substantial trading costs.
17
5
Manipulation of First-Stage Quotes
The last part of our analysis of the current CDS auction mechanism is to add the first-stage
quotes by dealers. In practice, these quotes determine the price cap or floor studied in the
previous section. The objective of this section is to illustrate that the net CDS positions of
dealers can create strong incentives to distort, or manipulate, the first-stage quotes. For this
purpose, we will make a few simplifying assumptions, elaborated shortly.
We assume that a finite number of infinitesimal traders, denoted j ∈ {1, 2, . . . , n}, are
dealers. (Because dealers have measure zero, it is inconsequential whether they are part of
the unit mass of traders or outside of it.) In the first stage, each dealer j makes a quote
bj ∈ [vL , vH ] on the defaulted bonds. (Restriction to [vL , vH ] is natural but not necessary
P
for our results.) The average quote is denoted b ≡ j bj /n. If R < 0, b is the price cap
in the second stage. If R > 0, b is the price floor in the second stage. (If R = 0, b is the
final auction price, but this does not happen on equilibrium path.) Although the practical
determination of price cap or floor involves adding or deducting a spread from the initial
market midpoint (see Section 2), we omit the spread for simplicity here.
We denote by pc the competitive price, which depends on the state. The utility of each
dealer is given by:

(1 − max(b, p∗ ))Q − c|E [pc ] − b |, if R > 0,
j
j
j
B
πj =
(1 − min(b, p∗ ))Q − c|E [pc ] − b |, if R < 0.
j
j
j
S
(21)
The first term of the utility πj is the payout from dealer j’s CDS position Qj , where the
final price is either unconstrained (p∗B if R > 0 and p∗S if R < 0) or constrained by the price
cap or floor b. The second term −c|Ej [pc ] − bj | is a reduced-form proxy for the reputation
cost of dealer j to quote a price different from his best estimate of the ex post competitive
price, where c > 0 is a given constant. Although a dealer’s best estimate of the competitive
price is typically a dealer’s private knowledge, this information may nonetheless be partially
observed by investors through informal communication.
In the utility function πj we have effectively assumed that dealers do not submit physical
settlement request rj and they do not buy or sell bonds in the second stage. This seemingly
strong assumption is actually reasonable because dealers often receive the bid-ask spread in
the secondary markets, but in the auction they receive zero spread. (By contrast, nondealers
would prefer trading in the auction at zero spread to paying bid-ask spreads in the secondary
markets.) That is, in our simplified model, the dealers’ only role is to determine the price
18
cap or floor.
From (21) it is clear that as long as Qj 6= 0, dealer j may have an incentive to report
bj that is different from Ej [pc ] in order to profit from his CDS position. Although a full
characterization of the equilibrium is complicated, in some special cases the equilibrium can
be solved explicitly. In particular, we will focus on cases in which dealers are homogeneous
in their CDS positions and the CDS positions are perfectly correlated with the underlying
state. These special cases look strong and stark, but they generate very simple predictions
that convey the main intuition.
Proposition 4. Suppose that conditional on the low state, Qj = Q > 0 for every dealer j;
and conditional on the high state, Qj = −Q < 0 for every dealer j.
1. In the low state, in equilibrium every dealer j quotes bj = pcL if Q ≤ nc and quotes
bj = vL if Q > nc. The final auction price is pcL if Q ≤ nc and is vL if Q > nc.
2. In the high state, in equilibrium every dealer j quotes bj = pcH if Q ≤ nc and quotes
bj = vH if Q > nc. The final auction price is pcH if Q ≤ nc and is vH if Q > nc.
By assumption, dealers perfectly infer pc from their CDS position Qj . In a sense, dealers
are “in the money” on their CDS contracts because they are CDS buyers in the low state
and CDS sellers in the high state. If the CDS positions are sufficiently small in magnitude
(Q ≤ nc), dealers “truthfully” quote pcL in the low state and pcH in the high state, leading to
an unbiased average quote and final price. If, however, dealers’ CDS positions are sufficiently
large (Q > nc), they would strategically manipulate the quotes to affect the final price. In
the low state, dealers are CDS buyers and profit from a low final price; hence, they quote
the lowest price possible, vL , which becomes the price cap. In the high state, dealers are
CDS sellers and profit from a high final price; hence, they quote the highest price possible,
vH , which becomes the price floor. The final price is downward biased if the open interest is
to sell (low state), and upward biased if the open interest is to buy (high state).
The next proposition starts from the opposite assumption from Proposition 4, that is,
dealers are “out of the money,” being CDS sellers in the low state and CDS buyers in the
high state.
Proposition 5. Suppose that conditional on the low state, Qj = −Q < 0 for every dealer j;
and conditional on the high state, Qj = Q > 0 for every dealer j.
1. In the low state, in equilibrium every dealer j quotes bj = pcL if Q ≤ nc and quotes
bj = p∗S if Q > nc. The final auction price is pcL if Q ≤ nc and is p∗S if Q > nc.
19
2. In the high state, in equilibrium every dealer j quotes bj = pcH if Q ≤ nc and quotes
bj = p∗B if Q > nc. The final auction price is pcH if Q ≤ nc and is p∗B if Q > nc.
In Proposition 5, if dealers’ CDS positions are small in magnitude (Q ≤ nc), all quotes
are equal to the competitive price, so is the final auction price. If Q > nc, dealers’ ability
to manipulate the final price is much weaker in Proposition 5 than in Proposition 4. In the
low state, dealers are CDS sellers and profit from a high final price, but the negative open
interest implies that b will be a price cap, which limits the dealers’ ability to push the price
up. Thus, the best the dealers can do is to quote {bj } such that the price cap b does not
binds, while minimizing the reputation cost −c|pcL − bj |. Quoting p∗S is optimal, and the final
price is p∗S . The case of high state is symmetric. The overall prediction from Proposition 5
is that if Q > nc, the final price is upward biased if the open interest is to sell (low state),
and downward biased if the open interest is to buy (high state). Because of the price cap
or floor is not binding, the equilibrium outcome of Proposition 5 when Q > nc is essentially
the same as Proposition 1 and Proposition 2.
Proposition 4 and Proposition 5 share one consistent prediction: if dealers are net CDS
buyers, the final price tends to be downward biased; if dealers are net CDS sellers, the
final price tends to be upward biased. This holds regardless of the open interest. But the
magnitude of price biases is larger in Proposition 4, namely if dealers are CDS buyers for
R < 0 and CDS sellers for R > 0.
Chernov, Gorbenko, and Makarov (2013) and Gupta and Sundaram (2012) document that
CDS auction prices under sell open interests tend to be too low relative to bond prices in
secondary markets, i.e., final auction prices are downward biased. Between Proposition 4 and
Proposition 5, Proposition 4 seems more consistent with this empirical pattern. This suggests
that the assumption of Proposition 4, namely dealer are “in the money” on their CDS
positions, is more supported by the data than the opposite assumption used in Proposition 5.
Without CDS position data on dealers, it is difficult to judge the strictness of the condition Q > nc. But the following heuristic might be helpful. In practice, the number of
participating dealers in each auction, n, is about 10. If c is of the same order of magnitude
as the committed quote size, it would be about $5 million. Thus, nc would be in the order
of $50 million. On the other hand, Vause (2011) documents that dealers are net protection
buyers in CDS markets, with a net long position exceeding $400 billion in notational amount
as of June 2011. These numbers suggest that the manipulation incentives of dealers can be
substantial, although it is hardly conclusive without CDS position data of individual dealers.
As a last remark of this section, we observe that the dealers’ (potential) manipulation
20
of the price cap or floor in CDS auctions shares a qualitatively similar mechanism with the
manipulation of the London Interbank Offered Rate (LIBOR), a key benchmark interest
rate. LIBOR is determined by asking a finite set of large banks to submit their costs of
unsecured borrowing and then taking the average after eliminating the highest and lowest few
submissions. Since trillions of dollars of interest rate derivatives are priced with reference to
LIBOR, banks may have incentives to misrepresent their borrowing costs in order to influence
the LIBOR fixing and make profits on their derivative positions. Market Participants Group
on Reference Rate Reform (2014) and Official Sector Steering Group (2014) recommend,
among other measures, that reference rates like LIBOR should be based on actual transaction
data as much as possible. In the concluding section, we propose a double auction mechanism
that mitigates price biases in CDS auctions.
6
Empirical Implications
This section discusses empirical predictions of our model.
Prediction 1.
1. In the first stage, dealers that are net CDS buyers quote lower prices,
and dealers that are net CDS sellers quote higher prices, both relative to the post-auction
bond prices.
2. If dealers are net CDS buyers, the final auction price is downward biased on average. If
dealers are net CDS sellers, the final auction price is upward biased on average. These
price biases are stronger if (i) dealers are CDS buyers under a sell open interest, and
(ii) dealers are CDS sellers under a buy open interest.
Prediction 1 follows from combining Proposition 4 and Proposition 5, as well as the
discussions following them. Although these two propositions are derived under the restrictive
assumption that all dealers have the same CDS position, we have no reason to believe that
the qualitative insight is changed if dealers have heterogeneous CDS positions.
Prediction 2.
1. If the open interest is to buy, low-value CDS traders get too much
allocation in the auctions and will sell bonds after the auctions. This effect is stronger
for CDS sellers.
2. If the open interest is to sell, high-value CDS traders get too little allocation in the
auctions and will buy bonds after the auctions. This effect is stronger for CDS buyers.
21
Prediction 2 follows from Proposition 3 and Figure 6. While CDS positions are verifiable
given the appropriate data, the value for owning a bond need to be approximated empirically.
A potential proxy may be obtained by examining the bids and offers in the second stage of
the auction. High-value traders would be those who offer higher prices, and our theory
predict that those traders wish to buy additional quantities after the auction. Low-value
traders do the opposite.
Testing Prediction 1 and Prediction 2 requires data that exist but are difficult to obtain
outside regulatory agencies. To test Prediction 1, one would need data on dealers’ CDS
positions, which exist in DTCC’s Trade Information Warehouse (at least for relatively recent
CDS auctions). Testing Prediction 2 faces a different challenge that bids and offers in the
second stage are aggregated at the dealer level. Nonetheless, tests of Prediction 2 are still
possible if (i) bids and offers under a dealer’s name in the second stage well represent those of
its customers, (ii) customers who submit bids and offers through a dealer are likely to trade
with the same dealer after the auction, and (iii) each dealer’s trading activity in the secondary
market is observable. Condition (i) and (ii) are reasonable assumptions, and condition (iii)
requires transaction data with counterparty identities that are part of TRACE, managed by
FINRA.
7
Conclusion: A Double Auction Proposal
We have shown in this paper that the current two-stage design of CDS auctions leads to
biased prices and inefficient allocations of defaulted bonds. Dealers may manipulate the
first-stage quotes to profit from their existing CDS positions. Even if manipulation does not
occur, various restrictions imposed on both stages of CDS auctions prevent certain investors
from fully participating in the price-discovery process.
We conclude this paper with a double auction proposal that mitigates these problems.
The double auction has the following features:
• Traders can submit both buy and sell limit orders in the second stage, regardless of the
open interest from the first stage. The final auction price is chosen to equate supply and
demand. Two-way orders allow maximum investor participation in the price-discovery
process.
• There is no price cap or floor. Dealers’ first-stage quotes no longer bind the secondstage prices, although the quotes can still be provided. Dropping the price cap and
floor reduces dealers’ incentives to manipulate the first-stage quotes.
22
• Physical settlement requests may still be submitted, and they are filled at the final
auction price.
This double auction proposal is nothing exotic or unusual. It is the standard and dominant
mechanism used in open auctions and close auctions in stock exchanges around the world,
with minor difference in details. The combination of a two-sided CDS auction and postauction bond trading resembles the combination of an open auction on a stock exchange and
subsequent continuous stock trading. Given the prevalence of the latter, we would reasonably
expect a double auction design of CDS auctions to function well.
Formally, in a second-stage double auction we allow trader i’s demand schedule xi (p) to
take both positive and negative values. Because traders are infinitesimal, given his physical
request ri , trader i’s optimal demand schedule is clearly:
xi (p) = −ri +
vi − p
.
λ
(22)
In the above expression, the optimal physical request ri is indeterminate because the second
stage limit orders xi can always adjust to it. One natural optimal physical request is ri =
−Qi , which fully settles trader i’s CDS position in the first stage. Another natural physical
request is ri = 0: there is no need to submit a physical request (market order) in the first
stage if traders can submit unrestricted limit orders in the second stage.
The following proposition shows that, regardless of the physical request, the final auction
price is p∗ = pcH in the high state and is p∗ = pcL in the low state. Moreover, the final
∗
allocation of trader i is ri + xi (p∗ ) = vi −p
, which is the efficient allocation.
λ
Proposition 6. Under the double auction design, there exists an equilibrium in which the
final price coincides with the competitive price, and the final allocation is efficient.
A seemingly innocuous challenge to a double auction proposal is why we should allow
additional limit sell orders to be submitted in the second stage, if the open interest is already
to sell? The answer, suggested by our analysis, is that if a double auction is implemented,
then investors and dealers would not use market sell orders in the first place, or use fewer of
them. After all, a market sell order is an extreme limit sell order—selling at any price. Since
traders are free to set the limit price to extreme values, we would expect a limit order to
perform no worse, and sometimes better, than a market order does.9 This also suggests that
9
A remotely related but revealing example is the Flash Crash of May 6, 2010. On that day, some market
buy orders were executed at extremely high prices, and some market sell orders were executed at extremely
low prices (e.g., $0.01). Clearly, these orders would not have been executed if they were protected by
reasonable limit prices.
23
the “price pressure” in a double auction, which has price-sensitive limit orders on both sides,
could be less than that in a one-sided auction, which has price-insensitive market orders on
one side.
The price-discovery property of a double auction applies in more general settings. For
example, a double auction also delivers the efficient price if (i) the traders are large and can
have a price impact on the final price, and (ii) they have arbitrary CDS positions (subject to
the constraint that all CDS positions sum to zero). In Appendix B, we characterize a double
auction equilibrium with features (i) and (ii), and this equilibrium achieves the competitive
price.10 In a more general setting Du and Zhu (2014) show that sequential double auctions
achieve the competitive price if new information arrives dynamically over time.
Allocative efficiency is not guaranteed under a static double auction if traders are large
and have price impact, but we show in Appendix B that the allocative inefficiency is of order
O(1/n), where n is the number of auction participants. In dynamic markets, allocative
inefficiency generally converges to zero exponentially over time (see Vayanos (1999) and Du
and Zhu (2014)).
10
If traders are large and can have price impact, CDS buyers or CDS sellers have an incentive to manipulate
the final auction price to increase the payout on their CDS positions. Because the second stage only allows
limit orders in one direction (which is opposite to the open interest), it can be shown that the manipulation
incentives of CDS buyers and CDS sellers do not offset each other. The resulting price is systematically biased.
In our model with infinitesimal traders, second-stage manipulation does not occur. We do not attempt to
bring in a large-trader extension because such model is already analyzed by Chernov, Gorbenko, and Makarov
(2013), and because a large-trader model is much more complicated in handling price discovery. Under a
double auction, however, manipulation incentives offset each other, and the resulting price is unbiased.
24
A
A.1
Proofs
Proof of Proposition 1
Let C ≡
vH −vL
.
λQ
The x in the equilibrium prices of Equations (12) and (13) is given by:

p
2

 1 − m/2 − (1 − m/2) − Cm(1 − m)



Cm/2





if m ≥ 8/(12 − C)







x = 4m2 − 5m + 2 + 4m(1 − m)2 C


q



2


−
(4m2 − 5m + 2 + 4m(1 − m)2 C) − C(8m3 − 12m2 + 5m) (4(1 − m)2 + C(1 − m)m(3 − 2m))





C(8m3 − 12m2 + 5m)





if m < 8/(12 − C)
(23)
A.1.1
Case 1: p∗S ≤ p∗B
We conjecture that vL ≤ p∗S ≤ p∗B ≤ vH and that in the high state R > 0, and in the low state
R < 0. We derive the equilibrium based on this conjecture, and then verify this conjecture
under some assumption about the parameter m.
We first show that the first-stage strategy in (11) is optimal, which has four cases.
1. If the trader has a high value for the bond and is a CDS seller, then he wants to
buy bonds but is constrained to buy at most −Qi unit in the first stage. Given this
v −αp∗ −(1−α)p∗S
constraint, it is optimal for him to submit ri = min( H B λ
, −Qi ) for any
α ∈ [0, 1], and in particular, for α = m. Indeed, if the open interest is to buy, then
v −αp∗ −(1−α)p∗S
v −p∗
he sells back some of his ri , i.e., xi = − min( H B λ
, −Qi ) + H λ B ≤ 0, which
gets him a total allocation of min((vH − p∗B )/λ, −Qi ), which is as close as possible
to his optimal allocation (vH − p∗B )/λ. If the open interest is to sell, then he buys
v −αp∗ −(1−α)p∗S
v −p∗
xi = − min( H B λ
, −Qi ) + H λ B ≥ 0 additional units in the second stage,
exactly achieving his optimal allocation (vH − p∗S )/λ.
2. The case for a low-value CDS buyer is analogous to case 1.
3. If the trader has a high value for the bond and is a CDS buyer, then he wants to buy
bonds but is constrained to sell in the first stage. Then clearly his optimal request is
to sell 0 in the first stage.
25
.
4. The case for a low-value CDS seller is analogous to case 3.
Given the strategies in Equations (7) and (11), R > 0 if the state is high and R < 0
if the state is low. Aggregating the allocations across the two stages we have the following
market-clearing condition given an open interest to buy (i.e., in the high state):
vL − p∗B m
+
(1 − m)
λ
2
vH − p∗B
1−
λQ
vH − p∗B m
+
λ
2
vH −p∗
B
λ
Z
0
Q
dQ = 0.
Q
(24)
In Equation (24), all low-value traders sell in the second stage and achieve their optimal
allocation of (vL − p∗B )/λ (the first term). The high-value CDS buyers do not trade in either
stage: they only want to buy, are constrained to sell in the first stage because of their CDS
positions, and are constrained to sell in the second stage because of the open interest to
buy. All high-value CDS sellers buy in the first stage; they sell in the second stage and
achieve their optimal allocation of (vH − p∗B )/λ if −Qi ≥ (vH − p∗B )/λ (second and third
v −p∗
terms in (24)). The upper limit of integration in the last term of (24) is H λ B , rather than
Q, because of the condition (3).
Define x such that
vH − p∗B
.
(25)
x≡
vH − vL
Equation (24) becomes:
− (1 − m)(1 − x) + m
i.e.,
− mx2
1
vH − vL
−x
2
2λQ
x + mx2
vH − vL
= 0,
4λQ
vH − vL m
+ 1−
x − (1 − m) = 0,
2
4λQ
(26)
(27)
i.e.,
1 − m/2 ±
x=
q
−vL
(1 − m/2)2 − m(1 − m) vHλQ
vH −vL
λQ
· m/2
> 0.
(28)
By the symmetry of our setup, we have vH − p∗B = p∗S − vL , and hence p∗S − vL =
x(vH − vL ).11 Therefore, to satisfy p∗B ≥ p∗S , we must have x ≤ 1/2. The parameter
condition (3) implies that the “+” root does not satisfy x ≤ 1/2. The “−” root in (28)
11
The market-clearing condition following a sell open interest (in the low state) is:
(1 − m)
vH − p∗S
m
+
λ
2
1+
vL − p∗S
λQ
26
vL − p∗S
m
+
λ
2
Z
0
vL −p∗
S
λ
Q
dQ = 0,
Q
(29)
satisfies x ≤ 1/2 if and only if
vH − vL 1 3m
− vL m
x − (1 − m)
= −m
≥ 0,
+ 1−
− +
2
2
4
4λQ
16λQ
x=1/2
2 vH
−mx
(30)
i.e.,
m≥
8
12 −
vH −vL
λQ
.
(31)
This completes the derivation and verification of the equilibrium in which p∗B ≥ p∗S .
A.1.2
Case 2: p∗S > p∗B
We conjecture that vL ≤ p∗B < p∗S ≤ vH and that in the high state R > 0, and in the low state
R < 0. We derive the equilibrium based on this conjecture, and then verify this conjecture
under some assumption about the parameter m.
We first show that the first-stage strategy in (11) is optimal, which has four cases.
1. If trader i has a high value for the bond and is a CDS seller, then he wants to buy
bonds but is constrained to buy at most −Qi unit in the first stage. For any ri ∈
[(vH − p∗S )/λ, (vH − p∗B )/λ], by (7) trader i buys 0 following a sell open interest, because
−ri + (vH − p∗S )/λ ≤ 0, and sells 0 following a buy open interest, because −ri + (vH −
p∗B )/λ ≥ 0, where we have used the conjecture that p∗B < p∗S . That is, trader i does
not trade in the second stage. By Bayes’ rule, trader i puts the probability m on the
high state (R > 0). Hence, his unconstrained optimal ri is (vH − mp∗B − (1 − m)p∗S )/λ,
and his constrained optimal ri is min((vH − mp∗B − (1 − m)p∗S )/λ, −Qi ).
2. The case for a low-value CDS buyer is analogous to case 1.
3. If the trader has a high value for the bond and is a CDS buyer, then he wants to buy
bonds but is constrained to sell in the first stage. Then clearly his optimal request is
to set ri = 0 in the first stage.
4. The case for a low-value CDS seller is analogous to case 3.
The strategies in Equations (7) and (11) imply that R > 0 in the high state, and R < 0
in the low state. Aggregating the allocations across the two stages, we have the following
which is symmetric to Equation (24) given a buy open interest.
27
market-clearing condition given a buy open interest (i.e., in the high state):
Z vL −p∗B
λ
1 − m vL − p∗B 1 − m −vL + p∗B vL − p∗B 1 − m
Q
dQ
(32)
+
+
vL −mp∗ −(1−m)p∗
2
λ
2
λ
2
S
B Q
λQ
λ
vL − mp∗S − (1 − m)p∗B vL − (1 − m)p∗B − mp∗S
1−m
1+
+
2
λ
λQ
Z vH −mp∗B −(1−m)p∗S
λ
m
vH − mp∗B − (1 − m)p∗S vH − mp∗B − (1 − m)p∗S m
Q
+
1−
+
dQ = 0.
2
λ
2 0
λQ
Q
In Equation (32), low-value CDS sellers trade only in the second stage and achieve their
optimal allocation of (vL − p∗B )/λ (the first term). There are three terms for the low-value
CDS buyers: (1) if ri = −Qi ≤ (vL − p∗B )/λ, they are able to sell more in the second stage
and achieve (vL − p∗B )/λ (the second term in (32)); (2) if (vL − p∗B )/λ < ri = −Qi or (3) if
ri = (vL − mp∗S − (1 − m)p∗B )/λ > −Qi , they do not trade in the second stage because they
have sold too much in the first stage (the third and fourth terms in (32)).
The high-value CDS buyers do not trade in either stage: they only want to buy, are
constrained to sell in the first stage because of their CDS positions, and are constrained to
sell in the second stage because of the open interest to buy. The high-value CDS sellers trade
only in the first stage, with a physical request to buy ri = min((vH −mp∗B −(1−m)p∗S )/λ, −Qi )
(the fifth and sixth term in Equation (32)).
Again, the upper limit of the two integrals in (32) is not Q because of condition (3).
Define x such that
vH − p∗B
x≡
.
(33)
vH − vL
Equation (32) then becomes:
vH − vL
(mx + (1 − m)(1 − x))2 − (1 − x)2 vH − vL
− (1 − m)(1 − x) − (1 − m)(1 − x)2
− (1 − m)
λ
λQ
2Q
(mx + (1 − m)(1 − x))(vH − vL )
− (1 − m) 1 −
(mx + (1 − m)(1 − x))
λQ
(mx + (1 − m)(1 − x))(vH − vL )
+m 1−
(mx + (1 − m)(1 − x))
λQ
(mx + (1 − m)(1 − x))2 vH − vL
+m
= 0,
(34)
λ
2Q
28
i.e.,
vH − vL
(mx + (1 − m)(1 − x))2 vH − vL
1−m
(1 − x)2
+ (2m − 1)
− (1 − m)(1 − x) −
2
2
λQ
λQ
vH − vL
+ (2m − 1) 1 − (mx + (1 − m)(1 − x))
(mx + (1 − m)(1 − x)) = 0,
(35)
λQ
i.e.,
− (1 − m)(1 − x) −
vH − vL
1−m
(mx + (1 − m)(1 − x))2 vH − vL
(1 − x)2
− (2m − 1)
2
2
λQ
λQ
+ (2m − 1)(mx + (1 − m)(1 − x)) = 0,
(36)
i.e.,
− vL
3
2
2
2 vH − vL
−x
(8m − 12m + 5m) + x 4m − 5m + 2 + 4m(1 − m)
2λQ
λQ
v
−
v
H
L
− 2(1 − m)2 −
(1 − m)m(3 − 2m) = 0,
2λQ
2 vH
(37)
i.e.,
4m2 − 5m + 2 + 4m(1 − m)2
s
±
x=
vH − vL
λQ
4m2 − 5m + 2 + 4m(1 − m)2
vH − vL
λQ
2
−
vH − vL
vH − vL
(8m3 − 12m2 + 5m) 4(1 − m)2 +
(1 − m)m(3 − 2m)
λQ
λQ
vH − vL
(8m3 − 12m2 + 5m)
λQ
(38)
By the symmetry of our setup, we have vH −p∗B = p∗S −vL , and hence p∗S −vL = x(vH −vL ).
The market-clearing condition in the low state (omitted) would lead to the same solution
for x.
Therefore, to satisfy p∗B < p∗S , we must have 1/2 < x ≤ 1. By condition (3), we can rule
out the “+” root, since (3) implies (40), which implies that the “+” root is larger than 1.
29
.
The “−” root in (38) satisfies 1/2 < x ≤ 1 if and only if
− vL
3
2
2
2 vH − vL
(8m − 12m + 5m) + x 4m − 5m + 2 + 4m(1 − m)
−x
2λQ
λQ
vH − vL
vH − vL
m
2
12 −
−2(1 − m) −
(1 − m)m(3 − 2m)
= −1 +
< 0, (39)
8
2λQ
λQ
x=1/2
2 vH − vL
3
2
2
2 vH − vL
−x
(8m − 12m + 5m) + x 4m − 5m + 2 + 4m(1 − m)
2λQ
λQ
vH − vL
m
vH − vL
2
−2(1 − m) −
(1 − m)m(3 − 2m)
= (2m − 1) 2 − m
> 0,
2
2λQ
λQ
x=1
(40)
2 vH
i.e.,
m<
8
12 −
vH −vL
λQ
,
(41)
since (40) is implied by condition (3).
This completes the derivation and verification of the equilibrium under which p∗B < p∗S .
A.2
Proof of Proposition 2
If m ≥ 8 12 −
vH −vL
λQ
−1
, then
1 − m/2 −
1−
q
−vL
(1 − m/2)2 − m(1 − m) vHλQ
vH −vL
λQ
· m/2
< m,
(42)
which implies that p∗B < pcH and p∗S > pcL .
−1
−vL
If m < 8 12 − vHλQ
, we have p∗B < p∗S , which implies that p∗B < pcH and p∗S > pcL ,
for otherwise we would have pcH ≤ p∗B < p∗S ≤ pcL , i.e., pcH < pcL , a contradiction.
The inefficiency of allocations directly follow from the biases in prices.
A.3
Proof of Proposition 3
We first conjecture that in the second stage there is pro-rata rationing at the price cap p if
open interest is to sell, and pro-rata rationing at the price floor p if open interest is to buy,
and following either direction of open interest, a same fraction ρ ∈ (0, 1) of the second-stage
orders is fulfilled. With probability ρ a trader i is selected to trade in the second stage, so he
30
still submits (7) in the second stage. Each trader optimizes his physical settlement request
ri taking the fraction ρ as given.
A high value trader’s problem is:
2 !
vH − p
vH − p
λ
max m (vH − p) ri + ρ min −ri +
,0
−
ri + ρ min −ri +
,0
ri
λ
2
λ
(43)
2 !
λ
vH − p
vH − p
+ (1 − m) (vH − p) ri + ρ max −ri +
,0
−
ri + ρ max −ri +
,0
,
λ
2
λ
subject to the constraint that 0 ≤ ri ≤ −Qi if Qi ≤ 0 and −Qi ≤ ri ≤ 0 if Qi ≥ 0.
First suppose p > p. This means that the efficient allocation is either (vH − p)/λ or
(vH − p)/λ. Clearly, setting ri < (vH − p)/λ is strictly dominated by setting ri = (vH − p)/λ.
Likewise, setting ri > (vH − p)/λ is strictly dominated by ri = (vH
− p)/λ. If ri∈ [(vH −
v −p
vH −p
vH −p
p)/λ, (vH − p)/λ], then max −ri + λ , 0 = −ri + λ and min −ri + Hλ , 0 = −ri +
vH −p
,
λ
so we can drop the max and min in (43) and take first order condition, which implies
that the unconstraint optimal ri is independent of ρ and is given by:
ri =
vH − mp − (1 − m)p
.
λ
(44)
Now suppose p < p. As before, ri > (vH − p)/λ and ri < (vH − p)/λ are dominated.
If ri ∈ [(vH − p)/λ, (vH − p)/λ], then the trader does not trade in the second stage since
v −p
max −ri + vHλ−p , 0 = 0 and min −ri + Hλ , 0 = 0. This is equivalent to ρ = 0 in the
case of p > p, so (44) is still optimal.
The case of a low-value trader is symmetric. Consequently, the optimal ri given the CDS
constraints are:



min((vH − mp − (1 − m)p)/λ, −Qi ) vi = vH , Qi < 0


ri (vi , Qi ) = max((vL − (1 − m)p − mp)/λ, −Qi ) vi = vL , Qi > 0 .
(45)



0
otherwise
We now verify that given the above physical settlement request and under the assumption
that p > p∗B there is indeed pro-rata rationing at the price p in the second stage in the high
state (the low state is symmetric). Let p∗S and p∗B be the equilibrium prices from Proposition 1
without price cap and floor. We have three cases.
31
−1
−vL
Case 1: m ≥ 8 12 − vHλQ
In this case we then have p > p, since from Proposition 1 we have p∗B ≥ p∗S .
In the high state the open interest with price cap and floor is:
vH −mp−(1−m)p
λ
!
vH − mp − (1 − m)p vH − mp − (1 − m)p
Q
dQ + 1 −
.
λ
Q
λQ
0
(46)
Following a buy open interest, the total order at price p in the second stage is:
2m − 1
R=
2
m
X≡
2
+
+
+
+
Z
vH −mp−(1−m)p
λ
vH − p dQ
−Q +
vH −p
λ
Q
λ
vH − mp − (1 − m)p
vH − mp − (1 − m)p vH − p
m
1−
−
+
2
λ
λ
λQ
Z 0
vL − p dQ
1−m
−Q
+
vL −mp−(1−m)p
2
λ
Q
λ
vL − mp − (1 − m)p
vL − mp − (1 − m)p vL − p
1−m
+
1+
−
2
λ
λ
λQ
v
−
p
1−m L
,
2
λ
Z
(47)
where the first two terms are the limit orders of the high-value CDS sellers who have bought
v −p
“too much” (more than Hλ ) in stage 1, the third and fourth terms are the limit orders of
the low-value CDS buyers, and the last term is the limit orders of the low-value CDS sellers.
It is easy to see that:
vL − p m
+
R + X = (1 − m)
λ
2
1−
vH − p
λQ
vH − p m
+
λ
2
Z
0
vH −p
λ
Q
dQ,
Q
(48)
which is Equation (24) in Case 1 of the proof of Proposition 1, with p replacing p∗B . By the
reasoning following Equation (24), p > p∗B implies that R + X < 0.
−1
−vL
Case 2-a: m < 8 12 − vHλQ
and p > p
With the price cap and floor in the second stage, the open interest R and total order X
at p in the second stage are still given by Equations (46) and (47) in the high state. On the
other hand, in Proposition 1 without price cap and floor, the open interest in the high state
32
is

R0 =
2m − 1 
2
Z
∗
vH −mp∗
B −(1−m)pS
λ
0

∗
∗
∗
∗
Q
vH − mpB − (1 − m)pS vH − mpB − (1 − m)pS 
,
dQ + 1 −
λ
Q
λQ
p∗B
(49)
in the second stage following a buy open interest
and the total amount of orders at price
is
Z
1−m 0
vL − p∗B dQ 1 − m vL − p∗B
0
X =
−Q +
.
+
vL −p∗
2
λ
2
λ
B
Q
λ
v −mp−(1−m)p
(50)
v −mp∗ −(1−m)p∗
B
S
We have R < R0 because p > p∗B and hence H
< H
. We have
λ
λ
X < X 0 by the last three lines in Equation (47). By construction, we have R0 + X 0 = 0.
Thus, we must have R+ X < 0. −1
−vL
Case 2-b: m < 8 12 − vHλQ
and p < p
The open interest with price cap and floor is still given by (46); the total order at price
p in the second stage is:
1−m
X=
2
Z
0
vL −p
λ
vL − p dQ 1 − m vL − p
−Q +
+
,
λ
2
λ
Q
(51)
so we still have X < X 0 (since p > p∗B ), and hence R + X < 0.
Inefficiency of the equilibrium allocation
If p 6= pcH or p 6= pcL , the equilibrium allocation with price cap and floor is inefficient
because the equilibrium prices are not competitive prices. If p = pcH and p = pcL , the
equilibrium allocation with price cap and floor is inefficient because of the pro-rata rationing
in the second stage.
A.4
Proof of Proposition 4
Without loss of generality, let us focus on the low state. The derivative of dealer j’s profit
πj with respect to bj is

P
∗

0
−1
b
+
b
≤
np
0
j
∂πj
S Qj /n + c
j 6=j j
P
=
∗
−1
∂bj
0 + bj ≤ np
b
0
j
S Qj /n − c
j 6=j
Clearly, if Qj > 0, then
∂πj
∂bj
if bj < pcL
if bj > pcL
.
(52)
< 0 for any bj > pcL . Thus in equilibrium we must have bj ≤ pcL .
33
∂π
If Qj = Q ≤ nc, then we have ∂bjj ≥ −Qj /n + c ≥ 0 for any bj < pcL . Hence, the optimal
quote is bj = pcL , for any quotes of the other dealers.
Suppose now that Qj = Q > nc. In equilibrium every dealer j 0 6= j quotes bj 0 ≤ pcL < p∗S .
∂π
Thus we have ∂bjj = −Qj /n + c < 0 for every bj < pcL , which means that the optimal quote
is bj = vL . Thus, the equilibrium is every dealer quoting bj = vL .
A.5
Proof of Proposition 5
Without loss of generality, let us focus on the low state. The case of Qj = −Q ≥ −nc is the
same as that in Proposition 4 and is omitted.
Suppose Qj = −Q < −nc. Fix a symmetric equilibrium in which every dealer quotes
bj = b. As argued for Proposition 4, we must have b ≥ pcL . Moreover, we have b ≤ p∗S ,
for otherwise each dealer would incur the marginal cost of manipulating the quote at b and
would enjoy no marginal benefit at b. For the marginal profit defined in Equation (52),
∂πj
= −Qj /n − c > 0 for any bj ∈ (pcL , p∗S ] if other dealers quote b ≤ p∗S . Thus, the optimal
∂bj
quote is bj = p∗S for all j.
A.6
Proof of Proposition 6
A trader’s total order (market plus limit) is ri + xi =
Z
i
vi −p
.
λ
By the market-clearing condition,
vi − p
di = 0,
λ
we get that p∗ is equal to the competitive price, pcH or pcL , depending on the state. Final
allocations are efficient because the equilibrium is competitive.
B
A More General Double Auction Model
In this appendix we generalize the double auction model in Section 7 by allowing a finite
number n of traders and general distributions of values and CDS positions. We show that
the double auction still has an equilibrium that produces the competitive price, even though
each trader has price impact. Although the allocations in this equilibrium are not fully
efficient, we show that the difference between the equilibrium allocation and the efficient
allocation for each trader is on the order of O(1/n), where n is the total number of traders.
34
Suppose that there are n ≥ 3 traders. As in Section 3, each trader i has a private
valuation vi for owning the defaulted bond, and a CDS position Qi , for 1 ≤ i ≤ n. We allow
for any joint probability distribution of values {vi }1≤i≤n and CDS positions {Qi }1≤i≤n (with
P
the condition that path-by-path, ni=1 Qi = 0). Trader i’s utility is still given by (6). The
double auction rule is described in Section 7. The final price p∗ is determined to clear the
market:
n
n
X
X
∗
xi (p ) +
ri = 0.
(53)
i=1
i=1
Notice that with a finite number of traders, each one of them can affect the final price p∗ by
changing his physical request ri or his demand schedule xi (p).
Proposition 7. In the double auction with n ≥ 3 traders, there exists an equilibrium in
which trader i submits a (arbitrary) physical request ri between 0 and −Qi in the first stage,
and submits in the second stage the demand schedule
xi (p) = −ri +
1
n−2
(vi − p) −
Qi .
λ(n − 1)
n−1
(54)
The equilibrium price is
n
1X
p =
vi .
n i=1
∗
(55)
Proof. Without loss of generality, suppose that ri = 0.
Fix a strategy profile (x1 , x2 , . . . , xn ). Given that all other traders use this strategy profile
and for a fixed profile of values (v1 , v2 , . . . , vn ), the payoff of trader i at the price of p is
!
Πi (p) = (1 − p)Qi + (vi − p) −
X
xj (p)
j6=i
1
− λ −
2
!2
X
xj (p)
.
j6=i
We can see that trader i is effectively selecting an optimal price p. Taking the first-order
condition of Πi (p) at p = p∗ , we have, for all i,
0 = Π0i (p∗ ) = −Qi − xi (p∗ ) + (vi − p∗ − λxi (p∗ )) −
X ∂xj
j6=i
∂p
!
(p∗ ) .
(56)
We conjecture a symmetric linear demand schedule:
xj (p) = avj − bp + cQi ,
35
(57)
where a, b and c are constants. Given this conjecture, trader i’s first order condition (56)
becomes
(n − 1)b(vi − p∗ ) − Qi
,
(58)
xi (p∗ ) =
1 + λ(n − 1)b
i.e., when trader i uses the demand schedule in (57) with
a=b=
1
n−2
, c=−
λ(n − 1)
n−1
(59)
trader i’s first order condition is always satisfied for every realization of (v1 , v2 , . . . , vn ).
It is easy to verify that under this linear strategy, Π00i ( · ) < 0 if n > 2.
A notable feature of the equilibrium in Proposition 7 is that it is independent of (and
hence robust to) assumptions about values and CDS positions.
The equilibrium in Proposition 7 clearly converges to the equilibrium in Proposition 6 as
the number n of traders tends to infinity. The factor n−2
in (54) captures the equilibrium
n−1
amount of “demand reduction” due to the traders’ price impact in the finite market. This
factor is cancelled out in the determination of the equilibrium price in (55), and hence
the equilibrium price is the same as the competitive price. The final allocation from this
equilibrium for trader i is
ri + xi (p∗ ) =
n−2
1
(vi − p∗ ) −
Qi ,
λ(n − 1)
n−1
(60)
and the efficient allocation for trader i is
1
(vi − p∗ ).
λ
The difference between the two allocations is on the order of O(1/n).
36
(61)
References
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