CDS and Bond Liquidity

8/3/2019 CDS and Bond Liquidity

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Credit and Liquidity Risk in Bond and CDS Markets

Abstract

In this paper, we develop a reduced-form credit risk model that incorporates

illiquidity in the bond and the credit default swap (CDS) market. Due to the

different nature of the two markets, we model the liquidity consequences in a

different way. In the bond market, illiquidity results in yield premia. In the

CDS market, the bid-ask spread constitutes a liquidity signal. This approachallows us to explore the liquidity spill-over between the bond and the CDS

market as well as the co-movement of credit and liquidity premia.

Our most important findings are threefold. First, we find that adding a

CDS-specific liquidity component to the model has the important consequence

of consistently positive credit risk and liquidity premia in bond markets. The

size of these premia relative to the yield spread is intuitively plausible. Second,

our analysis of the time series of the liquidity premia shows that the bond

markets liquidity dries up as default risk increases. The CDS market, on the

other hand, becomes more dominated by protection sellers during times of highdefault risk. Third, the liquidity of the bond market has a direct impact on

the liquidity of the CDS market but not vice versa.

Keywords: credit spread, credit default swap, illiquidity, reduced-form

model

JEL classification: G 13, G 14

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1 Introduction

Nowadays, credit derivatives markets provide a standardized alternative to bond markets

in taking on and selling off credit risk exposures. This development offers a new approach

to one of the most widely explored problems in fixed income analysis - the separation of

the corporate bond spread into its credit risk and liquidity component.

The corporate bond spread is usually defined as the difference between the bonds yield

to maturity and a given default-free interest rate such as the swap rate or the yield on

government bonds of the same maturity. Unarguably, credit risk is one of the spreads

most important determinants, but there is clear empirical evidence that liquidity also has

a significant impact. First, financial instruments such as AAA-rated bonds which are

practically default-free often trade at a significant positive spread in excess of the yield

on treasury bonds. This spread is mostly interpreted as a liquidity premium. Second,

government bonds and mortgage bonds of the same issuer with identical default risk butdifferent issuing volumes are traded in some markets, and they usually differ with regard to

their spreads. This spread differential is also attributed to differences in liquidity between

the two instruments.

These observations show that the separation of the total bond spread into its credit risk

and liquidity component constitutes a central question if an issuers credit risk has to be

quantified. When observed corporate bond spreads are directly used to estimate default

probabilities, neglecting systematic liquidity premia will lead to biased estimates of the

true default risk. Therefore, the problem is crucial for the correct pricing of bonds and

credit derivatives as well as for the calculation of the economic capital a bank has to holdbecause of credit risk. However, the identification of the pure credit risk component is

difficult since only the sum of the two risk premia can be observed in the market. In

addition, an interdependence between credit risk and liquidity cannot be ruled out.

The development of credit default swaps (CDS) markets presents a way to overcome

these difficulties. Since the credit risk of an issuer for which both bonds and single-name

CDS are traded is, at least in principle, the same for both instruments, the simultaneous

observation of two prices facilitates the identification of the pure credit risk premium. But

while there is broad agreement that bond spreads contain a positive liquidity premium, it

remains ambiguous whether the CDS premium is a pure measure of credit risk or whether

it also contains systematic liquidity premia. On the one hand, it can be argued that there

should be no liquidity-driven distortions in CDS premia as CDS are derivatives and not

assets. Why should the protection buyer pay a higher premium and the protection seller

earn more for taking on credit risk by liquidity reasons? This symmetry between the two

counterparties with regard to liquidity renders a systematic liquidity premium which is

always borne by one party less plausible. On the other hand, large relative bid-ask spreads

for single-name CDS can be observed which can at least partly be attributed to liquidity

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issues in the CDS market.

An extensive amount of empirical literature is concerned with identifying the components

of the corporate bond spreads which goes back to Fisher (1959). In a recent empirical

application, Elton et al. (2001) find that only approximately 25% of the corporate credit

spread

1

can be explained by default risk or expected default loss. They attribute theresiduals to a tax and a systematic risk premium which is also associated with liquidity.

Collin-Dufresne et al. (2001) analyse the impact of financial variables suggested by

structural-form models on credit spreads. They find that these factors only explain one

quarter of the variation and that a large systematic component seems to affect the spreads

which the authors are unable to determine from a variety of structural-form variables such

as liquidity, firm value and economic state variables. In contrast to this result, Ericsson and

Renault (2006) obtain a significant positive correlation between the default and liquidity

components of bond yield spreads in a cross-sectional regression analysis. This correlation

increases with credit risk. De Jong and Driessen (2005) explore the effect of liquidity riskon the expected excess returns of corporate bonds in a multi-factor model that accounts for

equity market liquidity and treasury bond market liquidity simultaneously. The estimated

liquidity risk premia are around 45 bp for investment-grade bonds and more than double

that size for subinvestment-grade bonds.

Aunon-Nerin et al. (2002) were among the first to combine information from the bond and

the CDS market to analyse the impact of typical factors used in structural-form models on

CDS premia and bond spreads. The authors find that these factors are significant while

liquidity measured by market capitalization does not seem to matter. The explanatory

power of structural-form models for CDS premia and bond credit spreads, however, issmall in their study. Houweling and Vorst (2005) also use information from bond and

CDS markets to implement a reduced-form model of the Duffie-Singleton type. They

estimate the model for CDS premia and use it to determine theoretical bond spreads, and

vice versa. This estimation procedure results in lower errors than assuming that the bond

spread (CDS premium) is exactly equal to the CDS premium (bond spread).

Expected liquidity in the CDS market has been investigated by Tang and Yan (2006) who

perform a panel regression analysis of CDS premia. Liquidity is proxied by the number

of trades and quotes in each month, the order imbalance and the bid-ask spread. The

results imply that CDS liquidity affects CDS premia and that there exists a liquidity

spill-over from the corporate bond, the stock and the equity options market to the CDS

market. Chen et al. (2005) proxy CDS liquidity by the quote updating frequency and

find evidence that expected liquidity and liquidity risk appears to be priced more strongly

for protection buyers. Bongaerts et al. (2007) extend the Acharya and Pedersen (2005)

model to a zero net supply market setup and calibrate it to a sample of CDS quotes and

1From now on, we will follow the standard in the literature and use the terms bond spread and credit

spread interchangeably.

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prices on US-based reference entities. Separating the expected liquidity and the liquidity

risk component, they observe that expected liquidity has a higher impact than liquidity

risk and that the liquidity premium is mainly borne by the protection buyers. Meng and

Ap Gwilym (2006) document that demand-supply pressure, inventory risk and clientele

effects have an impact on the size of the CDS bid-ask spread. None of these studies,

however, determines a time-varying CDS-specific liquidity premium for which the dynamic

relationship between credit risk premia and CDS liquidity premia can be explored.

There exists a vast theoretical literature that explicitly models credit risk for bonds and

credit derivatives. However, to the best of our knowledge, only Longstaff et al. (2005)

present a theoretical model that describes both credit and liquidity premia in bond spreads

and uses information from the CDS and the bond market simultaneously. In the empirical

part of the study, the authors are able to identify significant credit risk and non-default

components in corporate bond spreads. A delicate consequence of their reduced-form

model, however, is that they obtain partly negative liquidity premia in the bond market,i.e. the prices of illiquid bonds partly exceed the prices of liquid ones. We attribute this

puzzling finding to the fact that the CDS market is assumed to be perfectly liquid.

We contribute to the existing literature on the components of bond spreads and CDS

premia by exploring the idea that the bid and ask quotes for CDS premia contain

information on the liquidity of the CDS market. To this purpose, we develop a

reduced-form credit-risk model that incorporates the identical default risk but possibly

different liquidity in the CDS and in the bond market. We assume that illiquidity in the

bond market results in price discounts and yield surcharges. This bond-specific liquidity

has an effect on CDS premia since the potentially illiquid bond will be delivered underthe CDS contract. Therefore, the CDS premium in our model accounts for bond liquidity

as a source of bond price variation. In addition to this straightforward liquidity spill-over,

we include a CDS-specific liquidity which has a more intricate effect. We circumvent the

question of systematic liquidity premia in CDS mid premia by modelling the ask and

bid premia instead. From these, we are able to infer a theoretical pure credit risk CDS

premium which is unaffected by the CDS-specific liquidity. Our measure of liquidity then

naturally arises as the difference between this liquidity-free CDS premium and the mid

premium.

In the empirical part of our analysis, we calibrate the model in order to estimate the credit

and liquidity components of bond spreads and CDS premia. The data we employ are bond

prices and CDS premia for companies with ratings between AAA and CCC from a broad

range of sectors that were observed between June 1st, 2001 and June 30th, 2007. We then

analyse the relation between credit and liquidity premia in both markets as well as the

relation between the liquidity premia of the two markets.

Our most important findings are threefold. First, we find that adding a CDS-specific

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liquidity component to the model has the important consequence of consistently positive

credit risk and liquidity premia in bond markets. In particular, we obtain the result

that neglecting CDS-specific liquidity can result in negative bond liquidity premia. On

average, we attribute 65% of the total bond spread to credit risk and 35% to liquidity

while the credit risk component in the CDS market constitutes on average 93% of the

observed mid premium. In both markets, average liquidity premia increase for reference

entities with higher credit risk. We believe these percentages to be more plausible than

results in the literature where the reverse attribution is reported. Second, our model

sheds light on the strongly discussed relation between liquidity premia in the bond and

the CDS market if credit risk changes. We illustrate our results in a time-series analysis

of the credit and liquidity premia which shows that the bond markets liquidity dries up

as credit risk increases. The CDS market, on the other hand, becomes more dominated

by the protection sellers during times of high default risk. Third, we find evidence of

an empirical relation between the liquidity of the bond and the CDS market in excess

of the liquidity spill-over which is immanent to our model. Specifically, we are able to

demonstrate that higher liquidity premia in the bond market lead to increasing liquidity

premia in the CDS market as well but that the reverse effect does not apply.

The remainder of the paper is structured as follows. In the next section, we discuss the

characteristics of CDS contracts. We introduce our extended reduced-form model and our

measures of credit and liquidity risk in section 3. Section 4 presents the empirical results

of the model calibration and a detailed analysis of the estimated credit risk and liquidity

premia. A stability analysis is provided in section 5. Section 6 summarizes and concludes.

2 Credit Default Swaps

In this section, we repeat certain stylized facts regarding CDS contracts which have a

bearing on our model.

A credit default swap constitutes the exchange of a fee, paid by the default protection

buyer, for a payment by the default protection seller if a credit event on a given reference

asset occurs. The default protection can be purchased on a variety of different debt

instruments, including loans, bonds, sovereign and derivative contracts. The default

payment usually equals the difference between the notional value of the reference asset

and its recovery value.

CDS contracts can differ with regard to maturity, notional amount, definition of the credit

event, the protection buyers and sellers payments, and so forth. Naturally, buyers want

to interpret the scope of protection as widely as possible, while sellers prefer to interpret

it narrowly. The International Swaps and Derivatives Association (ISDA) has therefore

published standard guidelines regarding these properties in its 2002 Master Agreement

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and in the new 2003 credit derivatives definition. Both this publication and the growth of

the CDS market have led to a standard structure of the CDS contract. CDS premia are

paid quarterly on certain reference days (March, June, September and December 20th)

and are quoted in basis points of the nominal value. When a contract is entered into on a

non-reference day, the time until the next reference day is added to the specified contract

time such that all contracts mature on a reference day. If a default event occurs between

two payment dates, the protection buyer pays the CDS premium accrued since the last

payment date to the protection seller and physically delivers the reference debt entity.

The protection seller, in turn, pays the notional value of the debt to the protection buyer.

This physical settlement with a recovery of face value has been widely established in the

market2 with a standard specification of 30 days for the notice of physical delivery and

two to five business days for the termination of the contract upon the delivery notice.3

Cash settlement is only specified for approximately 15% of CDS contracts, see e. g. British

Bankers Association (2004). In this case, the post-default price of the defaulted referenceentity is determined through a dealer poll or with regard to a number of simplifying

valuation techniques specified in the ISDA guidelines. The settlement period is then used

by the protection buyer and seller to collect the information necessary to the valuation of

the debt and agree on a fair post-default price. While cash settlement can be beneficial

when there is a highly illiquid market for defaulted debt, determining the fair price is

often intricate.

In the following, we will assume that the CDS contract specifies recovery of face value and

that physical settlement occurs immediately after the default event.

3 The Credit Risk and Liquidity Model

In this section, we develop our approach to measure the size of the default and the liquidity

component in CDS premia and corporate bond prices.

3.1 Model Framework

We assume an arbitrage-free capital market in which default-free zero coupon bonds,default-risky coupon-bearing bonds and CDS are traded. The liquidity of these

instruments can differ, and we choose the liquidity of the default-free zero coupon bond

as the reference value of 1. This choice implies that the liquidity of each instrument is

2Approximately 85% of CDS contracts specify physical settlement.3The settlement period has been added to the ISDA guidelines as a reaction to the default of Adelphia

Communications in 2002. After the default event, numerous protection buyers who did not hold the

reference entity were forced to buy the defaulted debt contracts which led to a market squeeze. Eventually,

an auction was used to satisfy the buyers.

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measured relative to the default-free zero coupon bond. We can therefore circumvent the

problem of specifying a perfectly liquid instrument in comparison to which each illiquid

instrument trades at a discount.

Following Duffie and Singleton (1997), we let rt denote the instantaneous default-free

interest rate process that determines the price of the default-free zero coupon bond. trefers to the default-risk hazard rate which is assumed to be equally reflected in CDS and

corporate bonds. We use bt as the illiquidity process which determines the fraction of

the coupon-bearing bonds price due to liquidity deviations from the reference liquidity of

1. We also use the CDS-specific liquidity intensities caskt and cbidt to reflect the liquidity

which only affects the CDS ask and bid premium, respectively.

Then

P(t1, t2) = exp

t2t1

(s) ds

denotes the risk-neutral survival probability from time t1 to t2,

D (t1, t2) = exp

t2t1

r (s) ds

is the price of a default-free zero-coupon bond at time t1 with maturity in t2 and a notional

of 1, and Lb (t1, t2) = expt2t1

b (s) ds

equals the liquidity discount factor that measures the difference between a reference-liquid

default-free bond and a less liquid bond. We assume that rt, t and bt are stochastic with

a specific correlation structure which we introduce in section 3.2. Therefore,P,

D and Lb

are also stochastic. Besides, we assume that a bondholder recovers a fixed fraction R of

the face value F if default occurs.4

The (dirty) price of a coupon-bearing default-risky bond at time t with a fixed coupon of

c that is paid at times t1, . . . , tn, notional F and maturity in tn is then given by

CB (c,R,F,t,t1, . . . , tn) = c n

i=1

Et

P(t, ti) D (t, ti) Lb (t, ti)

+ F Et

P(t, tn) D (t, tn)

Lb (t, tn)

(1)

+ R F n

i=1

Et P(t, ti1) P(t, ti) D (t, ti) Lb (t, ti) ,where t0 := t and Et is the expectation operator under the risk-neutral measure. Note

that P(t, ti1) Pt (ti, ti) denotes the probability of surviving from t until ti1 and thendefaulting between ti1 and ti given that the current date is t. Equation (1) can be

interpreted as the expected present value of all future bond cash-flows: The first summand

4We have also explored the effect of recovery of treasury and recovery of market value, but our results

were not vitally affected.

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gives the expected present value of the coupon payments at each coupon date. The second

summand equals the expected present value of the principal payment in the last period.

The last term is the expected present value of the recovery rate payment. Each future

payment is therefore discounted with regard to the default risk of the bond and to the

illiquidity which affects the instrument.

As discussed above, it is not obvious whether liquidity should be included in a model for

CDS premia, and if so, in which way this should be done. A common solution to this

problem both in empirical studies and theoretical models, see e.g. Schueler and Galletto

(2003) and Longstaff et al. (2005), is to assume that the CDS mid premium reflects a

price which is entirely free of liquidity risk. This assumption neglects the possibility of a

liquidity-driven market pressure that causes the pure credit risk CDS premium which is

unaffected by the liquidity of the CDS market to be closer either to the ask or the bid

quote. In order to cope with this possibility, we focus directly on the CDS ask and bid

premia from which we extract the pure credit risk component of the CDS premium. Thisprocedure is equivalent to assuming that the bid and ask quote are affected by liquidity.

As a consequence, we model two values of the fixed leg of the CDS, one for the ask and

one for the bid side.

The value of the fixed leg of a CDS contract at time t with fixed arrear premium payment

sask at times T1, . . . , T n and maturity in Tn equals

CDSfix

sask, t ,T1, . . . , T n

= sask

ni=1

Et

P(t, Ti1) D (t, Ti) Lcask (t, Ti) , (2)

where Lcask is defined like Lb with the bond liquidity intensity b replaced by the CDSask liquidity intensity cask. Equation (2) suggests that the payment of all ask premia

sask from the protection buyer to the protection seller has to be discounted for the default

probability since the payment at time Ti1 only occurs with a probability P(t, Ti1). The

CDS-specific liquidity discount factor for the ask premium Lcask (t, Ti) is added since weassume that a part of the CDS ask premium is not due to default risk but to the fact that

the protection seller demands an additional premium because of illiquidity.

Analogously, we obtain for the CDS bid premium:

CDSfix sbid, t ,T 1, . . . , T n = sbid ni=1

Et P(t, Ti1) D (t, Ti) Lcbid (t, Ti) , (3)where Lcbid results from replacing the CDS ask liquidity intensity cask by the CDS bidliquidity intensity cbid .

The value of the floating leg, that is the payment of the protection seller contingent upon

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default, equals

CDSfloat (R,t,T1, . . . , T n) = F n

i=1

Et

P(t, Ti1) P (t, Ti)

D (t, Ti)

R

F

n

i=1 Et P (t, Ti1) P(t, Ti) D (t, Ti) Lb (t, Ti) .(4)

The first summand in (4) equals the discounted present value of the face value F which

we assume the protection seller pays out in cash. Since physical delivery of the defaulted

bond constitutes the CDS market standard, the second summand equals the discounted

present value of the defaulted bond which the protection seller has to sell in the market

upon the bonds delivery. Therefore, the second summand contains the discounting factor

for the bond liquidity in addition to the credit risk discounting factor.5

Setting equal (2) and (4) and solving for sask, we obtain

sask =F ni=1Et 1 R Lb (t, Ti) P(t, Ti1) P(t, Ti) D (t, Ti)n

i=1Et

P(t, Ti1) D (t, Ti) Lcask (t, Ti) , (5)

where, as in (1), T0 := t.

The closed-form solution for the CDS bid premium is identical to that for the ask premium

only with Lcask replaced by Lcbid:

sbid =

F

ni=1Et 1 R Lb (t, Ti) P (t, Ti1) P(t, Ti) D (t, Ti)n

i=1EtP (t, Ti1) D (t, Ti) Lcbid (t, Ti) . (6)

Equations (5) and (6) differ only with regard to the liquidity discount factor. If, as we

generally observe, the bid quote lies below the ask quote, we expect that Lcbid on averageexceeds Lcask. Since the reference liquidity equals 1, we expect that Lcbid > 1 > Lcask. Forlarge bid-ask-spreads, Lcask will be much smaller than 1 and Lcbid will be larger than 1,moving both the bid and the ask premium away from each other. If, on the other hand,

the CDS market were perfectly liquid, both

Lcask and

Lcbid would be identical to 1 and

the ask and bid premium coincide.

5In the above setting, the floating leg is not discounted with regard to the CDS-specific liquidity. We

believe this to be appropriate since the CDS premium is the variable fixed by the CDS protection seller

and buyer when they enter into the contract. Therefore, we simply capture the net effect of liquidity in the

fixed leg where we have directly observable variables. In addition, also discounting the floating leg for CDS

illiquidity is unnecessary since this would lead to the identical discounting factor eLl (t, Ti), l {cask, cbid},

only with different weights P(t, Ti1) D (t, Ti) andh

1 R fLb (t, Ti)i

h

P(t, Ti1) P(t, Ti)i

D (t, Ti),

in the floating and fixed payment leg. Overall, the effect of illiquidity can be captured in our setting while

keeping the model parsimonious.

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3.2 Specification of Intensity Processes

The second step in the specification of the model consists of choosing the dynamics of the

default and the liquidity intensities.6 As Longstaff et al. (2005), we assume a square-root

process for the default intensity (t). The liquidity intensities lt, l {b, cask, cbid}, for theliquidity discount factors of the bond (b), the CDS ask premium (cask) and the CDS bidpremium (cbid) follow an arithmetic Brownian motion.

7 The joint dynamics of the default

and the liquidity intensity are given by:d(t)

dl(t)

=

d(t) + fl dl(t)fl d(t) + dl(t)

=

(t) + fl

+ fl (t)

dt +

(t) fl fl

(t)

dW(t)

dWl(t)

(7)

with constants , , , l and l, correlation coefficients fl and fl , independent Brownian

motions W and Wl and independent processes

and l

. We do not specify thecovariance structure for the liquidity intensities l, l {b, cask, cbid}, for two reasons.First, it does not directly enter the model equations and second, it is implicitly defined by

the covariance matrix of each component with . Economically, a correlation between the

liquidity intensities that is not directly due to the implicit correlation with will allow us

to determine the way in which pure liquidity effects are translated from one market into

the other.

Given the above relation between the intensities, we deduce an analytical solution for the

joint expectation of P (t1, t2) and Ll (t1, t2) as well as P(t1, t2) and

Ll (t1, t3):

EtP(t1, t2) Ll (t1, t2) = Et expt2

t1

(s)ds expt4t3

l (s) ds=: Pt (f, t1, t2) Lt

lf, t1, t2

,

Et

P(t1, t2) Ll (t1, t3) = Et expt2

t1

(1 + f) s + (1 + f) s

ds t3t2

sds

=: Pt (f, t1, t2, t3) Lt

lf, t1, t2, t3

where the functions Pt (f, t1, t2), Pt (f, t1, t2, t3), Lt

lf, t1, t2

, Lt

lf, t1, t2, t3

and the

intensities f and lf are derived in the appendix A.

6We have also estimated the model for the case where both the default and the liquidity intensity are

correlated with the default-free short rate. The coefficient estimates, however, proved to be highly sensitive

to the estimation period, leading to an average coefficient estimate of 0 for 158 out of 171 companies.7This will allow the liquidity process to take on both positive and negative values. This specification of

the dynamics of the liquidity intensity can be favored over the simple White-Noise process in Longstaff et al.

(2005) since it additionally allows for liquidity trends which may be more appropriate for the maturing CDS

markets. In addition, the inclusion of a trend parameter results in a liquidity discount factor Ll (t1, t2)

that is concave in the time-to-maturity for positive values of l. This suggests that liquidity is over-

proportionately declining as the time-to-maturity increases which agrees with recent empirical evidence by

Chen et al. (2005).

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Substituting these expressions in (1) and (5) yields the analytical solutions for CB and s:

CB = c n

i=1

Pt (f, t , ti) Lbt (f, t , ti) Dt

+ F Pt (f, t , tn) Lt bf, t , tn

Dt

+ R F n

i=1

Pt (f, t , ti1, ti) Lt bf, t , ti1, ti Pt (f, t , ti) Lt bf, t , ti Dt,

(8)

sask/bid = F n

i=1

1 R Lt

bf, t ,Ti1, Ti,

Pt (f, t ,Ti1, Ti) Dtn

i=1 Pt (f, t ,T i1, Ti) Ltcask/cbidf , t ,Ti1, Ti,

Dt

F n

i=1

1 R Lt

bf, t ,T i

Pt (f, Ti, t) Dt

ni=1 Pt (f, t ,T i1, Ti)

Lt

cask/cbidf , t ,Ti1, Ti, Dt

,

(9)

where we suppress the arguments of the time-discount factor Dt for ease of presentation.

These closed-form solutions can then be calibrated to a set of bond prices and CDS ask

and bid quotes to obtain estimates of the default and liquidity intensities.

3.3 Measures for Credit Risk and Liquidity Premia

The model developed in sections 3.1 and 3.2 allows us to disentangle the bond spread

into a credit risk and a liquidity component. By an analogous procedure based on bidand ask quotes of CDS premia, we will be able to compute a credit risk and a liquidity

component for CDS. Basically, the credit risk premia are determined by model prices for

which the liquidity discount factors are set to the reference value of 1, i.e. the liquidity of

the bond and the CDS market are assumed to be identical to the reference liquidity. This

is identical to only considering the default intensity as a pricing factor. The liquidity

premia are computed from model prices with liquidity discount factors different from 1.

The pure credit risk premium csdef for a straight bond with coupon c and face value F

results from equation (8) for L b 1 and the definition of a spread as a premium onthe risk-free rate:c

ni=1

Pt (f, t , ti) Dt + F Pt (f, t , tn) Dt

+ R F n

i=1

[Pt (f, t , ti1) Pt (f, t , ti)] Dt

=:n

i=1

ccsdef+ D (t, ti)

1

tit

(tit) + Fcsdef+ D (t, tn)

1

tnt

(tnt) , (10)11


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where D (t, ti)

1

tit 1 equals the yield to maturity of a default-free zero coupon bondwith reference liquidity and time-to-maturity ti t.This pure credit risk premium csdef is directly comparable to the pure credit risk premium

of a CDS if the maturity of both instruments is identical and if, in addition, the clean

price of the bond equals its face value. The second condition is important to avoid thedifficulties discussed by Duffie (1999) and Duffie and Liu (2001) who show that the yield

spreads on fixed-coupon corporate bonds cannot directly be compared to CDS premia.

Credit spreads determined from par bonds are, apart from interest rate risk which we do

not consider, identical to theoretical asset swap spreads. These must coincide with the

premium of a CDS on this bond in arbitrage-free markets by definition.

The total bond spread cs is determined analogously to csdefwith the impact of the liquidity

intensity added:

c n

i=1Pt (f, t , ti) Lbt (f, t , ti) Dt + F Pt (f, t , tn) Lt bf, t , tn Dt

+ R F n

i=1

Pt (f, t , ti1, ti) Lt

bf, t , ti1, ti

Pt (f, t , ti) Lt

bf, t , ti

Dt

=:n

i=1

ccs + D (t, ti)

1

tit

(tit) + Fcs + D (t, tn)

1

tnt

(tnt) . (11)The liquidity premium is defined by the difference between the the total bond spread cs,

and the pure credit risk premium csdef:

csliq := cs csdef. (12)

The credit risk and liquidity components of a CDS are subsequently determined in the

following way. We first compute the pure credit risk premium in the CDS spread sdef by

assuming that the liquidity discount factors Lcask or Lcbid are equal to 1:

sdef := F n

i=1

1 R Lt

bf, t ,T i1, Ti,

Pt (f, t ,T i1) Dtn

i=1 Pt (f, t ,T i1) Dt

F ni=1 1 R Lt

bf, t ,T i Pt (f, Ti, t) Dtn

i=1Pt (f, t ,Ti1) Dt . (13)

Equation (13) shows that the pure credit risk CDS premium is exclusively determined by

the default-free interest rates, the survival and default probability and the bond liquidity.

Note that the bond liquidity affects the CDS both in the case of physical delivery and

cash settlement: a less liquid bond will have a lower post-default price compared to an

otherwise identical bond with higher liquidity. The CDS premium will therefore be higher

for the less liquid bond in order to compensate the protection seller for the lower value

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of the bond should default occur. This effect would pertain even if the CDS market were

perfectly liquid.

In a CDS market which has a liquidity that differs from the reference liquidity, the ask and

bid premia will differ from the pure credit risk premium sdef. The size of the bid-ask-spread,

however, is not necessarily an appropriate measure of illiquidity in our context for tworeasons. First, a comparison of (5) and (6) shows that the bid-ask-spread is clearly affected

by credit risk as well as liquidity since a higher default probability increases the ask

premium more strongly than the bid premium. Second, even if the bid-ask-spread were

taken relative to sdef, the resulting quantity would not be comparable to any liquidity

measure in the bond market since we only have access to mid prices in this market. We

therefore use the difference between the theoretical mid premium smid,

smid :=sask + sbid

2, (14)

and the pure credit risk premium sdef

, as a measure of CDS liquidity:sliq := smid sdef, (15)

where sask and sbid follow from equation (9). Note that sliq is positively related to the

asymmetry between the ask and the bid spread with regard to sdef. If there is a high

market pressure from the protection buyers side, that is if there is a high demand for

credit protection, protection sellers will be able to move the ask premium at which they

are willing to trade upwards. Since the pure credit risk premium sdef remains at its initial

value, sliq will increase. If, on the other hand, the market pressure from the protection

sellers side is high, protection buyers will set lower bid quotes. This will then result in

lower values ofsliq. Therefore, our measure of CDS liquidity is consistent with the measure

of bond liquidity: if a large number of investors seeks to sell bonds - which corresponds

to buying credit protection - the liquidity premium in the bond market will increase and

vice versa.

In the next section, we turn to an empirical application of the theoretical model derived in

this section. In particular, we estimate default and liquidity intensities from bond prices

and CDS premia. We then analyse the estimated time series of the credit risk and liquidity

components in CDS premia and bond credit spreads.

4 Empirical Analysis

In section 4.1, we first describe the data set of CDS premia and bond prices, respectively

credit spreads, to which we subsequently calibrate our model. We then discuss the

estimation procedure and the resulting credit risk and liquidity premia in credit spreads

and CDS premia. Eventually, we analyse the the time-series behavior of the estimated

premia.

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4.1 Data

As a dynamic proxy for the default-free interest rate, we choose the Nelson-Siegel-Svensson

curve whose parameters are estimated by the Deutsche Bundesbank on a daily basis

because our empirical analysis is focused on Euro-denominated bonds and CDS.8 The

resulting spot rates are used to compute prices of default-free zero-coupon bonds which

we assume to have the reference liquidity discount factor of 1. The recovery rate is assumed

to equal 40%.

All CDS and bond data is collected from Bloomberg. The daily CDS ask and bid closing

premia were made available to us by an international investment bank. We compute the

daily closing mid premia for CDS in order to compare them to the bonds credit spreads

which are computed from Bloomberg yields using mid prices. In order to avoid exchange

rate issues, we only collect prices of CDS denominated in Euro. As a starting and end

point, we use June 1st, 2001 and June 30th, 2007 which yields a total of 1,548 trading

days during which we observe CDS premia with a time-to-maturity of 5 years. All CDS

which are not quoted with a reference time-to-maturity of 5 years are excluded in order to

obtain a sample with a homogenous CDS liquidity. To account for the time conventions

in the CDS market described in section 2, we compute the distance between the quoting

day and the next reference date and add this to the quoted time-to-maturity to obtain the

true CDS maturity.

Bloomberg records CDS prices for companies from 10 industry sectors. We collect all

bond mid prices for companies from these industry sectors which had at least 2 bonds

outstanding at some point-in-time during the observation interval. We drop companieswith fewer than 20 observations on consecutive trading days on which at least two bond

prices as well as the bid and ask CDS premium were available.

For each of the remaining companies, we collect a rating history from Bloomberg for the

period during which we observe bond prices and CDS premia. Both the Standard&Poors

rating and the Moodys rating are collected and mapped on a numerical scale ranging

from 1 to 50 where 1 corresponds to a AAA Standard&Poors rating (Aaa Moodys

rating). 50, on the other hand, corresponds to a CCC+ Standard&Poors rating (Caa1

Moodys rating) and is the worst rating which we observe during the entire observation

interval. If the resulting numerical rating differs by 2 or more, we take the average of

the two ratings. If the rating differs by 1, we choose the Standard&Poors rating and

ignore the Moodys rating. If no rating for the company could be found for at least 20

observations on consecutive trading days on which at least two bond prices as well as the

bid and ask CDS premium were available, we drop the company from our sample.

8As an alternative, we also used the swap curve which is, on average, 10 basis points higher than the

NSS curve for a time-to-maturity of 5 years. Since the dynamics are almost identical, we here only present

the results for the NSS curve.

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The above procedure leaves us with a set of 171 companies from the sovereign and 9

corporate sectors and a numerical rating history that consistently lies between 1 and 50.

A detailed overview is given in table 1. For ease of exposition, we first compute the average

numerical rating of a company for all days during which there are a sufficient number of

observations. We then map the numerical value to the Standard&Poors rating and use

this as the column heading.

Table 1 shows that the majority of companies has an average investment-grade rating over

time; only 13 companies lie in the subinvestment-grade range. The largest industry group

consists of the sector Financials with a total of 54 companies. At the same time, these

companies are also among the top-rated ones. Overall, table 1 demonstrates that the

sample with which we work is skewed toward financial companies and investment-grade

rating. We will therefore conduct the estimation procedure on a single-firm level and

aggregate the data subsequently only.

In order to present the time-series of the credit spreads and CDS premia, we compute the

average credit spread and CDS mid premium for each rating class at every date of our

observation interval. To do so, we first identify the rating for a particular company on

each day. We then compute the credit spread for each bond of that particular company

as the difference between the yield of this bond and the yield of a synthetical default-free

bond with identical coupon and time-to-maturity. Next, we interpolate the resulting credit

spreads for each company to obtain a synthetical time to maturity of 5 years. We then take

averages of the obtained credit spreads and the observed CDS mid premia for all companies

with an average investment, respectively subinvestment grade rating. The resulting timeseries of the average credit spread and CDS mid premia for the investment-grade and

subinvestment grade rating classes are depicted in figure 1.

As we see from figure 1, the time series of the average investment-grade bond credit spreads

(depicted in the solid blue line) consistently exceeds the mid CDS premia. Overall, the

mean investment-grade credit spread has an average of 89.42 bp with a standard deviation

over time of 23.03 bp. The lowest average credit spread of 33.54 bp is attained on August

22, 2001, the highest one which equals 178.86 bp on February 18, 2002. CDS premia

fluctuate between 15.86 bp and 143.76 bp with a mean of 45.42 bp and a standard deviationof 27.96 bp.

Credit spreads for the subinvestment-grade sector are, as expected, clearly higher with an

average of 369.62 bp and a time-series standard deviation of 188.70 bp. Overall, the credit

spread fluctuates between 87.60 bp and 1320.33 bp which are attained on March 1, 2005

and on October 9, 2002. The average subinvestment-grade CDS premium fluctuates above

and below the credit spread with a standard deviation of 224.53 bp, but the time-series

average of 341.33 bp lies below that of the credit spread.

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Figure 1: Average Bond Credit Spreads and Mid CDS Premia Time Series

The figure shows the average bond credit spreads and mid CDS premia between June 1,

2001 and June 30, 2007. Average mid CDS premia are denoted in black, credit spreads in

blue. The solid line is used to depict the investment-grade, the dashed line to depict the

subinvestment-grade time series.

0

200

400

600

800

1000

1200

June-01 June-02 June-03 June-04 June-05 June-06 June-07

Date

Premiainbp

Av. CDS Premia IG

Av. Bond Spreads IG

Av. CDS Premia Sub-IG

Av. Bond Spreads Sub-IG

4.2 Estimation Procedure

To ensure that the 9 parameters ,,,b, b, cask, cask , cbid, cbid of the 4 intensityprocesses, the starting values , b, cask, cbid (t), t = 1, . . . , 1583, and the 6 correlation

coefficients can be identified, we demand that the instantaneous default and liquidity

intensity are equal for bonds of the same issuer with identical seniority but different time-

to-maturity and coupon rate.9 This identification assumption will make our parameters

issuer-specific.

The procedure consists of three steps. Initially, we set all intensity correlation

coefficients to zero. This corresponds to the case of independent credit risk and

liquidity intensities. In the first step, we then choose a value for the process parameters

,,,b, b, cask, cask, cbid, cbid and then determine the values , b, cask, cbid (t)which minimize our objective function, the squared sum of the deviation between theobserved and the theoretical prices, on each observation date t.10 We then repeat

9It seems rather reasonable that bonds of the same seniority have the same default probability during

the next infinitesimally small time interval. The liquidity discount, on the other hand, may well depend

on the time to maturity and the coupon of a bond. We hope, however, that the homogeneity of the bonds

of the same issuer will limit the differences induced by this assumption. In addition, the functional form of

the stochastic liquidity process results in larger liquidity premia for bonds with a longer time-to-maturity.10CDS premia are matched at the basis point level, bond prices at a level that translates to basis point

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this procedure of choosing process parameters and determining an optimal associated

time-series of the intensities until we arrive at a minimum of the objective functions in the

process parameter set. In the second step, we estimate the correlation of the estimated

intensity time series for the discretized version of equation (7) which gives us our new

starting point for the correlation coefficients. We then return to the first step and repeat

the estimation procedure until our estimate of the correlation coefficients changes by less

than 0.01 in two steps.11

Given the estimates of the process parameters, the intensities and the correlation

coefficients, we then compute the credit risk and liquidity premia for CDS and for bonds

in the third step as explained in section 3.3.

4.3 Credit Risk and Liquidity Premia: Cross-Sectional Results

Our estimates for the correlation coefficients imply that the credit risk intensity increasesboth the bond liquidity intensity and the CDS ask and bid liquidity intensity. affects

b

significantly for 136 out of 171 reference entities. 149 of the significant correlation

coefficients are positive and have a mean of 17.39%. The 7 negative correlation coefficients

are obtained for sovereign reference entities which had a AAA, respectively a AA, rating.

The impact of on cask is significant for 148 and positive for 147 reference entities

with a mean of 26.22%. cbid is, in turn, significantly affected by for only 76 reference

entities with a coefficient which is negative for 44 reference entities. The mean values for

the positive and the negative coefficient estimates are very similar at -20.73% and 20.32%,

respectively. The impact of the liquidity intensities on , on the other hand, is almostnegligible: we obtain only one significant coefficient estimate for b

, three for cask out

of which two are positive and two for cbid with a positive and a negative one. These

results show that credit risk premia increase liquidity premia in the bond market but not

vice versa. We can also conclude that higher credit risk leads to a higher distance between

the pure credit risk CDS premium and the ask premium. CDS bid premia, on the other

hand, are not as symmetrically affected. We display the premium components in table 2.

Overall, we see that the credit risk and liquidity premia increase as the rating deteriorates.

For the AAA rating class, the pure credit risk premium in credit spreads csdef hasan average of 6.11 bp which approximately doubles for each rating downgrade in the

investment grade sector. The subinvestment grade sector exhibits values of csdef which

are about five times as large as for the investment grade sector. For the liquidity premia

csliq, the increase is less steep than for the credit risk premia. Nevertheless, we obtain

strictly positive estimates for the liquidity premia. It is also interesting to note that

accuracy at the yield spread level.11Convergence in the correlation coefficients is usually achieved in less than 10 iteration steps.

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the minimum of csliq is not monotonously increasing as credit risk increases. These

findings indicate that even though more risky bonds also contain higher absolute liquidity

premia, the interdependence between credit risk and liquidity cannot be fully captured

cross-sectionally.

Comparing the CDS credit risk premia CDS sdef

, we find that they almost consistentlyexceed csdef but that the difference, caused by the bond liquidity premia, is very limited.

The difference is smallest for the AAA rating class with 0.27 bp and maximal for the BBB

class with 3.40 bp on the absolute level. This agrees with the increasing average level of

the bond liquidity premium csliq.

The most noteworthy results of table 2 concern the CDS-specific liquidity premia. As

explained in section 3.3, we measure the liquidity of the CDS market by the asymmetry

between bid and the ask quotes relative to the credit risk premium. On average, the

liquidity risk premium sliq is positive which suggests that the CDS market is mostly

dominated by protection sellers. On an absolute level, the asymmetry increases as therating deteriorates. Taken relative to the pure credit risk premia, however, the pure

liquidity premia are smaller for the subinvestment grade sector, and the difference is

particularly pronounced for the gap between the investment and the subinvestment grade

sector where the ratio descends from 7.14% to 3.82%. In addition, we find that 19.24% of

the liquidity premia in the subinvestment grade sector are negative, suggesting that the

asymmetry may actually be smaller.

From an economic point of view, investors who have to maintain a given default risk level

threshold or face a value-at-risk constraint will demand more credit protection through

CDS if a reference entity is closer to the subinvestment grade barrier. Since the liquidity of

the bond market is on average lower than that of the CDS market, selling the bond would

be associated with a loss which can be avoided by buying the CDS. For the subinvestment

grade sector, these investors will not increase the demand pressure, but the low liquidity of

the bond market makes it more attractive to take on credit risk synthetically as a protection

seller. Therefore, the demand pressure decreases and the supply pressure increases. In the

next section, we will further explore how credit risk and liquidity premia behave during

times of high and low credit risk.

4.4 Credit Risk and Liquidity Premia: Time-Series Results

In this section, we first present the time series of the premia estimates for the investment

and the subinvestment grade sector. We then explore the relation between the liquidity

premia in the bond and the CDS market. The section concludes with an empirical analysis

of the impact of market-wide credit and liquidity measures on the estimated premia time

series.

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The estimated credit risk and liquidity premia are depicted in figure 2. For ease of

presentation, the investment and subinvestment-grade rating classes are summarized into

a single time series each.

Figure 2: Estimated Credit Risk, Liquidity and Correlation Premia Time Series

The figure shows the estimated credit risk, liquidity and correlation components in CDS

premia and credit spreads for all rating classes. The estimates are computed with regard to

a constant time-to-maturity of 5 years and a synthetical par bond.

Investment Grade Premium Components

-20

0

20

40

60

80

100

120


Date

Premiainbp

CDS Credit Risk

CDS Liquidity

Bond Credit Risk

Bond Liquidity

Subinvestment Grade Premium Components

-200

0

200

400

600

800

1000

1200


Date

Premiainbp

CDS Credit Risk

CDS Liquidity

Bond Credit Risk

Bond Liquidity

Figure 2 shows that the pure credit risk premia both in credit spreads and CDS premia,

depicted in the solid black lines, can hardly be distinguished both for the investment grade

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and the subinvestment grade sector. Particularly for the investment grade sector, there

are two distinct spikes in late 2001 and late 2002 which can be associated with the defaults

by Enron and WorldCom. The reaction of the subinvestment grade rating sector to the

Enron default is almost negligible which may be due to the fact that only two companies

have a subinvestment grade rating between June 2001 and February 2002. Overall, we

observe a flattening of the pure credit risk premia curves of csdef and sdef over time with

much lower average levels at the end of the observation interval. Since the premia are

computed with regard to par bonds with a constant time-to-maturity of 5 years, we can

directly attribute the lower premium level to a lower amount of risk.

We observe that the bond liquidity premia csliq also exhibits a similar behavior across

the rating classes. During the high credit risk periods, the liquidity premia are high

and rather volatile and become much lower and rather flat during the latter part of the

observation interval. Visual inspection of the CDS-specific liquidity premia sliq is more

difficult since the absolute values are small. Overall, the level of s

liq

is closer to 0 nearthe end of the observation interval for both the investment and the subinvestment grade

sector. During times of high credit risk on the other hand, the liquidity premium exhibits

a reverse behavior across the two sectors. For the investment grade sector, sliq is higher

during times of high credit risk. In the subinvestment grade sector, sliq becomes more

negative when credit risk is high. This finding suggests that the demand pressure which

characterizes the CDS market for investment grade debt is replaced with a supply pressure

for the subinvestment grade sector.

In order to study the dynamic interaction between the liquidity premia in the bond and

the CDS market, we perform a vector error correction model (VECM) analysis. Theaugmented Dickey-Fuller test cannot reject the hypothesis of a unit root in csliq and sliq for

162 reference entities. The Johansen procedure cannot reject cointegration of the liquidity

premia across the two markets for 159 companies. The VECM-specification which we use

joins the models for credit risk and liquidity premia in credit spreads and CDS premia

and is of the form

csliqi = w1 csliqi1 + w2 csliqi1 + w3 sliqi1 + w4 sliqi1,sliqi = z1 csliqi1 + z2 csliqi1 + z3 sliqi1 + z4 sliqi1.

We demand that the parameters are identical for firms in the same rating class. Time lags

up to degree 5 are considered in order to capture at least a weekly time interval, and the

resulting parameter estimates are transformed into a single estimate using the approach

of Fowler and Rorke (1983). The results of the estimation are displayed in table 3.

As table 3 shows, the interdependence between the pure liquidity premia in bond and

CDS markets differs between the investment and the subinvestment grade sector. With

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regard to their own previous level and change, bond and CDS liquidity premia exhibit a

consistently negative coefficient, suggesting that the liquidity premia are mean-reverting

and on average declining. This is particularly pronounced for the CDS market. Regarding

liquidity spillover effects, we observe that the bond market liquidity is not affected by the

CDS liquidity premia for the investment grade sector. For the subinvestment grade sector,

we obtain a positive coefficient estimate for sliq and a negative one for sliq which suggests

that a higher level of CDS liquidity premia leads to lower levels of bond liquidity premia

and that changes occur in opposite directions. Even though this observation supports

our hypothesis that the CDS market constitutes a possible substitute in the taking on of

credit risk for subinvestment grade debt, both the statistical and the economic significance

of the effect are limited. The bond liquidity premia, on the other hand, significantly

affect CDS premia both for the investment and the subinvestment grade sector. For the

investment-grade segment, the association between the level of csliq and changes in sliq is

positive which shows that higher liquidity premia in the bond market lead to increasing

liquidity premia in the CDS market. This suggests that protection sellers increase their

CDS ask quotes relative to sdef if the bond market becomes less liquid. As expected, the

sign of the coefficient for csliq becomes negative for the subinvestment grade sample: the

CDS market becomes a more attractive substitute to the bond market in the taking on

of credit risk. In addition, changes in csliq coincide with subsequent changes ofsliq in the

opposite direction, further strengthening this hypothesis.

To conclude our empirical analysis, we determine the impact of market-wide credit risk and

liquidity measures on the premia estimates. As a proxy for credit risk, we choose the J.P.

Morgan Aggregate Index Euro (MAGGIE) for which we obtain daily yield spreads from

Bloomberg. The MAGGIE index comprises a total of roughly 1,750 Euro-denominated

bonds and Jumbo Pfandbriefs which are included on the basis of their liquidity. Liquidity

is proxied by the European Central Bank (ECB) financial market liquidity indicator for

which daily values were made available to us through the ECB. The indicator is designed

to mirror dynamic patterns in the overall liquidity of the Euro area financial market and

combines information from the stock, the bond and the equity options market as well as

European interest rate data.12 A higher value marks higher aggregate financial market

liquidity.

Since the premium estimates are not stationary, we perform a Johansen cointegrationanalysis with the credit risk and liquidity premia as dependent variables, i.e. the premium

change is regressed on its own previous change and that of the explanatory variable. The

results of the analysis are displayed in table 4.

As table 4 shows, the credit risk premia for bonds and CDS show a similar dependency on

12For a detailed description of the indicator, see European Central Bank Financial Stability Report June

2007.

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the aggregate market-wide measures. Both csdef and sdef depend positively on credit risk

and negatively on the liquidity measures, and the impact is stronger for the investment

grade sector. This suggests that credit risk premia in the subinvestment grade sector

mainly depend on the reference entitys idiosyncratic default risk. The liquidity premia,

on the other hand, exhibit a different sensitivity to the explanatory variables depending

on the rating. For the investment grade sector, csliq and sliq both react positively to

increases of credit risk and negatively to increases of liquidity, but the effect on the bond

liquidity premium is much more pronounced. We partly attribute this to the fact that

the CDS market is, on average, rather liquid, and partly to the increasing overall liquidity

in the CDS market throughout the observation interval. In the subinvestment grade

sector, on the other hand, csliq reacts with strong increases to increases in the credit risk

and steep decreases to increases in overall market liquidity. For CDS liquidity premia, we

observe a negative dependency on market-wide credit risk and expected liquidity premium

decreases if the market-wide liquidity decreases. Comparing these results with our findings

in figure 2, the signs of the coefficients agree with the hypothesis that the market for CDS

on subinvestment grade reference entities becomes less dominated by protection sellers if

credit risk increases, possibly because taking on credit risk synthetically becomes more

attractive.

Summarizing the results of this section, we find clear evidence that bond credit spreads

computed from mid prices contain strictly positive credit and liquidity premia. Both

decrease over time and are subject to frequent changes. For CDS, the pure liquidity

premia can either be positive or negative but tend to be positive. Their decrease over

time documents the maturing of the CDS market as a whole. In addition, our results showthat liquidity in the bond market dries up as the default risk increases. CDS liquidity is

somewhat more difficult to grasp, but we find that the asymmetry between protection

buyers and sellers increases with default risk for investment-grade debt. We attribute this

to a higher pressure on investors to insure against further downgrades. The subinvestment

grade CDS market partly exhibits the reverse behavior. The size of the liquidity premia

is also very different in both markets. This result raises the question whether an explicit

modelling of the CDS liquidity premium is necessary or whether the CDS mid premium

can consistently be used as a proxy for sdef. Since the difference between the mid CDS

premium and our estimate of this premium is rather small on average, we further explore

this issue in section 5.1.

5 Stability of Credit Risk and Liquidity Premia

In this section, we perform a stability analysis of the estimated credit risk and liquidity

premia for bonds and CDS. To this purpose, we first explore how bond premia react if we

ignore liquidity in the CDS market. We then compare the pure credit risk premia which

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are obtained if the default and liquidity intensities are estimated from CDS ask or bid

quotes only to the estimate which uses both observations.

5.1 The Effect of Excluding CDS Illiquidity

We first explore whether the positive bond liquidity premia csliq we obtain in our

estimation are a result of including stochastic liquidity in CDS ask and bid premia or

simply a property of our data set. To do so, we propose the following modification of our

model: First, we shift our focus to the CDS mid premia in our estimation procedure since

there is no theoretically compelling reason why sdef must differ systematically from the

mid premium. Second, we re-estimate the default and bond liquidity intensity time-series

under the restriction c = c = c = 0 which is basically the approach by Longstaff et al.

and which makes the liquidity of the CDS market constant and equal to the reference

liquidity. We last compute csdef and csliq as explained above and compare them to the

results from the initial estimation which included liquidity in CDS premia. Since the effect

only pertains when a companys bonds are liquid relative to the CDS on the company, we

show the estimated time-series for a representative company, the Dutch communications

company The Nielsen Company (formerly VNU Group B.V.).13 The results are displayed

in figure 3.

As we see from figure 3, the estimated default risk premium in the bond credit spread

csdef has similar dynamics whether CDS liquidity is included or not, but there are also

clear differences. Overall, when stochastic CDS liquidity is excluded (blue solid line),csdef is higher, fluctuating between 34.98 bp and 537.71 bp with a mean of 110.02 bp and

a standard deviation of 97.56 bp. When stochastic CDS liquidity is included, csdef lies

between 35.74 bp and 429.29 bp, the mean equals 95.94 bp and the standard deviation

73.36 bp. The differences are especially noteworthy during the beginning of our observation

interval when the CDS market was still relatively illiquid.

Conversely, the bond liquidity premium csliq that results from excluding stochastic CDS

liquidity is consistently lower than when CDS liquidity is modelled with a mean of 3.74

bp versus 24.69 bp, a minimum of -95.95 bp (5.26 bp), a maximum of 81.82 bp (79.78

bp) and a standard deviation of 32.55 bp (16.11 bp). This suggests that neglectingstochastic CDS liquidity can yield overestimates of liquidity in the bond market and,

for above time-series, in bond price surcharges instead of discounts. At the same time, the

default risk is overestimated when the bond liquidity becomes negative, and this results in

overestimates of a companys default probability. Since neglecting CDS liquidity attributes

yield differences between the bond and the CDS market directly to bond liquidity, the effect

13The Nielsen Company is active in marketing and media information, business publications and trade

shows in more than 100 countries and has a total of 42,000 employees.

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Figure 3: The Effect of Excluding Stochastic CDS Liquidity

The figure shows the bond credit risk and liquidity premia estimated for the communications

company Nielsen when stochastic CDS illiquidity is ignored (blue line) and included (black

line).

-200

-100

0

100

200

300

400

500

600


Date

Premiainbp

Bond Credit Risk w/ CDS Liquidity

Bond Liquidity w/ CDS Liquidity

Bond Credit Risk w/o CDS Liquidity

Bond Liquidity w/o CDS Liquidity

will be especially prominent when the bond liquidity is high relative to the CDS liquidity.

As the CDS market matures, the erroneous results of neglecting CDS liquidity becomes

less striking as long as the net liquidity premium in the bond market remains positive.

5.2 Estimation from Ask or Bid CDS Premia

In section 4, we use the CDS ask and bid premia simultaneously in order to extract the

pure credit risk and liquidity components from CDS premia and bond credit spreads.

However, only the sum of these components can be observed in practice and our estimate

could therefore differ significantly from the true values. As a robustness test, we repeat the

firm-specific estimation procedure described in section 4.2 once using only CDS ask premia

and once using only CDS bid premia instead of both. We then compare the resulting

estimates of csdef, csliq, sdef and sliq with those we obtained for the entire sample. Inparticular, we compute the mean, the standard deviation and the mean absolute difference

between the estimates which are obtained using only one CDS premium and the estimates

which use both simultaneously. The results are displayed in table 5.

Table 5 shows that the estimates of the credit risk and liquidity components are almost

identical regardless of which CDS premia are used in the estimation. On average, the mean

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estimate of csdef from the CDS ask premium of 47.75 bp falls below the one using both

premia by 0.01 bp, but the similar standard deviation and the mean absolute deviation of

0.48 bp imply that the sign is not indicative of a systematic error. The same is true for

the estimate which uses only the bid premia with a mean difference between the credit

risk premia of 0.02 bp and a mean absolute error of 0.45 bp. For the bond liquidity

premia csliq, we observe the reverse result, the mean estimates which only use ask premia

are slightly higher and the ones using only bid premia are slightly lower. The difference,

however, does not appear to be systematic in this case either which is supported by the

low absolute mean deviation of 0.13 bp and 0.14 bp, respectively. The results for the CDS

credit risk premia sdef and liquidity premia sliq are similar to those for the bond. Again,

the use of ask premia leads to a very slight underestimation of the credit risk premia and

overestimation of the liquidity premia while bid premia yield slightly higher values for sdef

and lower ones for sliq.

Overall, we find that the choice of ask or bid premia in the CDS market does notsignificantly affect the size and the dynamics of the estimated credit risk and liquidity

premia. Since these premia are not directly observable in the market, we take the robust

behavior of the estimates as a sign that our estimation does not result in a systematic

deviation from the true premia.

6 Summary and Conclusion

The purpose of our paper was to develop a credit risk model that simultaneously accounts

for stochastic liquidity in CDS and bond markets. While there is broad agreement

in the literature that modelling liquidity in bond prices is an important issue both in

structural and reduced-form models, CDS markets are often assumed to be perfectly

liquid. Therefore, default probabilities or, in the context of reduced-form models, default

intensities, are often estimated directly from observed CDS mid premia and the results

are used to measure the size of the default component in corporate bond prices and credit

spreads.

In our paper, we develop a model that explicitly allows for stochastic liquidity in CDS

ask and bid premia. As CDS are derivatives and not assets, it is not clear whether

illiquidity should consistently increase mid premia or whether it should only result in

larger bid-ask-spreads. We avoid this issue by directly modelling the CDS ask and bid

premium. This approach allows for closed-form solutions for bond prices and CDS premia

which are affected by the same default risk but by a different liquidity risk. Specifically,

we are able to compute a theoretical, liquidity-free credit spread for par bonds and CDS

premia which are unaffected by the CDS-specific liquidity. These credit risk premia can

be compared to the corresponding liquidity premia and the CDS mid premium as well as

25


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the credit spread. The latter two can be - and our empirical analysis shows them to be -

affected by illiquidity.

Eventually, we estimate the model using bond mid prices and CDS ask and bid premia

for companies with ratings between AAA and CCC from a broad range of sectors. As

the default-free reference market which is assumed to be of a constant reference liquidity,we use the market for German government bonds. Our results show that the corporate

bond and CDS markets as a whole reacted strongly to the WorldCom default in late 2002.

The estimated credit risk component in CDS premia and bond credit spreads is almost

identical. We also find that the period of highest credit risk coincides with a period of low

liquidity in the bond market. Liquidity in the CDS market exhibits a less straightforward

behavior, but it can be observed that the estimate of the pure credit risk component

in the CDS market becomes more biased towards the bid in times of high credit risk

for investment-grade CDS. Economically, this suggests that protection sellers demand an

additional premium in excess of the fair credit risk premium when setting their askquotes. In the subinvestment-grade sample, the evidence is mixed. On the one hand, the

CDS market shows a higher average liquidity premium than for the investment-grade CDS

market. On the other hand, liquidity premia can become negative as credit risk rises for

badly rated debt. This implies that investors increasingly use the CDS market to take on

synthetic credit risk as the liquidity of the bond market dries up.

Overall, we observe declining default risk, a slightly increasing bond market liquidity and

an increase in liquidity in conjunction with a more symmetrical distribution of liquidity

premia in the CDS market. From an economic perspective, this agrees with the evolution

and the standardization of the CDS market. The asymmetrical distribution of the liquiditypremia between protection buyers and sellers in the CDS market indicates that CDS mid

premia are not an appropriate measure of the pure credit risk component. In a firm-specific

analysis, we show that restricting stochastic liquidity to the bond market can result in

price surcharges and yield discounts for corporate bonds. This effect, however, becomes

less apparent as CDS market liquidity evolves.

An issue not addressed in this paper is that CDS contracts are usually designed to allow

for a number of bonds deliverable upon the default of a given reference asset. Before

default, however, it is not clear which of the admissible bond will be cheapest to deliver.

The choice option of the protection buyer should also be priced in CDS premia.

As a second extension of our model, it is also possible to add information from the

stock market. Blanco et al. (2005) and Norden and Weber (2004) have explored the

information spillover between stock, CDS and bond markets, and their results suggest

that incorporating stock market information may facilitate the estimation of the default

intensity. It may be interesting to explore whether including this information in a

reduced-form model will render explicit liquidity-modelling in CDS premia unnecessary

26


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or whether, as our results suggest, our proposed extension of the existing reduced-form

models is vital if information from the CDS market is used.

27


28/38

References

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D. Aunon-Nerin, D. Cossin, T. Hricko, and Z. Huang. Exploring for the Determinants

of Credit Risk in Credit Default Swap Transaction Data: Is Fixed Income Markets

Information Sufficient to Evaluate Credit Risk? Working Paper, HEC-University of

Lausanne and FAME, 2002.

R. Blanco, S. Brennan, and I. W. Marsh. An empirical analysis of the dynamic relationship

between investment-grade bonds and credit default swaps. Journal of Finance, 60(5):

22552281, 2005.

D. Bongaerts, F. De Jong, and J. Driessen. Liquidity and liquidity risk premia in the cds

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British Bankers Association. 2003/2004 Credit Derivatives Report.

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R. Chen, X. Cheng, and L. Wu. Dynamic interactions between interest rate, credit, and

liquidity risks: Theory and evicence from the term structure of credit default swap

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P. Collin-Dufresne, R. S. Goldstein, and J. S. Martin. The determinants of credit spread

changes. Journal of Finance, 61(6):21772207, 2001.

F. De Jong and J. Driessen. Liquidity risk premia in corporate bond markets. Working

Paper, University of Amsterdam, 2005.

D. Duffie. Credit swap valuation. Financial Analysts Journal, 55:7387, 1999.

D. Duffie and J. Liu. Recursive valuation of defaultable securities and the timing of

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corporate bonds. Journal of Finance, 56(1):247277, 2001.

J. Ericsson and O. Renault. Liquidity and credit risk. Journal of Finance, 61(5):22192250,

2006.

European Central Bank Financial Stability Report June 2007. 2007.

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Appendix A. Analytical Solutions for the Discount Factors

The dynamics of the default and liquidity intensities are defined as follows. First, we

define the independent intensities and through the following system of stochastic

differential equations:dt

dt

=

t

dt +

t 0

0

dW,t

dW,t

, (16)

with constants , , , and and independent Brownian motions W and W. The

correlated intensities and are then defined asdt

dt

=

dt

+ f dtf dt + dt

= t + f

+ f t dt +

t f

f t dW,t

dW,t .We can rewrite the sum of the correlated intensities as a weighted sum of the independent

intensities:

t + t = 0 + 0 +

t0

ds +

t0

ds

= 0 + 0 +

t0

s + f

ds +

t0

+ f

t

ds

+t

0

s + f

s

dW,s +

t0

(f + )dW,s

= 0 + 0 +

t0

(1 + f) s

ds +

t0

(1 + f) ds

+

t0

(1 + f)

sdW,s +

t0

(1 + f) dW,s

=

=:(1+f)0

0 +=:(1+f)0

0 + (1 + f) t

0

ds

+ (1 + f)

t

0

ds

= (1 + f) t + (1 + f) t.

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Since and are independent, the joint expectation for the discount factors P(t1, t2)

and L (t1, t2) equals:

Et

P(t1, t2) L (t1, t2)

= Et

exp

t2t1

sds

exp

t2t1

sds

= Et expt2t1

s + sds= Et

exp

t2t1

(1 + f) s + (1 + f) sds

= Et

exp

t2t1

(1 + f) sds

Et

exp

t2t1

(1 + f) sds

=: P

(1 + f) , t1, t2

L

(1 + f) , t1, t2.

The dynamics of the scaled intensities f := (1 + f) and f := (1 + f) areidentical to those of the independent intensities with the process parameters adjusted:

tf = (1 + f) t

= (1 + f) 0 + (1 + f) t0

ds

= (1 + f) 0 =0

f

+

t0

(1 + f) =:f

(1 + f) s =s

f

ds

+

t0

1 + f =:f

(1 + f) s

=

sf

dW,s

= 0f + t0f sf ds + t

0fsfdW,s

dtf =f tf

dt + f

t

f dW,t,

tf = (1 + f) t

= (1 + f) 0 + (1 + f) t0

ds

= (1 + f) 0

=0f+

t0

(1 + f)

=:fds +

t0

(1 + f)

=:fdW,s

dtf = f dt + f dW,t ,

that is the scaled intensities also follow a square root process, respectively an arithmetic

Brownian motion. Thus, the following well-known analytical solutions arise for

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Pf, t1, t2

and L

f, t1, t2

:

Pf, t1, t2

:= a1 (t1, t2) exp

t1f a2 (t1, t2)

,

Lf, t1, t2 := a3 (t1, t2) exp t1

f

a4 (t1, t2) ,

a1 (t1, t2) =

1

1 exp[ (t2 t1)] 2

2

exp

(+ )

2(t2 t1)

,

a2 (t1, t2) = f2

+2

f2 ( exp[ (t2 t1)] 1) ,

a3 (t1, t2) = exp

f

2 (t2 t1)36

+f (t2 t1)2

6

,

a4 (t1, t2) = t2 t1,

=

2f2 + 2,

=+

.

The bond and CDS pricing equations do not only contain the expectation of the

simultaneous default risk and liquidity discount factors but also Et

P(t, ti1) L (t, ti)

.

Since and are correlated, we also have to determine the expectation of thisnon-simultaneous discount factor. Wlg, assume that t = t1, ti1 = t2 and ti = t3. Then,

the definition of P and L implies that

P(t1, t2) L (t1, t3) = exp

t2t1

sds

exp

t3t1

sds

= exp

t2t1

sds t2t1

sds t3t2

sds

= exp

t2t1

(1 + f)s + (1 + f) s

ds t3t2

sds

.

We also know that

s = 0 +

s0

du

= 0 + f 0 +

s0

du + f du

= s + f s.

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Therefore, we can rewrite the above exponential function as

exp

(1 + f) t2t1

sds

=:I (1 + f)

t2t1

sds

=:II f

t3t2

sds

=:II It3t2

sds

=:IV

for which we need to determine expectation at time t1. Applying the law of iterated

expectations, we obtain

Et1 [exp(I II III IV)] = Et1 [Et2 [exp(I II III IV)]]= Et1 [exp(I II) Et2 [exp(II I IV)]]

= Et1

exp(I II) Et2 [exp(II I)]

=P(f,t2,t3)

Et2 [exp (IV)]

=L(,t2,t3)

= Et1 exp(I) Pf , t2, t3

Et1

exp(III) L, t2, t3

=: P(f, t1, t2) L

lf, t1, t2

,

where the first equality follows from the law of iterated expectations, the second from the

fact that I and II are known at t2 and the third and fourth from the independence of

and . The last two expectation terms are the moment-generating functions of and

with the following well-known solutions:

Et1exp(I) Pf , t2, t3 = a1 (t2, t3) Et1 exp (1 + f) t2

t1

sds

expa2 (t2, t3) f t2

= a1 (t2, t3) b1 (t2, t3) exp

t1 b2 (t2, t3)

,

Et1

exp(II I) L

, t2, t3

= a3 (t2, t3) Et1

exp

(1 + f)

t2t1

sds

expa4 (t2, t3) t2= a3 (t2, t3) b3 (t2, t3) exp

t1 b4 (t2, t3)

,

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b1 (t2, t3) =2 exp

t3t22 ( + )

2 f a2 (t2, t3)(exp[ (t3 t2)] 1) + + exp [ (t3 t2)] ( + )

2

2

,

b2 (t2, t3) =

f

a2 (t2, t3) [ + + exp [ (t3

t2)] (

)] + 2 (1 + f)(exp[ (t3

t2)]

1)

2 f a2 (t2, t3)(exp[ (t3 t2)] 1) + + exp [ (t3 t2)] ( + ) ,

b3 (t2, t3) = exp

1

62 (1 + f)

2 (t3 t2)3 + (1 + f)2

2a4 (t2, t3) +

(t3 t2)2

+2a4 (t2, t3) + 2

2a4 (t2, t3) (t3 t2)

,

b4 (t2, t3) = a4 (t2, t3) + (1 + f) (t3 t2) .

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35/38

Table1:ReferenceEnt

itiesbyRatingClassandIn

dustrySector

Thetableshowsth

enumberofreferenceentitiesineach

ratingclassandindustrygroup.

Rat

ingsareaveragesforthereferenceent

ityover

timewhenbothCD

Spremiaandatleast2bondyields

wereobserved.

Thelastcolumnsand

rowsshowthenumberofobservedm

idbond

yieldsandmidCD

Spremia.

ThenumberofsyntheticalbondyieldsmatchedtotheCDSc

ontractmaturityequalsthenumber

ofCDS

observationsandis

thereforesuppressed.

AAA

AA

A

BBB

BB

B

All

#

Obs.Bonds

#

Obs.CDS

Basi

cMaterials

-

2

4

7

2

1

16

33,3

93

13,0

79

Com

munication

-

1

7

8

3

-

19

73,2

11

20,4

81

Cycl.Cons.Goods

-

2

3

9

2

-

16

47,4

97

15,6

34

Diversified

-

-

2

2

-

-

4

6,5

36

3,0

96

Financial

-

22

28

4

-

-

54

175,8

70

38,0

46

Industrial

-

-

4

5

-

-

9

40,6

24

9,5

31

Noncycl.Cons.Goods

-

-

5

8

1

-

14

40,5

19

12,3

19

Sovereign

5

1

3

3

2

2

16

55,1

45

6,5

94

Utility

1

5

13

4

-

-

23

79,6

04

19,0

36

All

6

33

69

50

10

3

171

552,3

99

137,8

16

#O

bs.Bonds

40,4

14

137,7

02

193,1

95

161,1

82

16,3

99

3,5

07

552

,399

#

Obs.CDS

2,9

46

29,1

29

55,2

21

41,5

27

7,9

62

1,0

31

137

,816

35


36/38

Table 2: Estimated Credit Risk and Liquidity Premia

The table shows the mean, standard deviation, minimum and maximum for

the credit risk and the liquidity premia components for each rating class. All

values are in basis points.

AAA AA A BBB BB B CCC All

csdef 6.11 13.05 28.53 55.33 249.52 349.97 250.83 47.76

Std. Dev.(csdef) 4.46 10.60 28.35 65.59 242.82 167.45 34.96 83.71

min(csdef) 2.96 2.96 8.32 11.20 33.08 32.60 111.92 2.96

max(csdef) 52.91 260.85 260.85 1,201.93 1,948.43 1,175.58 397.29 1,948.43

csliq 1.55 12.60 18.12 42.81 62.58 83.68 71.23 26.54

Std. Dev.(csliq) 3.10 30.33 43.56 58.57 64.51 47.13 28.94 48.10

min(csliq) 0.66 3.09 3.09 4.87 21.07 14.50 1.50 0.66

max(csliq) 30.78 508.81 495.96 349.34 537.66 296.79 194.23 537.66

sdef

6.38 13.72 28.95 56.89 252.92 351.04 252.20 49.38Std. Dev.(sdef) 4.41 18.30 28.36 64.67 2

Documents

CDS and Bond Liquidity