derivative pricing with liquidity risk

8/7/2019 Derivative Pricing With Liquidity Risk

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THE JOURNAL OF FINANCE VOL. LXVI, NO. 1 FEBRUARY 2011

Derivative Pricing with Liquidity Risk: Theory

and Evidence from the Credit DefaultSwap Market

DION BONGAERTS, FRANK DE JONG, and JOOST DRIESSEN

ABSTRACT

We derive an equilibrium asset pricing model incorporating liquidity risk, derivatives,

and short-selling due to hedging of nontraded risk. We show that illiquid assets can

have lower expected returns if the short-sellers have more wealth, lower risk aversion,

or shorter horizon. The pricing of liquidity risk is different for derivatives than forpositive-net-supply assets, and depends on investors net nontraded risk exposure.

We estimate this model for the credit default swap market. We find strong evidence

for an expected liquidity premium earned by the credit protection seller. The effect of

liquidity risk is significant but economically small.

THE RELATION BETWEEN LIQUIDITY and asset prices has received considerableattention recently. It is often thought that illiquid assets always trade at dis-counted prices. In this paper we show that short-selling dramatically changesthe effect of liquidity on asset prices. In particular, illiquid assets can tradeat higher prices than liquid assets. Short-selling is relevant in many markets.For example, it is of crucial importance when analyzing derivative marketssince derivatives are in zero net supply, but short-selling can also be presentin markets with positive net supply for example due to hedging pressures.We derive new and testable implications for the pricing of liquidity and testthese empirically for one of the key derivative markets, the credit default swapmarket.

Our paper has two main contributions. The first contribution is a new equi-librium asset pricing model that incorporates liquidity risk and short-selling

Bongaerts is with Finance Group, RSM Erasmus University Rotterdam, and De Jong and

Driessen are with the Department of Finance, Netspar, Tilburg University. A previous version of

this paper was circulated under the title Liquidity and Liquidity Risk Premia in the CDS Market.

We are grateful to Moodys KMV and Graeme Wood for providing the MKMV EDF data. We would

like to thank Andrea Buraschi, Long Chen, Mike Chernov, Lasse Pedersen, Olivier Renault, Ilya

Strebulaev, Marti Subrahmanyam, Akiko Watanabe, Avi Wohl, and participants in the Gutmann

Symposium 2007, EFA 2007, Workshop on Default Risk Rennes 2007, the WFA meeting 2008, 2008

SIFR conference on credit markets, CREDIT 2008 Venice conference, Inquire Europe Symposium

2008, Erasmus Liquidity Conference and seminars at Imperial College, Maastricht University, Free

University of Amsterdam, University of Kaiserslautern, University of Rotterdam, ABN-AMRO,

Bank of England, the European Central Bank, Yale University, and Lancaster University. We

especially thank Campbell Harvey, an anonymous associate editor, and two anonymous refereesfor many helpful comments and advice.

203


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204 The Journal of Finance R

due to hedging of nontraded risk. This model builds on the Liquidity Capi-tal Asset Pricing Model (CAPM) of Acharya and Pedersen (2005) (AP), whoonly consider investors with long positions in assets that are in positive netsupply, in which case illiquidity always leads to lower asset prices and higher

expected returns. We show that if some investors optimally short assets, theeffects of liquidity are more complicated and can be zero, positive, or negative.Our theory thus contributes to the literature on asset pricing and liquidity.1

In our equilibrium framework, heterogeneous investors use some assets tohedge exposure to a nontraded risk factor. We derive that the expected returnon these hedge assets can be decomposed into priced exposure to nonhedgeasset returns, hedging demand effects, an expected liquidity component, andseveral liquidity risk premia. Our first theoretical result is that the sign ofthese liquidity effects depends on heterogeneity in investors nontraded riskexposure, risk aversion, horizon, and wealth. For example, illiquid assets can

have lower expected returns if the short-sellers are more aggressive (due to ei-ther higher aggregate wealth or lower risk aversion) or have a shorter horizonthan the long investors. The intuition for this result is that the aggressive in-vestors are most sensitive to transaction costs and thus need to be compensatedfor these costs in equilibrium. This result holds for both positive-net-supply as-sets and derivatives. Our second main theoretical result is that, if the hedgeassets are derivatives in zero net supply, some of the liquidity risk covari-ances have a zero premium. Specifically, the covariance between the return ona derivative (net of transaction costs) and the return on the total derivativemarket return (net of market-wide transaction costs) has a zero premium in

case of zero-net-supply assets. However, another liquidity risk covariance, thecovariance between transaction costs on a given derivative and the return onthe nontraded risk factor, does carry a risk premium, whose sign depends oninvestors nontraded risk exposures.

Our model also generates interaction effects between hedging demand andliquidity premia. We show that, on the one hand, increased hedging demandimplies that some investors will short hedge assets for hedging purposes, butthat, on the other hand, this hedging pressure pushes up expected returns,which increases the speculative demand of investors. The net effect can beeither an increase or a decrease in the net aggressiveness of the long versus

short investors and the premium on expected liquidity. If hedging demandvaries across assets, the model thus generates cross-sectional effects in thepricing of liquidity and we test for such effects empirically. Of course, we alsoget a standard hedge pressure effect in our model because the covariance ofan assets return with the nontraded risk factor is priced. We thus add toexisting work on hedging pressures in derivative markets (see, e.g., De Roon,

1 For the equity market, the pricing of liquidity risk has been studied by Amihud (2002), Acharya

and Pedersen (2005), Pastor and Stambaugh (2003), Bekaert, Harvey, and Lundblad (2007), and

Korajczyk and Sadka (2008), amongst others. de Jong and Driessen (2007), Downing, Underwood,

and Xing (2005), and Nashikkar and Subrahmanyam (2006), amongst others, study the pricing ofliquidity in corporate bond markets.


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Derivative Pricing with Liquidity Risk 205

Nijman, and Veld (2000) and Garleanu, Pedersen, and Poteshman (2009)) byincorporating liquidity effects.

Our second main contribution is an empirical test of the theoretical frame-work above for an important class of derivative assets, namely credit default

swaps (CDS). The CDS market is one of the largest derivative markets, witha total notional amount of approximately 31 trillion USD as of August 2009according to the Depository Trust & Clearing Corporation. This has inducedresearchers and practitioners to use CDS spreads as pure measures of defaultrisk (see, e.g., Longstaff, Mithal, and Neis (2005) and Blanco, Brennan, andMarsh (2005)). However, as we discuss further, we find that part of the CDSspread reflects liquidity effects.

Empirical work on liquidity and derivative assets is scarce. Two recent pa-pers empirically assess the impact of liquidity on CDS spreads, Tang and Yan(2007) and Chen, Cheng, and Wu (2005). Tang and Yan (2005) regress CDS

spreads on variables that capture expected liquidity and liquidity risk, andfind that illiquidity leads to higher spreads. Chen et al. (2005) estimate theimpact of liquidity and other factors on CDS spreads using a term structureapproach, and find that premia for liquidity risk and the expected liquiditypremium are earned by the CDS buyer. Notice that in Chen et al. the identi-fication of the liquidity risk premium comes from the term structure of CDSspreads, whereas our method follows the standard procedure of identifying riskpremia from expected excess returns. In other recent work, Das and Hanouna(2009) develop a framework in which lower equity market liquidity leads tohigher CDS prices and confirm this mechanism empirically. Buhler and Trapp

(2009) estimate a reduced-form CDS pricing model that allows for the inter-action of CDS and bond liquidity effects. Turning to other derivative markets,Cetin et al. (2006) add illiquidity to the standard Black-Scholes frameworkand Brenner, Eldor and Hauser (2001) investigate the effect of nontradabilityon currency derivatives. Deuskar, Gupta, and Subrahmanyam (2009) find em-pirically that illiquid interest rate options trade at higher prices than liquidoptions. Our paper contributes to this literature by developing an equilibriumpricing model and testing this structural asset pricing model on the CDS mar-ket. Our use of an asset pricing model allows for an immediate interpretationof our results in terms of hedging pressure effects, liquidity, and liquidity risk

premia.In addition to our two main contributions, the empirical analysis makesseveral methodological contributions. We derive expressions for realized andexpected excess returns on CDS positions. In particular, we show how onecan construct expected excess returns from the CDS spread level, correctedfor the expected loss (where we use Moodys-KMV Expected Default Frequen-cies (EDFs) to assess default probabilities). As argued by Campello, Chen,and Zhang (2008) and de Jong and Driessen (2007), this procedure gives moreprecise estimates of expected returns than averaged realized returns. On theeconometric side, we use a repeat sales method to construct unbiased and

efficient estimates of portfolio CDS returns from the unbalanced panel of indi-vidual CDS quotes.


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We use a representative data set of CDS bid and ask quotes for U.S. firmsand banks over a relatively long period (20042008). We only rely on the moststandard 5-year contracts. Applying the repeat sales method to these data, weconstruct CDS returns and bid-ask spreads for portfolios sorted on a range of

variables capturing default risk and liquidity. The level and innovations of thebid-ask spreads are used to construct measures of expected liquidity and liquid-ity risk. We estimate our asset pricing model using the Generalized Method ofMoments and find significant exposure to expected liquidity and liquidity risk.Our key empirical finding is that sellers of credit protection receive illiquiditycompensation on top of the compensation for default risk. Most of the liquidityeffect comes from the expected liquidity component, and the effect of expectedliquidity implies that credit protection sellers are more aggressive than protec-tion buyers, either due to more wealth, lower risk aversion, or shorter horizons.The theoretical model can fully explain the empirical effect of expected liquid-

ity provided that holding periods are not too long. We also confirm that severalparameter restrictions imposed by the theory are not violated by our empiricalestimates. For example, the theory predicts that the covariance between deriva-tive transaction costs and derivative market-wide transaction costs carries azero premium, and we cannot reject this empirically.

We perform an extensive series of robustness checks that support our results.Our benchmark results imply that the expected liquidity effect on expected re-turns is 0.175% per quarter on average. Across all robustness checks, the lowestexpected liquidity effect is 0.060% per quarter, still strongly statistically sig-nificant. We also control for corporate bond market liquidity and find evidence

of hedging demand effects that are in line with our theory, with some evidencefor the interaction between hedging demand and liquidity effects as discussedearlier.

Our results are robust to including the last two quarters of 2008. In thiseconomic crisis period, CDS spreads and bid-ask spreads increase dramatically,in part due to a strong increase in counterparty risk and deleveraging of manyfinancial institutions. The liquidity effects remain statistically significant andeven become larger in some cases.

The remainder of this paper is structured as follows. Section I introducesour theoretical model. Section II discusses in detail how the model can be

applied to and estimated for the CDS market. Section III discusses the data andconstruction of CDS returns and liquidity measures, and Section IV presentsthe results of the empirical analysis. Section V concludes.

I. A Model for Pricing Liquidity in Asset and Derivative Markets

A. Motivation

In this section, we derive an asset pricing model in a setting with transactioncosts, heterogeneous agents, and multiple assets. The model has two key ingre-

dients that differentiate it from the liquidity CAPM of Acharya and Pedersen(2005). First, some of the assets can be derivative assets in zero net supply.Second, agents have exposure to a nontraded risk factor. We introduce this


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nontraded risk factor so that, in equilibrium, a fraction of the agents optimallyhold short positions in some assets in order to hedge the nontraded risk ex-posure. This contrasts with AP, whose assumptions imply that all agents holdlong positions in equilibrium.

After deriving the implications of this benchmark setup, we study two ex-tensions. First, we incorporate long-horizon investors who are not exposed totransaction costs. Second, we allow part of the nontraded risk to be idiosyn-cratic.

In the empirical analysis, we apply our model to a setting where agentshave a nontraded exposure to credit risk. For example, commercial banks haveexposure to nontraded bank loans and illiquid corporate bonds, which theycan partially hedge using CDS contracts. Other agents, such as hedge fundsor insurance companies, have little or zero exposure to nontraded credit riskand may sell CDS contracts to commercial banks, thus earning a credit risk

premium.Even though we test our model empirically on a derivative market, the theory

applies more generally to asset markets with positive supply, but where someagents short assets to hedge their nontraded risk exposure. A pricing modelsimilar to the liquidity CAPM of AP is a special case of our model, since in

AP all assets are in positive net supply and agents have no nontraded riskexposure.

B. Assumptions and Notation

1. Agents. The economy has N agents with CRRA preferences. Agents livefor one period, invest at time t 1, and consume at t. This assumptionis relaxed in one of the model extensions. We allow for heterogeneity inrelative risk aversion i and initial wealth wi.

2. Traded assets. There areKtraded assets, divided into two subsets withoutloss of generality. For the first subset of Kb assets, all agents optimallyhold long positions in equilibrium. We refer to these assets as basic assetsor nonhedge assets. The second subset has Kh assets, and some agentsoptimally hold short positions in these assets. We refer to these assets ashedge assets. For simplicity, each investor can be either optimally long

in all hedge assets or short in all hedge assets.2

Again, this assumptionis relaxed in one of the model extensions. Using i to denote a diagonalmatrix with the sign of the hedge asset positions of investor i on thediagonal, we thus have either i = I or i = I.

3 Aggregate supply is de-noted by Sb for nonhedge assets and Sh for hedge assets. Obviously, fornonhedge assets, supply has to be positive. For hedge assets, aggregatesupply could be zero, in which case these are derivative assets, but it can

2 This assumption is made only to generate tractable expressions that allow for a simple empir-

ical test of liquidity effects. It is straightforward to include more than two groups in the theory.3 Of course, the sign of the optimal portfolio weights is determined in equilibrium. In the Ap-

pendix, we show the necessary conditions (equation (23)). Essentially, the long group of investorsshould have zero or small hedging demand while the short group needs to have sufficient hedging

demand to ensure that they always short derivatives.


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also be nonzero. To maintain tractability, we assume that all asset returnsare jointly lognormally distributed.4

3. Nontraded risk exposure. Agent i has an exposure ofqi to nontraded back-ground risk with log-return equal to rn,t, where rn,t is the return on a

single nontraded risk factor. This assumption is necessary to generatenonzero positions in hedge assets. If the exposure qi varies across agentsor if agents have different risk aversion, their hedging demands will dif-fer and they will hold different hedge asset positions if the hedge assetreturns correlate with rn,t. For example, agents with large positive qi mayhold short positions in hedge assets, while agents with zero or negativenontraded exposure take long positions in the hedge assets. We think ofrn,t as the return on a diversified portfolio of very illiquid assets that arenot traded in equilibrium. For our credit risk application, these can bebank loans or illiquid corporate bonds.

4. Transaction costs. Following AP, the one-period investors investing at t 1pay transaction costs proportional to the current price when closing theposition at time t. Transaction costs are stochastic. The percentage costsare denoted by the Kb-dimensional vector Cb,t for nonhedge assets and theKh-dimensional vector Ch,t for hedge assets. Transaction costs representboth the bid-ask spread and search costs, which are relevant in over-the-counter markets (see Duffie, Garleanu, and Pedersen (2005)). Both longand short holders pay transaction costs. This can be motivated by theexistence of (implicit) market makers who only play a role as interme-diary, that is, they earn the bid-ask spread and hold zero net positions

in the hedge assets. Alternatively, our assumption that both long andshort investors pay transaction costs is appropriate in a setting withoutmarket makers if transaction costs fully represent search costs. The netlog-returns of the agents (in excess of the risk-free rate) are then given bythe vectors rb,t cb,t for the basic assets, rh,t ch,t for the investors who arelong in the hedge assets, and rh,t ch,t for the investors who are short inthe hedge assets, with log-transaction costs defined as ch,t = ln(1 Ch,t).

C. Asset Pricing Implications: Benchmark Model

We now derive the main asset pricing implications for this benchmark econ-omy. We focus on the implications for the hedge asset returns, since these arethe relevant assets for our empirical analysis. Using the standard approxi-mation of log-portfolio returns (Campbell and Viceira (2002), p. 29), investor iwith CRRA preferences and relative risk aversion i trades off the mean andvariance of the log-portfolio return. As shown in the Appendix, investor i thusmaximizes

4 In this one-period setup, similar asset pricing implications are obtained without distributional

assumptions if we assume mean-variance investors. The assumption of CRRA investors and log-

normal returns allows us to extend the model to a setting with long-horizon investors in a simpleway.


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xiE

rb cb +

1

22b

+ yiE

rh ich +

1

22i,h

1

2ivar(x

i(rb cb)

+yi(rh ich) + qirn), (1)

over positions xi in nonhedge assets and positions yi in hedge assets, where 2bis the diagonal of var(rb cb) and

2i,h the diagonal of var(rh ich). To simplify

notation, we drop all time subscripts. In particular, note that the expectationand variance in (1) are conditional on the information set at time t 1.

Imposing the market clearing condition and the optimality conditions foreach investor, we can derive the asset pricing equation for the hedge assets.

THEOREM 1: Given Assumptions 14, and N investors maximizing the func-tion in (1), and using a Taylor expansion of var (rh ch) var(rh i ch)

1 around

ch = 0, in equilibrium the vector with expected excess log-returns on the hedge

assets, E(rh), satisfies

E (rh) +1

22rh + Fh AE(ch) =

cov(rh, rn) Acov(ch, rn)

+ 0

E(ch) AE(rh +

1

22rh )

+ 0

cov(ch, rn) Acov(rh, rn)

+ (I H1)

1cov(rh ch, rm cm),

(2)

where Fh is a vector of convexity corrections (see the Appendix) and

rh = rh rh,rb (rb cb), ch = ch ch,rb (rb cb), (3)

where rh,rb = var(rb cb)1cov(rh, rb cb) and ch,rb = var(rb cb)

1cov(ch, rb cb). The market-wide hedge asset return and transaction costs are defined asrm = Sh

rh/ Sh and cm = S

hch/

Sh if Sh = 0 and are equal to zero otherwise. Thefollowing asset pricing parameters are scalars, and are defined as

=1

Ni=1

wiqi

0 = 1

i:i =I

wi 1

i

i:i =I

wi 1

i

0 =1

i:i =I

wiqi

i:i =I

wiqi

=1

Sh 0

=

N

i=1

wi 1

i > 0

(4)

where A = (I H1)1H1, H1 = (C + C

)var(rh)1, and C = cov(ch, rh).


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Proof: See the Appendix.

Note that the expected return on the hedge assets also enters on the right-hand side of equation (2), and hence the expected returns are implicitly defined

in (2).5This theorem shows that expected returns on the hedge assets are deter-

mined by five terms. First, from the definition of rh in equation (3) we obtainthe standard result that covariance with the nonhedge asset returns is pricedif these nonhedge assets carry a risk premium (E(rb cb) > 0).

The second term concerns the covariance of hedge asset returns (orthogo-nalized for nonhedge asset returns) with the nontraded risk factor returns rn.Given that > 0, the sign of the coefficient on this covariance depends on thesign of

i wiqi . For example, if aggregate exposure to nontraded risk is pos-

itive, we obtain the intuitive result that assets that have positive covariance

with the nontraded risk factor have high expected returns, in a similar way asin De Roon, Nijman, and Veld (2000).

The third term captures expected liquidity. The sign of this effect dependson 0. The model implies that if the more wealthy or less risk-averse investorshave long positions in the assets, the coefficient on expected transaction costsis positive and the long holders earn an expected liquidity premium. In thenext section, we illustrate this in a simple example.

The fourth term contains a coefficient 0 on the covariance of hedge assettransaction costs (orthogonalized for nonhedge asset returns) with rn, whichdepends on the nontraded risk exposure of the short versus long agents. If,

for example, the short agents have positive nontraded risk exposure whilethe long agents have zero nontraded risk exposure, 0 is positive. In thiscase, only the short agents care about covariance with rn. If the covariancebetween costs ch and rn is negative, the nontraded risk is amplified by the costsbecause the return on the short position equals rh ch. This weakens thehedge quality of the hedge asset, thus decreasing the short agents demandfor hedge assets, which in turn lowers the expected return. This is the effectthat we find empirically.

The final term in (2) only enters if aggregate supply Sh is positive. In thiscase, the covariance with the net market-wide return of hedge assets is priced

since the agents have to hold the positive supply in equilibrium. This term isequal to zero if the hedge assets are in zero net supply ( Sh = 0).All expectations and covariances in (2) are corrected for the covariance ma-

trix between the hedge asset returns and costs through the matrix A. In theexample later we provide more intuition for the presence of this correction ma-trix A. If the matrix cov(ch, rh) is equal to zero, we have A = 0 and (2) simplifiesto the linear asset pricing equation

5 It is straightforward to show that equation (2) reduces to a model similar to APs Liquidity

CAPM ifSh > 0,Kb = 0, qi = 0, and i = Ifor all agents. In this case and 0 are equal to zero andall the liquidity risk premia are in the term (I H1)

1cov(rh ch, rm cm).


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E (rh) +1

22rh + Fh = 0E(ch) + cov(rh, rn) + 0cov(ch, rn) + cov(rh ch, rm cm).

(5)

In Section II.B, we discuss in detail how the general model in equation (2) canbe estimated using the Generalized Method of Moments (GMM).

D. Example

We provide more intuition for the expected liquidity effect by a simple exam-ple with constant transaction costs c. Let there be one hedge asset with returnr and variance var(r) = 2, and let there be two investors. One investor (agent1) has positive exposure to the nontraded risk factor rn and the other investor(agent 2) has zero exposure, hence q1 > 0 and q2 = 0. There are no other as-

sets. If agent 1 is short and agent 2 is long, the investors asset demands (as afraction of wealth) are

y1 =E(r) +

1

22 + c

12

cov(r, rn)q1/2

y2 =E(r) +

1

22 c

22.

(6)

In a zero-net-supply market, the wealth-weighted demands have to add up to

zero, w1y1 + w2y2 = 0, which gives the equilibrium expected returns

E(r) +1

22 =

cov(r, rn)w1q1

w11

1 + w21

2

+

w2

12 w1

11

w11

1 + w21

2

c = + 0c, (7)

with the compensation for transaction costs determined by the coefficient 0.For this equilibrium to hold, we need to make sure that y1 < 0 and y2 > 0in equilibrium. These inequalities are satisfied if cov(r, rn)q11 > 2c, which re-quires that the hedge demand has to be sufficiently large relative to the costs cand the speculative demand (which depends on 1). In the Appendix, we show

the restrictions needed in the general setting.Figure 1 illustrates the equilibrium for two cases, c = 0 and c > 0. The figuredepicts (minus) the asset demand of agent 1 (w1y1) and the asset demandof agent 2 (w2y2), and is drawn such that agent 2 is more aggressive (i.e.,less risk averse or more wealthy, w2

12 > w1

11 ) and hence her speculative

asset demand is steeper than that of agent 1. The equilibrium expected returnis obtained at the point where the two lines cross, w1y1 = w2y2 or w1y1 +

w2y2 = 0. For c > 0, both lines shift downward as transaction costs make theinvestors want to invest less (in absolute value). The lines now cross at adifferent point, where the expected return is higher by 0c. Due to higher wealth

and/or lower risk aversion, the demand of agent 2, the more aggressive agent,is more sensitive to transaction costs than that of agent 1. Therefore, when


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Figure 1. Asset pricing equilibrium with transaction costs. The figure illustrates the asset

pricing equilibrium with constant liquidity costs, as in the example of Section I.D. The figure

graphs (minus) the asset demand of an investor with hedging needs (w1y1) who is short in the

asset, and of a less risk-averse investor with no hedging demand ( w2y2) who is long. The solid

lines reflect the situation without transaction costs; the dashed lines reflect the situation with

transaction costs c. And is the equilibrium expected return without transaction costs; + 0c is

the equilibrium expected return with transaction costs.

transaction costs increase, the asset demand of agent 2 decreases more stronglythan the asset demand of agent 1. To persuade agent 2 to invest a sufficientamount in the risky asset, the expected return needs to increase so that 0cmust be positive. Obviously, if instead agent 1 were more aggressive than agent2, we would get the opposite effect: 0 is negative, implying a negative relationbetween expected returns and costs. The Internet Appendix illustrates thesecond case.6

Allowing for stochastic transaction costs, this two-investor example canbe used to shed light on the role of the correction matrix A for expectedreturns and costs in equation (2). In this simple example we have A =

(1 2cov(c, r)/V(r))1

2cov(c, r)/V(r). This leads toE(r)

E(c)=

0 + A

1 + 0A=

0 + 1

1 + 2(0 1)cov(c, r)/V(r) 1, (8)

which has positive dependence on cov(c, r) since |0| 1. For example, em-pirically we typically have cov(c, r) < 0. This negative covariance lowers thespeculative demand of agent 2, since it increases the variance of her net returnr c, while the speculative demand of agent 1 increases, since the return onher short position is r c. In the context of Figure 1, negative covariance thus

6

An Internet Appendix for this article is available online in the Supplements and Datasetssection at http://www.afajof.org/supplements.asp.


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flattens the demand of agent 2 and steepens the demand of agent 1, leading toa lower coefficient on E(c) and a lower expected return for agent 2. This effectoccurs even when both agents have the same wealth and risk aversion (i.e., if

0 = 0).

E. Model Extension I: Long-Horizon Investors

We extend the model presented earlier in two directions. Our first extensionis that we allow some investors to have a long horizon. Specifically, we add M

investors with CRRA preferences and an investment horizon up to the expi-ration of the contract, so that they are not exposed to transaction costs. Forsimplicity, we assume that any intertemporal rebalancing is costless for theseinvestors.7 In addition, in each period N one-period investors enter the marketand close their positions at the end of the period, incurring transaction costs

at that moment. In order to generate a stationary equilibrium, we assume thatall returns and costs are independent and identically distributed (IID) overtime and that the characteristics of the one-period investors are the same eachperiod.

This setup captures in a simple way the fact that some investors essentiallyhold buy-and-hold portfolios and thus incur no transaction costs, while otherinvestors trade frequently in financial markets, due to factors such as externalconstraints, liquidity shocks, investor-specific preferences, or investor beliefs.

Given these assumptions, the optimal portfolio choice of the long-horizoninvestors is constant over time and equal to the myopic portfolio rule. The

equilibrium thus generates the following asset pricing implications:

THEOREM 2: Given Assumptions 14, N investors maximizing the functionin (1), and M long-horizon investors, and using a Taylor-expansion of var(rh ch)var(rh i ch)

1 around ch = 0, in equilibrium the Khdimensional vectorwith expected excess log-returns on the hedge assets, E(rh), satisfies

E(rh) +1

22rh + Fh =

1

1 + 0

EBM

rh +

1

22rh

+ Fh

+

1

1 + 0cov(rh, rn), (9)

where EBM(rh) denotes the expected returns that hold in the benchmark model

(Theorem I),

0 =1

N+Mi=N+1

wi 1

i

1 =1

N+Mi=N+1

wiqi,

(10)

7

As shown later, these investors have constant portfolio weights in equilibrium. We thus neglectthe transaction costs needed to maintain constant portfolio weights.


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and, as before, =N

i=1

wi 1

i .


The effect of the presence of long-term investors is twofold. First, given thatthe long-term investors do not care about transaction costs, all terms involv-ing expected transaction costs and covariation with costs (liquidity risk) aredepreciated by the term 1

1+0, with 0 capturing the aggressiveness of long-

term investors relative to short-term investors. Second, the risk premium onthe nontraded risk factor rn is adjusted to reflect the exposure of long-terminvestors.

An interesting feature of this extension is that it can generate liquidityeffects even if investors are identical in terms of wealth and risk aversion. The

coefficient on expected liquidity is

0 =

i:Ti =1,i =I

wi 1

i

i:Ti =1,i =I

wi 1

i

i:Ti =1

wi 1

i +

i:Ti =L

wi 1

i

, (11)

where Ti = 1 denotes one-period investors and Ti = L denotes long-horizoninvestors. In the model with only short-term investors, the expected liquidity

coefficient 0 is zero if buyers and sellers have the same aggregate wealth-weighted risk aversion. However, if the sellers are long-term investors and thebuyers are short-term investors, the expected liquidity effect is positive. In thiscase, only the short-term buyers care about transaction costs and hence needto be compensated for these costs by the long-term sellers.

F. Model Extension II: Idiosyncratic Background Risk and Hedging Demand

The second extension incorporates the possibility that (part of) the nontradedbackground risk is idiosyncratic. To do so, we assume that each investor i has

exposure qi to background risk with log-return equal to rn,t + i,t, where i,t isindependent from rn,t and independent across investors but is potentially cor-related with asset returns. In addition, we introduce a third group of investorsto generate heterogeneity in hedging demand across hedge assets. This groupof investors is long in a fixed subset of the hedge assets and short in the otherhedge assets. For this long/short group we have i = D, where D is a diagonalmatrix with diagonal elements equal to 1 or 1. This leads to an interactioneffect between hedging demand and liquidity effects as discussed further.

THEOREM 3: Given Assumptions 14, the presence of idiosyncratic back-

ground risk, the presence of long/short investors, all investors maximizingthe function in (1) and using a Taylor expansion of var(rh ch)var(rh i ch)1


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around ch = 0, in equilibrium the Khdimensional vector with expected excesslog-returns on the hedge assets, E(rh),satisfies

E(rh) = EBM(rh)

+ 1

B1E(ch) B2E

rh +

1

22rh

+ 1[B1cov(ch, rn) B2cov(rh, rn)]

+1

(I H1)

1

i

wiqiI 2H11i =I + (H2 H1)1i =D

cov(rh ch, i)

,

(12)

where EBM(rh) denotes the expected returns that hold in the benchmark model(Theorem I), 1 is the indicator function,

1 =1

i:i =D

wi 1i 0

1 = 1

i:i =D

wiqi ,

(13)

and H2 = (DC + CD)var(rh)

1, B1 = (I H1)1(D + H2D H1 H1D), and

B2 = (I H1)1H2. If C = cov(ch, rh) = 0, equation (12) simplifies to

E(rh) = EBM(rh) + 1DE(ch) + 1Dcov(ch, rn) +1

i

wiqicov(rh ch, i)

.

(14)


Theorem III adds three new terms to the model. First, the final term inequation (12) and (14) shows how the presence of idiosyncratic background riskgenerates asset-specific and investor-specific hedging pressures. For example,if a wealthy investor has large and positive qicov(rjh cjh, i) for asset j, thisinvestor will likely short the asset, paying a premium to those investors willing

to buy the asset and thus increasing the assets expected return.The second key implication of Theorem III is that the introduction oflong/short investors generates additional effects for expected liquidity and liq-uidity risk. There are several reasons why some investors are optimally longin some hedge assets and short in others. For example, if they have only mod-erate nontraded risk exposure, they may short those hedge assets that covarymost strongly with the nontraded risk factor. In addition, the investor-specifichedging effects in Theorem III can also generate strong hedging demand foronly a subset of hedge assets.

The effect of investor-specific hedging effects on liquidity premia can be

seen most easily when C = 0. In this case, assets for which the long/shortinvestors are long have a coefficient on expected costs equal to 0 + 1 =


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(

i:i =Ii =Dwi

1i

i:i =I

wi 1

i )/. For the remaining hedge assets, thelong/short investors have a short position, and the expected liquidity coefficientis 0 1 = (

i:i =Iwi

1i i:i =Ii =D

wi 1

i )/. In both cases the coefficientcaptures the net aggressiveness of long versus short investors. A similar effect

occurs for liquidity risk.The assets in which the long/short investors are long or short depends on

a complex interaction between hedging demands and expected returns. Onthe one hand, high hedging demand for an asset implies that some investorswill short hedge assets to hedge. On the other hand, this hedging pressurepushes up expected returns, which increases the speculative demand of otherinvestors. The net effect in equilibrium can be either a long position or a shortposition. For example, if high hedging demand for an asset is caused by only afew agents who have large idiosyncratic background risk that covaries with thishedge asset, the net effect will be that most other investors will be long in this

hedge asset to exploit the higher expected return (resulting from the hedgingdemand). In this case, we expect the long/short investors to be long in theassets with high hedging demand, leading to a larger coefficient on expectedliquidity for these assets. Alternatively, it is possible that several investorsshare a high hedging demand for a particular hedge asset. Especially if theseinvestors have high risk aversion, their speculative demand will be small andthe net effect of the high hedging demand may be that most investors optimallyshort the hedge asset, thus decreasing the coefficient on expected liquidity. Inthe empirical analysis, we construct proxies for hedging demand to capturethis cross-asset heterogeneity and to assess which of the two effects prevails.

II. Applying the Theory to the CDS Market

A. Motivation and Theoretical Predictions

We apply the theory developed earlier to the market for credit risk. In thismarket, the key background risk is the systematic credit risk of illiquid corpo-rate bonds and bank loans. We focus on CDS contracts as assets to hedge thisrisk. In principle, one could also short liquid corporate bonds to hedge creditrisk exposure. However, in practice shorting corporate bonds may be difficult

and costly (see Nashikkar and Pedersen (2007)). In addition, Garleanu andPedersen (2009) document that margins on (unfunded) corporate bond posi-tions are considerably higher than those for CDS trades. We therefore abstainfrom including corporate bonds as hedge assets and focus on the pricing of CDScontracts.

Table I summarizes figures reported by Mengle (2007), providing an overviewof buyers and sellers in the CDS market. It shows that insurance companiesand funds (pension funds, hedge funds, and mutual funds) are net protectionsellers in the CDS market, while banks are net protection buyers for their loanportfolio. This suggests that banks typically use the CDS market to hedge credit

exposure, while institutional investors try to enhance their returns by buyingcredit risk and earning a credit risk premium. This supports our assumption


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Table I

Buyers and Sellers in the CDS MarketThis table shows the fraction of CDS contracts held by various parties.

Buy Protection (%) Sell Protection (%) Net Position (%)

BanksLoan portfolio 20 9 11

BanksTrading activity 39 35 4

Insurers 6 17 11

Funds 32 39 7

Hedge funds 28 32 4

Pension funds 2 4 2

Mutual funds 2 3 1

Corporates and other 3 2 1

Source: 2006 Survey of the British Bankers Association, reported in Mengle (2007).

that investors are either hedgers (always buying insurance) or speculators whohave little or no exposure to nontraded risk and take on credit risk to earn arisk premium.

Linking this market to our theoretical model, the background credit risk isproxied by a general credit index. CDS contracts are the hedge assets in zeronet supply. We include as the nonhedge asset (with positive net supply) the U.S.equity market index. We also construct a proxy to capture hedging demand.The precise definitions of these variables are discussed further.

For this application to the CDS market, we can deduce the expected sign for

most parameters of the structural model. We define a long position in a CDS tobe the case in which the investor takes on credit risk, in other words, when hesells protection and receives the CDS premium. The nontraded risk factor isgiven by the return on a portfolio exposed to credit risk, thus giving a negativereturn in the case of default or other credit events. Given these definitions,we expect that investors have either zero exposure qi to nontraded credit risk(e.g., institutional investors) or positive exposure (commercial banks). We thusexpect 0 and 1 0. In addition, investors with zero qi are likely the longinvestors (taking on credit risk and earning a risk premium), while those withpositive qi are the short investors (hedging credit risk). This implies 0 > 0.

Finally, as shown in equation (4), the theory always implies 0 1 1, |0| 1,and, in the case of zero-net-supply markets, = 0. In sum, we expect thefollowing signs and restrictions for the model parameters

|0| 1, 0 1 1, 0, 0 0, 1 0, and = 0. (15)

B. Estimation Method

For estimation purposes, it is useful to write the asset pricing model in

Theorem III as a beta-pricing model and invert the model to back out theexpected hedge asset return. This gives the estimation equation


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E(rh) +1

22rh = rh,rb + X + W( ){(A + 0I)E(ch) + B1E(ch)1

+ [rh,rn Ach,rn ] + [ch,rn Arh,rn ]0 + [B1ch,rn B2rh,rn]1}

(16)

with

W( ) = (I+ 0A + 1B2)1

rh,rn = cov(rh, rn)/var(rn)

ch,rn = cov(ch, rn)/var(rn)

(17)

and = E(rb cb), = var(rn), 0 = var(rn)0, and 1 = var(rn)1. The earlierestimation equation imposes Fh = 0 since this term is numerically small (see

the Appendix). All expectations and covariances are conditional upon the in-formation set at time t 1, implying that we have to focus on the innovationsin transaction costs when calculating betas. The matrix X denotes any addi-tional explanatory variables (for example, hedging demand or bond marketcharacteristics), including an intercept.

This theoretical model can be estimated using GMM, since equation (16)defines a system of Kh moment conditions (with Kh denoting the number ofhedge assets) and parameter vector ( , , 0, 1, , 0, 1). In the estimation, wereplace all expectations and betas with their sample counterparts in (16) andthen perform GMM estimation by minimizing the (weighted) sum of squared

errors over the parameters. The system in (16) is nonlinear because W( )depends on the model parameters. However, given an initial estimate of ,the equations are linear in the parameter vector and can be estimated byweighted least squares. Iterating this procedure yields the GMM estimates.We weight the errors according to the diagonal of the error covariance matrixof (16).8 Following Shanken (1992), we correct the standard errors for the factthat betas, expected costs, and expected returns are estimated (see the Internet

Appendix for an outline of this procedure).In asset pricing tests, expected returns are typically estimated by the sam-

ple average of realized excess returns. However, given the short sample period,

this will give noisy estimates of expected returns in our case. Instead, follow-ing Campello et al. (2008) and de Jong and Driessen (2007), we use an exante measure of expected returns by correcting CDS spread levels for the ex-pected default losses (the exact procedure is discussed in detail further). Thisprocedure gives the expected simple return E(Rh) on a CDS contract. Usingthe lognormality assumption, we back out the expected log-return from thisexpected simple return via E(rh) +

12

2rh = ln(1 + E(Rh)).

8 We do not perform the full GMM where the weighting matrix is the inverse of the covariance

matrix of pricing errors because these pricing errors exhibit considerable cross-correlations (seeCochrane (2005), p. 194, for a discussion).


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III. Data and Construction of Main Variables

In this section, we describe in detail our data and the construction of (ex-pected) CDS returns, liquidity measures, and the nontraded risk factor, all of

which are needed to estimate the asset pricing model.

A. Data Description

We use CDS data from Datastream, which in turn sources its data fromCredit Market Analysis (CMA).9 The data start in January 2004 and containdaily consensus bid, mid, and offer CDS spreads from a panel of 30 leadingfinancial institutions. In addition, the data contain a so-called veracity score,which indicates whether the quotes are based on market quotes/trades or arederived. We collect these data for the U.S. market (corporates and financials)from January 2004 to December 2008.

Since we only want to base our analysis on actual market data, we discard allderived quotes. Moreover, we remove all quotes for which both the bid and theask are stale. We further limit attention to CDS contracts of U.S. companiesin U.S. dollars on senior unsecured debt with a maturity of 5 years and aModified Restructuring (MR) clause. These filters yield a sample of 422,779daily observations on 595 entities.

Our CDS sample captures a significant part of the market for credit risk.This follows from the fact that the 595 firms for which we have CDS contractscover 46% of the corporate bond market in terms of amount issued.

To create characteristics for our regressions and a basis for our sorts, wematch our CDS data to several other data sources. First, we match the CDSdata to issuer rating data in Moodys Default Risk Services Corporate Database.To obtain leverage and total debt outstanding, we also match our data to Com-pustat. In addition, we use corporate bond data from TRACE to construct sev-eral proxies for corporate bond liquidity. For each firm with CDS quote data, wecalculate the firms total amount of corporate bonds issued, total corporate bondtrading volume, and total bond turnover (volume divided by amount issued).

Finally, we construct a proxy for hedging pressure effects. For each firmwith CDS quote data, we calculate the total dollar amount of syndicated loansoutstanding by this firm for each quarter in our sample. Data on syndicated

loans come from Dealscan. Since credit lines are likely to be largely drawn indistress scenarios, we assume 100% usage at default.

B. Portfolio Construction

As is usual in the asset pricing literature, we fit the model to different testportfolios rather than to individual assets. We perform five different sequentialsorts on the cross-section of CDS contracts. In each sequential sort we always

9 The leading Bloomberg data platform obtains its CDS data also from CMA. Nashikkar, Sub-

rahmanyam, and Mahanti (2009) compare these CMA data with CDS data provided by GFI Groupand find that they are substantially in agreement.


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sort first on credit rating, and then on one of the following variables: (i) firmleverage, (ii) total syndicated loan amount, (iii) total debt outstanding, (iv) CDSbid-ask spread, and (v) CDS quote frequency. Credit rating and leverage shouldgenerate variation in market risk exposure, the bid-ask spread and quote fre-

quency are included to capture CDS liquidity variation, and total loan amountcaptures hedging pressure effects. In order to have enough observations in eachportfolio, we construct five rating categories: Aaa to Aa, A, Baa, Ba, and B toCaa. Within each rating category, we create quartile portfolios based on the sec-ond sorting variable. In total, we have five sequential sorts with five times fourportfolios per sort. We re-sort portfolios each quarter, and calculate the sortingvariables over the previous quarter. Given that our data start in January 2004,our first portfolio observations are for the second quarter of 2004.

To illustrate the patterns in the CDS spread and bid-ask spread, we presentthe (geometric) average CDS spread across all firms in the Internet Appendix.

Before mid-2007, the market-wide CDS spread showed only moderate varia-tion, but spreads increased dramatically after July 2007, with a peak aroundNovember 2008. The Internet Appendix also shows the bid-ask spreads for thesort on rating and bid-ask spreads, where we average each bid-ask spread quar-tile across all ratings. We see a small peak at the Ford/GM downgrade10 anda strong increase in bid-ask spreads during the crisis, with the most extrememovements around the Lehman default in September 2008.

C. CDS Portfolio Return and Transaction Costs

Calculating CDS portfolio returns is nontrivial. For some CDS contracts, weobserve quotes every day, while for other contracts, missing observations occur.We therefore adopt a technique called weighted repeat sales that originatesfrom the real estate literature (see, e.g., Bailey, Muth, and Nourse (1963) andCase and Shiller (1987)). This method calculates a mid-quote CDS spread indexfor all CDS contracts in a specific portfolio. Letk(i) be the portfolio that containsconstituent i. We assume that the spread mid-quote of a 5-year CDS contractpi,t is given by

pi,t = C DSk(i),t + ui,t, (18)

where C DSk(i),t is the portfolio CDS spread level (which is to be estimated) andui,t is a CDS-specific error term. The repeat sales method employs regressionanalysis to estimate the innovations in the portfolio CDS spread at differentpoints in time as regression coefficients.11 In the Internet Appendix, we providemore details on the repeat sales procedure. The portfolio bid-ask spread is

10Acharya, Schaefer, and Zhang (2008) provide evidence that liquidity risk for Ford and GM

bonds increased correlations within the entire CDS market.11 To construct portfolio CDS spread levels from the CDS spread innovations obtained by the

repeat sales procedure, we match the accumulated spread change to the level of the portfolio spreadat the start of each calendar year.


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obtained by equally weighting all bid-ask spreads of all CDS contracts in aportfolio in each week.

We next transform the portfolio CDS spreads to excess returns. In the CDSmarket, bid and ask quotes are given in terms of the CDS spread. Denote

this bid-ask spread sk,t. Now, consider an investor at time t t who sellsprotection using CDS contract k at a CDS spread of C DSk,tt

12sk,tt, paid

in quarterly periods. One week later at time t, the investor buys an offsettingcontract at C DSk,t +

12sk,t, which generates a net cash flow of

14

(C DSk,t

C DSk,tt) 14

( 12sk,tt +

12sk,t) each quarter until default or maturity. The value

of this stream at time t is the value of a portfolio of defaultable zero-couponbonds, which gives the excess holding return net of costs 12

Rk,t ck,t = 1

4 C DSk,t +1

2sk,tt +

1

2sk,t

Tt

j=1

Bt(t + j)QSV

k,t (t + j)

+t

4

C DSk,tt

1

2sk,tt

, (19)

where C DSk,t = C DSk,t C DSk,tt,QSV

k,t (t + j) is the risk-neutral survivalprobability up to time t + j, and Bt(t + j) is the price of a risk-free zero-couponbond maturing at t + j.13 We also include the accrued spread in (19), althoughthis term is very small for our weekly return frequency. Time is measured inquarters (the payment frequency) and T is the final payment date. Since weinitiated the contract at zero cost, equation (19) directly defines an excess

return. Equation (19) also shows that proportional transaction costs are equalto

ck,t =1

4

1

2sk,tt +

1

2sk,t

Ttj=1

Bt(t + j)QSV

k,t (t + j) +t

4

1

2sk,tt. (20)

We thus obtain transaction costs by multiplying the bid-ask spreads with aduration factor. This shows that the CDS bid-ask spread (scaled by duration)is already a proportional transaction cost and that it is not necessary to dividethis bid-ask spread by the CDS spread to obtain proportional costs: for anynotional amount N, the dollar payoff equals N(Rk,t ck,t).

14 Note that part ofthese costs (due to sk,tt) are known at t t. To assess liquidity risk, we

12 Of course, if default occurs between t t and t, the excess return on the CDS is equal to

(minus) the loss given default. However, if we assume that these individual jumps-to-default are

not priced, we can ignore these cases for estimating portfolio betas.13 This method is very close to the one used by Longstaff et al. (2009), who discount each cash

flow by the risk-free rate plus CDS spread, whereas we discount by the risk-free rate and multiply

by the risk neutral survival probabilities. The Internet Appendix provides details on the risk-free

rate data and the construction of the risk-neutral default probabilities.14 To see this in a numerical example, consider a CDS with bid CDS spread of 50 basis points

and ask CDS spread of 60 basis points. The bid-ask spread is then equal to 10 basis points. A

round-trip CDS transaction would then cost 10 basis points (scaled by the duration) times thenotional amount. A similar example is given by Pires, Pereira, and Martins (2008).


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therefore only include the part ofck,t that is not known at t t, while the fullcosts ck,t are included for the expected liquidity calculation.

D. Nontraded Credit Risk Factor and Nonhedge AssetFor our empirical analysis, we also need to measure the return on the non-

traded credit risk factor rn. As mentioned earlier, we think ofrn as the returnon bank loans and illiquid corporate bonds. Since, by their very nature, thesereturns are not observable, we construct a proxy for this systematic credit riskusing returns on all commonly available credit instruments: corporate bondindices and CDS portfolios. Weekly bond index holding returns are obtainedfor the intermediate-maturity Lehman Brothers U.S. corporate bond indicesper credit rating. The average maturity of all indices is approximately 5 years.Five-year benchmark Treasury returns from Datastream are subtracted from

the corporate bond returns to construct excess returns.In order to capture the systematic credit risk factor in these bond indices

and CDS portfolios, we perform principal component analysis (PCA). We takethe excess holding returns of all our CDS portfolios and the excess holdingreturns on Lehman corporate bond indices for the ratings AAA through CCCand calculate the first principal component of these returns based on theircorrelation matrix. Since all these instruments have exposure to systematiccredit risk, we expect the first principal component to be a good measure ofsystematic credit risk. The loadings on this first component are all close toeach other. This first component explains 52% of the variation in corporate

bond returns. As a robustness check, we use the return on the investment-gradeCDX index as a proxy for the nontraded credit risk factor and find qualitativelysimilar results (see Section IV.D).

For the nonhedge asset we use returns on the S&P 500 equity index. We donot include transaction costs for this index since S&P futures can be tradedwith costs that are negligible for our purpose.15

E. Time-Series Model of Liquidity

The betas in the asset pricing model are defined as the ratio of conditional co-

variances and variances, that is, the (co)variances of the shocks (innovations) inreturns and costs. As in Acharya and Pedersen (2005), we assume that returnshave no serial correlation and we correct for persistence in the liquidity levelby taking the residuals of an autoregressive model as the liquidity innovations.We incorporate the lagged portfolio CDS spread as an explanatory variable inthe time-series model for liquidity, because, as shown in the Internet Appendix,it seems that bid-ask spreads are higher when CDS spreads increase. We thusestimate for each portfolio the regression

ck,t = a + b1ck,t1 + b2C DSk,t1 + k,t, (21)

15

Karagozoglu, Martell, and Wang (2003) report bid-ask spreads on S&P 500 futures of 1.5 to 2basis points.


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where ck,t denotes the portfolio-level transaction costs (bid-ask spread timesduration factor). Empirically, we find that the coefficient on the lagged CDSspread is indeed positive at 0.047 with a t-statistic of 4.5 (both averagedacross all portfolios), which shows that it is important to correct for this level-

dependency. The coefficient on the lagged transaction cost b1 equals 0.74 witha t-statistic of 15.5 (again averaged across portfolios).

F. Expected CDS Returns

To calculate the expected excess return Rk,t,T on a CDS portfolio at time t,we calculate the expectation under the real-world measure of all cash flowsresulting from the CDS contract when held to maturity T, discounted at therisk-free rate:

Et(Rk,t,T) =14

C DSk,tTtj=1

Bt(t + j)PSV

k,t (t + j)

L

Ttj=1

Bt(t + j)PSV

k,t (t + j 1)Pdef|SVk,t (t + j),

(22)

where L is the expected loss rate in the case of default, which we set at themarket-standard rate of 60%, PSVk,t (t + j) is the real-world survival probability

up to time t + j, and Pdef|SVk,t (t + j) is the probability of a default in period

t + j conditional on survival up to time t + j 1. Constructing expected excessreturns in this way rather than averaging realized excess returns allows usto achieve more accurate estimates and lower standard errors for risk premia.Since we use the expected return to maturity for this calculation, the underlyingassumption we make here is that the term structure of expected CDS returnsis flat.

Real-world default probability estimates, needed to construct the expectedexcess returns, are obtained from Moodys-KMV EDF Database. We have dataon 1-year and 5-year EDFs, which we use to construct 1-year and 5-year defaultprobabilities, and we interpolate linearly to obtain the intermediate default

probabilities. We prefer using Moodys-KMV EDFs over rating-implied defaultprobabilities because, especially in the last 2 years of the sample (2007 and2008), we observe a strong increase in the EDFs; it is not obvious how to adjustfor these new market circumstances when using rating-based historical defaultfrequencies.

We construct these default probabilities at the portfolio level by takingweighted averages of all individual default probabilities, where the weightis the number of quotes per issuer relative to the total number of quotes in theportfolio.

We construct the expected return in equation (22) over a 5-year holding

period for each week in the sample. An estimate of the unconditional expectedexcess return is then obtained by averaging these weekly expected returns over


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all weeks in the sample. These unconditional expected returns are then usedin the estimation of the asset pricing model.

G. Alternative CDS Liquidity MeasuresIn the benchmark analysis, we use the bid-ask spread as our liquidity mea-

sure. As a robustness check, we use two alternative liquidity measures. Ourdata set consists of bid and ask quotes. Due to the absence of volume data, wecannot construct Amihuds (2002) ILLIQ measure of price impact directly, butwe can create proxies for price impact. Our first alternative measure (ILLIQ-1)divides the absolute value of the weekly portfolio CDS return by the number ofquotes observed for this portfolio in the same week. If the number of quotes iscorrelated with actual trading volume, this measure captures the price impactof trade in a similar way as Amihuds ILLIQ. Second, we construct a similar

measure at the individual CDS level. We take the sum of the absolute values ofall observed changes in the CDS spread in a given week, and divide this by thenumber of quotes observed for this CDS contract in that week. We then aggre-gate this measure to the portfolio level (ILLIQ-2). The advantage of this lattermeasure is that it takes the average of individual CDS data, which reducesthe noise in the portfolio-level liquidity estimate. On the other hand, individ-ual CDS spread changes are more volatile than portfolio-level CDS returns,which increases the noise in the ILLIQ-2 measure. Since these liquidity mea-sures should capture transaction costs on CDS contracts, we normalize thesemeasures such that they have the same overall mean and variance as the bid-

ask spreads. Note that these two alternative liquidity measures are similar toTang and Yans (2007) volatility-to-volume ratio; they divide the volatility ofCDS spreads by the total number of quotes and trades observed in each month.

IV. Empirical Results

This section presents the results of estimating the model on CDS returns.We first discuss the estimated betas, expected returns, and transaction coststhat go into the asset pricing model. We then present the benchmark estimatesof the structural asset pricing model. We provide a range of robustness checks

and present two extensions of the empirical model by including bond liquidityand hedging pressure variables.

A. Betas, Costs, and Expected Returns

For the sake of brevity, we show expected CDS returns, costs, and betasonly for the portfolios sequentially sorted on credit ratings and bid-ask spreadsin Tables II and III. Expected returns are reported for a quarterly holdingperiod and costs are equal to the bid-ask spread times the duration factor(see (20)). Table II shows that bid-ask spreads are typically higher for lower-

rated firms. We also see that expected returns are higher for lower ratings,as expected. In addition, for each rating category, expected returns are always


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Table II

Descriptive Statistics CDS Returns and Transaction CostsThe table presents summary statistics for the CDS returns and transaction costs for the sequential

sort on credit rating (five categories) and bid-ask spread (four quartiles). Panel A presents expected

CDS returns for a quarterly horizon in percentages, constructed by averaging equation (22) overall weeks in our sample. Panel A also contains average proportional transaction costs per portfolio,

defined as the bid-ask spread times a duration factor (equation (20)). Square brackets give t-

statistics of the weekly averages, calculated using NeweyWest with 24 lags. Panel B contains the

correlation matrix of the time series of CDS returns and transaction cost innovations, Corr(rh, ch),

presenting both the diagonal elements and the average of each row of this 20 20 matrix. The

sample period is April 2004 until June 2008.

Panel A: Descriptive Statistics: Expected Transaction Costs and Expected CDS Returns

Expected CDS Return (%/Quarter) Expected Transaction Cost (%)Rating/Bid-Ask

Spread Low Q2 Q3 High Low Q2 Q3 High

Aaa to Aa 0.038 0.036 0.047 0.104 0.121 0.143 0.175 0.299

[0.88] [2.17] [2.06] [1.73] [24.62] [25.28] [29.22] [5.08]

A 0.047 0.055 0.060 0.104 0.158 0.185 0.205 0.289

[4.35] [4.28] [3.56] [1.97] [44.6] [46.78] [40.76] [14.63]

Baa 0.077 0.100 0.131 0.202 0.178 0.207 0.243 0.356

[6.52] [4.91] [5.02] [3.45] [56.36] [39.48] [35.43] [19.81]

Ba 0.174 0.241 0.308 0.403 0.260 0.348 0.443 0.618

[5.49] [5.21] [6.12] [5.78] [36.80] [36.96] [31.52] [14.11]

B to Caa 0.434 0.527 0.570 0.627 0.392 0.532 0.619 1.120

[5.24] [7.45] [5.10] [3.61] [21.85] [26.31] [27.28] [3.18]

Panel B: Time-Series Correlations: Transaction Cost Innovations and CDS Returns

Diagonal AverageRating/Bid-Ask


Aaa to Aa 0.121 0.129 0.061 0.376 0.004 0.010 0.101 0.108

A 0.187 0.083 0.020 0.240 0.038 0.018 0.040 0.085

Baa 0.102 0.012 0.117 0.142 0.005 0.002 0.026 0.067

Ba 0.014 0.075 0.024 0.026 0.005 0.017 0.027 0.099

B to Caa 0.012 0.023 0.039 0.135 0.082 0.087 0.055 0.127

monotonically increasing with bid-ask spread. Averaged across ratings, thehigh-liquidity portfolios have an expected return of 0.14% per quarter versus0.29% per quarter for the low-liquidity portfolios. This provides some firstinformal evidence that illiquidity may affect expected returns in a positive way.This is further illustrated in the Internet Appendix, where we plot expectedreturn pricing errors for a two-factor model with equity and credit risk only (andno liquidity variables) against the expected liquidity per portfolio for all 100portfolios used in our analysis. The scatter plot shows that a model without

liquidity variables generates substantial pricing errors between 0.3% and0.3% per quarter. Most importantly, these pricing errors exhibit a positive


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relation with expected liquidity, in line with the strongly positive estimate for

0 reported further.Table II also gives t-statistics for the expected CDS returns using

NeweyWest with 24 lags. These are generally high despite our relatively

short sample period, which is due to our forward-looking way of backing outexpected returns from CDS spreads. Finally, Table II reports estimated valuesfor Corr(rh, ch). We see that both the diagonal and off-diagonal elements of thismatrix vary across portfolios, but in some cases differ substantially from zeroand are typically negative. Hence, for our empirical analysis, we do not imposecov(rh, ch) = 0, but instead rely on the general case of Theorem I.

Turning to the betas in Table III, CDS returns have positive exposure toequity market returns (rh,rb > 0), which is higher for CDS contracts with lowerratings and higher bid-ask spreads. Since we define a long return as the returnfor a credit protection seller, a positive sign for rh,rb can be expected: when eq-

uity prices increase, CDS spreads go down. As expected, all betas are less thanone, since especially high-rated bonds have small market risk (see also Eltonet al. (2001)). These betas are always significant for the reported sort. Acrossall sorts, these betas are significant at the 5% level for 97 out of 100 portfo-lios. We observe a similar pattern for the exposure to the proxy for systematicnontraded credit risk (rh,rn > 0), significant at the 5% level for all portfolios.

As expected, the return on selling credit protection is positively related to ourproxy for systematic credit risk.

The exposure of transaction costs to equity market returns and systematiccredit risk is typically negative, suggesting that bid-ask spreads increase in bad

times. This is mainly the case for portfolios with low ratings and high bid-askspreads, and significant at the 5% level for 30 out of 100 portfolios for equityindex returns (ch,rb ) and for 44 out of 100 portfolios for the credit risk factor(ch,rn).

B. Benchmark Results

We estimate the asset pricing model for a benchmark setting where we use (i)the bid-ask spread to construct expected liquidity and liquidity risk variables,(ii) a sample period from April 2004 until June 2008, and (iii) a holding period

assumed to be one quarter. The choice of holding period, which is required asthere are no data on the typical turnover frequency in CDS markets, impliesthat we insert expected returns over a quarterly horizon in equation (16). Inthe benchmark setup, we exclude the two final quarters of 2008. Because of theevents around the default of Lehman Brothers in September 2008, these twoquarters exhibit different behavior compared to the rest of the sample (see theInternet Appendix). Further, we perform robustness checks on all choices forthis benchmark setup. To assess the fit of the model, we define a cross-sectionalR2 as one minus the ratio of the variance of pricing errors divided by thevariance of the expected CDS returns, where we weight both pricing errors and

returns with the optimal GMM weights. Hence, this R

2

measures the improve-ment over a model with constant expected CDS returns across portfolios.


25/38


Table III

Beta EstimatesThe table presents beta estimates for the sequential sort on credit rating (five categories) and bid-

ask spread (four quartiles). The upper left panel contains the beta of CDS portfolio returns with

respect to S&P 500 index returns (rh,rb ). The upper right panel contains the beta of transactioncost innovations (residuals of the AR-X model in equation (21)) with respect to the S&P 500 index

returns (ch,rb ). The lower left panel contains the beta of CDS returns (orthogonalized for S&P 500

index returns) with respect to the PCA factor capturing systematic credit risk ( rh,rn). The lower

right panel contains the beta of transaction cost innovations (orthogonalized for S&P 500 index

returns) with respect to the PCA systematic credit risk factor (ch,rn ). The sample period is April

2004 until June 2008. t-statistics are given in square brackets.

rh,rb ch,rbRating/Bid-Ask


Aaa to Aa 0.041 0.048 0.054 0.190 0.001 0.002 0.001 0.015

[8.52] [7.34] [6.85] [5.27] [1.38] [2.2] [0.91] [2.17]A 0.046 0.045 0.061 0.134 0.001 0.001 0.001 0.005

[8.59] [9.72] [6.86] [6.56] [1.18] [1.58] [1.6] [2.97]

Baa 0.062 0.069 0.077 0.12 0.001 0.001 0.001 0.001

[8.64] [9.57] [8.48] [8.82] [1.69] [1.12] [0.97] [1.11]

Ba 0.113 0.159 0.198 0.245 0.000 0.002 0.005 0.001

[7.17] [8.76] [7.32] [7.79] [0.08] [1.08] [1.4] [0.23]

B to Caa 0.285 0.271 0.218 0.396 0.003 0.005 0.007 0.024

[7.72] [5.63] [4.62] [5.97] [1.00] [1.61] [2.24] [1.43]

rh,rn ch,rnRating/Bid-Ask


Aaa to Aa 0.186 0.315 0.310 1.401 0.004 0.003 0.006 0.106

[8.94] [12.8] [9.21] [9.04] [1.23] [0.79] [1.86] [3.06]

A 0.267 0.244 0.349 0.821 0.005 0.000 0.004 0.024

[13.62] [14.89] [9.15] [9.48] [2.17] [0.11] [1.32] [2.67]

Baa 0.331 0.397 0.473 0.648 0.002 0.002 0.004 0.008

[11.75] [16.54] [14.63] [12.36] [0.83] [0.89] [1.31] [1.32]

Ba 0.762 0.969 1.095 1.000 0.004 0.008 0.002 0.065

[12.62] [15.21] [9.64] [7.01] [0.43] [1.10] [0.12] [2.51]

B to Caa 1.713 2.121 1.838 2.482 0.022 0.019 0.019 0.207

[11.83] [10.86] [9.13] [8.61] [1.62] [1.35] [1.22] [2.49]

Table IV presents the parameter estimates and R2 obtained by applyingGMM to the moment conditions in equation (16) for the case without heteroge-neous pricing of liquidity (1 = 1 = 0) and with an intercept 0. The key resultis that we find a strong positive effect of expected liquidity on expected CDSreturns. This effect is both statistically and economically significant. For thebenchmark case with quarterly holding periods, the coefficient 0 equals 0.69with a t-statistic of 13.5. The high statistical precision illustrates the useful-ness of constructing forward-looking expected returns from CDS spreads. Note

that |0| < 1, in line with the theory (equation (15)). The coefficient on liquidityrisk 0 is also significantly positive, again in line with theoretical predictions.


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Table IV

Benchmark GMM Estimation ResultsThe theoretical asset pricing model in Theorem I is estimated by applying GMM to the moment

condition (16) for 100 CDS portfolios (sorted first on credit rating, and then either on leverage,

total debt, total syndicated loan amount, bid-ask spread, or quote frequency). Moment conditionsare weighted by the diagonal of the optimal GMM weighting matrix. The model also includes an

intercept. t-statistics (in square brackets) are calculated with GMM using NeweyWest with 24

lags, and incorporate estimation errors in expected returns, costs and betas, following Shanken

(1992); see the Internet Appendix. The table also reports a cross-sectional R2, where pricing errors

are weighted using the GMM weights. Finally, the table reports results for a benchmark holding

period of one quarter, and for holding periods of one month and 1 year. The sample period is April

2004 until June 2008.

(1) (2) (3) (4) (5)

Holding Period Quarterly Quarterly Quarterly Annual Monthly

Intercept 0.106 0.113 0.012 0.333 0.040

(0) [11.59] [13.18] [0.39] [10.98] [12.32]Equity risk 0.224 0.512 0.032 0.660 0.073

() [4.46] [4.38] [0.82] [3.53] [3.16]

Credit risk 0.066 0.226 0.462 0.015

() [2.48] [8.58] [4.69] [1.66]

Expected liquidity 0.694 0.763 2.050 0.270

(0) [13.51] [14.04] [12.61] [14.42]

Liquidity risk 1.819 1.72 5.978 0.753

(0) [3.25] [2.80] [3.40] [4.13]

R2 0.899 0.892 0.203 0.880 0.652

Turning to the premia on equity market risk () and credit risk (), we findpositive and significant estimates for both risk factors. However, the exposuresto these two risk factors are strongly correlated cross-sectionally. This becomesevident when we exclude the credit risk premium (specification (2) in Table IV):this leads to a higher estimate for the equity risk premium (increasing from0.22% to 0.51% per quarter), but the decrease in R2 is minimal. Overall, themodel explains about 90% of the cross-sectional variation in expected returns.Table IV, specification (3) shows that, if we exclude expected liquidity and liq-uidity risk from the model, this R2 goes down to 20%, which shows that the

liquidity effects are not simply picking up the effects of equity and credit riskpremia.The intercept is significantly negative at about 11 basis points per quarter.

It turns out that, without an intercept, the model has problems fitting theexpected returns on high-rated and liquid portfolios. These portfolios had verylow CDS spreads during the period before the crisis, which is in line withanecdotal evidence that high-rated assets were overpriced in the pre-crisisperiod. A negative intercept helps to fit such portfolios better.

To assess the economic significance of these estimates, we decompose thetotal expected CDS portfolio return into parts explained by market risk

premia, expected liquidity, liquidity risk, and a residual. We use equation(16) to explicitly solve for the expected CDS return. Figure 2 presents this


27/38


Figure 2. Decomposition of expected CDS returns: Benchmark model. Using the bench-

mark GMM estimates in Table IV, specification (1), for the model for expected CDS returns in

equation (16), the graph decomposes expected portfolio CDS returns into an intercept, market risk

premiums (sum of equity risk premium and credit risk premium), expected liquidity, liquidity risk,

and a pricing error. Results are presented for the sequential sort on credit rating (five categories)

and bid-ask spread (four quartiles). All returns are in percentages for a quarterly period.

decomposition for the rating/bid-ask spread sort. The figure shows that boththe expected liquidity effect and market risk effect increase steadily for lowerratings and less-liquid portfolios, while the effect of liquidity risk is only rele-vant for portfolios with the lowest liquidity. Given that ch,rn is negative for theseportfolios and 0 > 0, liquidity risk decreases the expected CDS return. This isline with our argument in Section II.A. In particular, a negative ch,rn lowersthe hedge demand of short investors (who care most about this covariance) andthis decreases expected returns.

Taking the average across the 20 portfolios in Figure 2 (weighted by theiroptimal GMM weights), we find that equity and credit risk premia togetheraccount for 0.060% of quarterly returns, expected liquidity for 0.175%, and liq-uidity risk for 0.005%. This shows the large economic significance of expectedliquidity.

C. Backing Out Structural Parameters

With the earlier parameter estimates, we can back out the relative weightsof the two investor groups. We consider two cases. First, we use the benchmark


28/38


setting presented in Theorem I. Using the estimates of specification (1) inTable IV and the definition of 0 in equation (11), the estimates imply that thelong investors (who take on credit risk) represent 84.7% of the total wealth-weighted risk tolerance , while the short investors (hedgers) represent 15.3%.

This allows us to interpret the strong positive coefficient on expected liquidity0. Given that the long investors are much more aggressive (due to more wealthor lower risk aversion), they require large compensation for expected liquidity,as explained in the simple example in Section I.D.

Alternatively, we can relate the parameter estimates to the extension ofthe theory with long-term investors (Theorem II). For example, we can makethe identifying assumption that investors that are short in all hedge assets(hedgers) are always long-term investors. This seems plausible: commercialbanks hedge credit risk using CDS contracts and may not need to tradeafterwards. Mathematically, this implies that i:i =I wi 1i = 0, and from0/(1 + 0) = 0.694, it follows that the long-term hedgers represent 69.4% ofthe total wealth-weighted risk tolerance (equal to (1 + 0)), where the short-term risk takers represent 30.6% of the risk tolerance.

In sum, the positive expected liquidity coefficient is consistent with two set-tings, one where protection sellers are more aggressive than hedgers, and analternative setting where protection sellers have shorter horizons than hedgers.

An important assumption in these two explanations is the holding period ofthe short-term investor. The benchmark setting uses a quarterly holding period.From Theorem I, it follows that if cov(rh, ch) = 0, the empirical model can betranslated to a different frequency simply by multiplying the coefficients by

the holding period (relative to the quarterly period). Theorem I also shows thatthis is not exactly true in the general setting, but given that the correlationsbetween rh and ch are only moderately different from zero, we do not expectlarge deviations from the simple multiplication effect. Table IV reports resultsfor monthly and annual holding periods, thus measuring expected returns overshorter or longer horizons. The table confirms our intuition: the parametersestimates are approximately scaled by the holding period. This implies thatthe relative importance of the different premia remains the same irrespectiveof the holding period and that the empirical model has similar explanatorypower across holding periods. The holding period does, however, affect the

interpretation of the coefficients in terms of the theoretical model. The theoryconstrains the coefficient on expected liquidity 0 to be less than one. For theannual holding period, the estimate for 0 is larger than one. This impliesthat the empirical estimates are in line with the theoretical predictions aslong as the holding period is not too long (roughly less than 6 months). If theactual holding period were really 1 year, setting 0 at its maximum level of one(as opposed to the unrestricted estimate of 2.05) implies that the theoreticalmodel explains about 1/2.05 100% 49% of the empirically observed effect ofexpected liquidity. It is also useful to note that the bid-ask spread probably doesnot capture all transaction costs of a CDS trade. For example, search costs are

not explicitly incorporated. Also, our bid-ask spreads are for on-the-run 5-yearCDS contracts, and unwinding a shorter-maturity CDS contract after some


29/38


time may be more costly. This could partially explain the high coefficient on thebid-ask spread.

D. Robustness ChecksTable V provides a range of robustness checks on the benchmark results of

Table IV.First, we use an alternative proxy for systematic credit risk: the returns on

the investment-grade CDX index. This leads to similar effects for liquidity riskand expected liquidity, but a smaller and less significant estimate on the creditrisk premium. The overall R2 is essentially unchanged.

Second, we change the sample period. We add the two final quarters of 2008and report estimates for the full sample period April 2004 to December 2008.This leads to an even larger coefficient on expected liquidity, but a somewhat

smaller effect of liquidity risk (still significant). We also consider a pure crisissample, namely, July 2007 to December 2008, and again find very similarresults for the liquidity variables. The cross-sectional R2 is a bit lower when weinclude the crisis period, which is not surprising since there are some extrememovements in CDS spreads and bid-ask spreads in this period.

Third, instead of the bid-ask spread, we use the two alternative measuresfor liquidity discussed in Section III.G, that is, the proxies for price impact

ILLIQ-1 and ILLIQ-2. The results show that similar liquidity effects obtainfor both alternative liquidity measures. Compared to the bid-ask spread, bothalternative liquidity measures generate slightly lower values for the R2.

Fourth, we use different time-series models for transaction costs. Our bench-mark model has an autoregressive specification with the lagged transactioncost and the lagged CDS spread as explanatory variables (denoted the ARXmodel in the table). As an alternative, we use pure autoregressive models withtwo or four lags but without the lagged CDS spread (denoted the AR(2) and

AR(4) models in the table). We find that all results are very similar to thebenchmark case.

Fifth, we remove the intercept from the asset pricing model. It turns out thatthis is the only robustness check that does substantially affect the estimatefor the expected liquidity premium, lowering it from 0.694 to 0.243, but the

estimate remains strongly statistically significant, with a t-statisitic of 7.2.The economic effect equals 0.060% per quarter across the 20 rating/bid-askspread portfolios, compared to 0.175% in the benchmark case. The estimate forthe equity risk premium becomes negative and statistically significant. Thisis because in the absence of a constant term, the model exploits the strongcross-sectional correlation between the equity and credit risk betas to mimicthe constant term. The sum of the market and credit premia is positive for allportfolios, however, at an average of 0.076% per quarter across portfolios. TheInternet Appendix shows the decomposition of expected CDS returns acrossrating/bid-ask spread portfolios for the specification without an intercept.

Finally, we incorporate chcm = cov(ch, cm)/var(cm) as an explanatory variable.According to the theoretical model, this CDS-market liquidity risk should not


30/38


TableV

Robustnessan

dSpecificationTests

Thetablereportsestimatesoftheas

setpricingmodelinTheorem

I,o

btainedbyapplyingGMM

tothe

momentcondition(16)for100C

DSportfolios

(sortedfirstoncreditrating,andthenonleverage,totaldebt,totalsyn

dicatedloanamount,bid-askspread,orquotefrequency).Thesettingdescribed

inTableIV(includedasspecificatio

n(1)forconvenience)ischanged

inthefollowingways.First,thePCAfactorisreplacedbyretur

nsontheIG

CD

Xindex.Second,resultsarepre

sentedforalternativesamplepe

riods:April2004toDecember2

008(full)andJuly2007toDe

cember2008

(cr

isis).Third,twoalternativemea

suresforliquidityareusedinste

adofthebid-askspread(seeSec

tionIII.G.Fourth,theARXtime-seriesmodel

for

transactioncostsisreplacedbye

itheranAR(2)modeloranAR(4

)model.Fifth,theinterceptisex

cludedfrom

themodel.Finally,a

premium

on

the

cost-costbeta(ch,cm)fortheCDSmarketisincluded.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

Nontradedrisk

PCA

CDX

PCA

PCA

PCA

PCA

PCA

PCA

PCA

PCA

Sampleperiod

Base

Base

Full

Crisis

Base

Base

Base

Base

Base

Base

Liq

uiditymeasure

BAS

BAS

BAS

BAS

ILLIQ-1

ILLIQ-2

BAS

BAS

BAS

BAS

Costinnovations

ARX

ARX

ARX

ARX

ARX

ARX

AR(2)

AR(4)

ARX

ARX

Int

ercept

0.106

0.115

0.114

0.135

0.290

0.154

0.108

0.108

0.106

(0

)

[11.59]

[13.06]

[17.06]

[7.01]

[24.52]

[10.49]

[

10.95]

[11.08]

[8.34]

Equityrisk

0.224

0.428

0.104

0.202

0.527

0.183

0.279

0.302

0.179

0.221

()

[4.46]

[7.12]

[1.77]

[3.76]

[5.15]

[1.46]

[4.51]

[4.75]

[4.18]

[3.64]

Cre

ditrisk

0.066

0.023

0.001

0.020

0.093

0.020

0.057

0.054

0.162

0.066

()

[2.48]

[1.23]

[0.16]

[1.02]

[3.28]

[0.81]

[2.05]

[1.90]

[6.88]

[2.15]

Expectedliquidity

0.694

0.71

derivative pricing with liquidity risk

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