Specification Analysis of Structural Credit Risk Models

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Specification Analysis of Structural Credit Risk Models Jing-zhi Huang and Hao Zhou 2008-55 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Specification Analysis of Structural Credit Risk Models ∗ Jing-zhi Huang Penn State University Hao Zhou Federal Reserve Board First Draft: November 2005 This Version: September 2008 Abstract In this paper we conduct a specification analysis of structural credit risk models, using term structure of credit default swap (CDS) spreads and equity volatility from high-frequency return data. Our study provides consistent econometric estimation of the pricing model parameters and specification tests based on the joint behavior of time-series asset dynamics and cross-sectional pricing errors. Our empirical tests reject strongly the standard Merton (1974) model, the Black and Cox (1976) barrier model,andtheLongstaffandSchwartz(1995)modelwithstochasticinterestrates. The double exponential jump-diffusion barrier model (Huang and Huang, 2003) improves significantly over the three models. The best one among the five models considered is thestationaryleveragemodelofCollin-DufresneandGoldstein(2001),whichwecannot reject in more than half of our sample firms. However, our empirical results document the inability of the existing structural models to capture the dynamic behavior of CDS spreads and equity volatility, especially for investment grade names. This points to a potential role of time-varying asset volatility, a feature that is missing in the standard structural models. ∗ We would like to thank Viral Acharya, Ren-Raw Chen, Thomas Dangl, Jan Ericsson, Jean Helwege, Rangarajan Sundaram, Yi Zhou, and seminar participants at CEIBS, Cheung Kong Graduate School of Business, Lehigh, MIT, SHUFE, SWUFE, UESTC, the 2006 Econometric Society North American Winter MeetinginBoston,the2006DerivativesandRiskManagementConferenceatFDIC,the2007FederalReserve Board Conference on Credit Risk and Credit Derivatives, Moody’s 4th Annual Credit Risk Conference in Copenhagen, the 2007 Gutmann Center Symposium on Credit Risk and the Management of Fixed-Income Portfolios in Vienna, the 2007 CICF in Chengdu, the 2008 Mitsui Symposium at Michigan, and the 2008 Singapore International Conference on Finance for helpful comments and suggestions. Huang is at Smeal College of Business, Penn State University, University Park, PA 16802; jxh56@psu.edu. Zhou is at Division of Research and Statistics, Federal Reserve Board, Washington, D.C. 20551; hao.zhou@frb.gov. Huang acknowledges a Smeal Summer Research Grant for partial support. The views presented here are solely those of the authors and do not necessarily represent those of the Federal Reserve Board or its staff.

Specification Analysis of Structural Credit Risk Models Abstract In this paper we conduct a specification analysis of structural credit risk models, using term structure of credit default swap (CDS) spreads and equity volatility from high-frequency return data. Our study provides consistent econometric estimation of the pricing model parameters and specification tests based on the joint behavior of time-series asset dynamics and cross-sectional pricing errors. Our empirical tests reject strongly the standard Merton (1974) model, the Black andCox(1976)barriermodel,andtheLongstaffandSchwartz(1995)modelwithstochasticinterest rates. The double exponential jump-diffusion barrier model (Huang and Huang, 2003) improves significantly over the three models. The best model is the stationary leverage model of Collin- Dufresne and Goldstein (2001), which we cannot reject in more than half of our sample firms. However, our empirical results document the inability of the existing structural models to capture the dynamic behavior of CDS spreads and equity volatility, especially for investment grade names. This points to a potential role of time-varying asset volatility, a feature that is missing in the standard structural models. JEL Classification: G12, G13, C51, C52. Keywords: Structural Credit Risk Models; Credit Default Swap Spreads; High Frequency Equity Volatility; Consistent Specification Analysis; Pricing Error Diagnostics.

1 Introduction Credit derivatives markets have been growing exponentially over the past several years. According to the most recent biennial survey by the British Bankers’ Association, the global credit derivatives market is expected to exceed $8 trillion in 2006. Credit default swaps (CDS) are currently the most popular credit derivatives instrument and account for about half of the credit derivatives market. Under a CDS contract the protection seller promises to buythereferencebondatitsparvaluewhenapre-defineddefaulteventoccurs. Inreturn,the protectionbuyermakesperiodicpaymentstotheselleruntilthematuritydateofthecontract or until a credit event occurs. This periodic payment, usually expressed as a percentage of the notional value underlying a CDS contract, is called the CDS spread. Compared with corporate bond spreads, CDS spreads are a relatively pure pricing of default risk of the underlying entity, abstracting from numerous bond characteristics, such as seniority, coupon rates, embedded options, and guarantees. As a result, there is a growing literature on testing credit risk models using the information from the CDS market. A widely used approach to credit risk modeling in practice is the so-called structural method, originated from Black and Scholes (1973) and Merton (1974). Whereas there have been many empirical studies of structural models, especially recently, based on corporate bond data, the empirical testing of these models using CDS spreads is quite limited. Such a testing is desirable especially given the recent empirical evidence based on the corporate bond market that existing structural models have difficulty either fitting corporate bond spreads(e.g., Jones, Mason, andRosenfeld(1984), LydenandSaraniti(2000), Delianedisand Geske (2001), Eom, Helwege, and Huang (2004), Arora, Bohn, and Zhu (2005) and Ericsson and Reneby (2005)) or explaining both spreads and default frequencies simultaneously (the so-called credit spread puzzle documented in Huang and Huang (2003)). If CDS spreads are considered to be a purer measure of credit risk than corporate bond spreads, then the existing structural models (purely default risk based) may perform better in capturing the behavior of CDS spreads than they do for corporate bond spreads. In this article we test five representative structural credit risk models using a sample of 93 single name CDS contracts during the period January 2002 - December 2004. The models we consider are the standard Merton (1974) model, the Black and Cox (1976) model with a flat barrier, the Longstaff and Schwartz (1995) model with stochastic interest rates, the Collin-Dufresne and Goldstein (2001) model with a stationary leverage, and the double exponential jump diffusion model 1

used in Huang and Huang (2003). More specifically, we formulate a specification test based on the pricing solutions of CDS spreadsandequityvolatilityimpliedbyaparticularstructuralmodel. Byassumingthatboth equityandcreditmarketsareefficientandthattheunderlyingstructuralmodeliscorrect, we obtain the identifying moment restrictions on the model parameters, such as asset volatility, the default barrier, and the speed of mean-reverting leverage. Such a GMM estimator with an ensuring J-test is a consistent econometric method, for parameter estimation and specification analysis of the structural credit risk models. One advantage of such a test is that it provides us with a precise inference on whether a particular structural model is rejected or notin thedata, unlike theexisting studiesbased oncalibration, rolling estimation or regression analysis. Furthermore, unlike the existing studies that focus on 5-year CDS contracts, we use the entire term structure of CDS spreads. Such a method provides us a tighteridentificationofstructuralmodelparametersandminimizestheeffectofmeasurement error from using bond characteristics, and thus attributes the test results mostly to the specification error. More importantly, by focusing on the equity volatility measured with high frequency data, instead of low frequency daily data, our approach speaks directly to the recent finding that volatility dynamics has a strong potential in better explaining the credit spreads.1 Our empirical tests reject strongly the following three standard models: the Merton (1974) model, the Black and Cox (1976) model, the Longstaff and Schwartz (1995) model. However,thedoubleexponentialjump-diffusionbarriermodeloutperformssignificantlythese three models. The stationary leverage model of Collin-Dufresne and Goldstein (2001) is the best performing one among the five models examined in our analysis and more specifically, is not rejected by the GMM test for more than half of the 93 companies in our sample. In addition, the test results allow us to gain a better understanding of the structural models, whichotherwisedoesnotobtaineasilyfromad hoccalibrationsorrollingestimationanalysis. For example, when allowing the default barrier to be different from the total liabilities, we discoveranegativerelationshipbetweentheobserveddebt/assetratioandtheimplieddefault 1Campbell and Taksler (2003) find that idiosyncratic equity volatility can explain a significant part of corporate bond yield spreads cross-sectionally. Huang and Huang (2003) conjecture that a structural credit risk model with stochastic asset volatility may solve the credit spread puzzle. Huang (2005) considers an affine class of structural models with both stochastic asset volatility and L´evy jumps. Based on regression analysis, Zhang, Zhou, and Zhu (2006) provide empirical evidence that a stochastic asset volatility model may improve the model performance. 2

boundary trigger. Moreover, when a dynamic leverage or a jump component is allowed for, the overall fitting of average CDS term structure is improved with a much smaller pricing error. Further more, for the best performing dynamic leverage model, the individual firms sensitivity to interest rate or varies dramatically from significant positive for investment grade names to significant negative for speculative grade names, suggesting a great deal of heterogeneity in each firm’s exposure to systematic risk. Finally, our empirical analysis sheds some light on how to improve the existing structural models in order to fit better CDS prices. One implication from our results is that a term structure model more flexible than the one-factor Vasicek (1977) model – used in Longstaff and Schwartz (1995) and Collin-Dufresne and Goldstein (2001) – may reduce the pricing error. Also judging from several pricing error diagnostics, jump augmentation seems to improve the investment grade names, while dynamic leverage seems to improve the speculative grade names. We also find that for the junk rated names, the observed spot leverage is very close to the long-run mean of the risk-neutral leverage implied by the Collin-Dufresne and Goldstein (2001) model; while for investment grades the spot leverage is much lower than the risk-neural leverages. This mirrors the recently documented low leverage puzzle for high rating firms (Strebulaev and Yang, 2006; Chen and Zhao, 2006). Our analysis also documents the inability of the standard structural models in fitting time-series of both CDS spreads and equity volatility. Given that equity volatility in structural models is time-varying, this result provides a direct evidence that a structural model with stochastic asset volatility may improve the model performance (Huang and Huang, 2003; Huang, 2005; Zhang, Zhou, and Zhu, 2006). There are several empirical studies of structural models based on CDS data that are directly related to ours. For instance, Predescu (2005) examines the Merton (1974) model and a Black and Cox (1976) type barrier model with a rolling estimation procedure combined with the MLE approach proposed in Duan (1994). Hull, Nelken, and White (2004) study the Merton model using a calibration approach. Chen, Fabozzi, Pan, and Sverdlove (2006) investigate the Merton, Black-Cox, and Longstaff-Schwartz models (however, their implementation of the latter model is based on an approximated solution). Examples of studies that link CDS premiums with variables from structural credit risk models using a regression analysis include Cossin and Hricko (2001); Ericsson, Jacobs, and Oviedo (2005); Houweling and Vorst (2005). Our analysis differs from these studies in that we consider three more recent models (Longstaff-Schwartz, CDG, and Huang-Huang) and conduct a rigorous 3

GMM-based specification analysis. Structural credit risk models have also been examined empirically using information from the corporate bond market. Examples include Jones, Mason, and Rosenfeld (1984); Huang andHuang(2003);Cremers, Driessen, Maenhout, andWeinbaum(2004);Eom, Helwege, and Huang (2004); Schaefer and Strebulaev (2004). These studies have indicated that structural modelshavedifficultypredictingcorporatebondyieldspreadsaccurately. Onelineofreasoning is that structuralmodels may be able to do a better job in fitting CDS prices, presumably because CDS prices are a purer measure of default risk and corporate bond prices (Longstaff, Mithal, and Neis, 2005; Ericsson, Reneby, and Wang, 2006). One implication of our analysis is that structural models still have difficulty predicting credit spreads even if when a purer measure of credit risk is used in the empirical analysis, although a better measure of credit spread can help us rank order the extent structurally models more consistently. Finally, notice that like many other studies such as Jones, Mason, and Rosenfeld (1984), Lyden and Saraniti (2000), Delianedis and Geske (2001), Eom, Helwege, and Huang (2004), and Ericsson and Reneby (2005), here we examine the implications of structural models for pricing only (and ignore the implications for default probabilities under the real measure). In another word, we do not examine the credit spread puzzle here. The rest of the paper is organized as follows. Section 2 briefly outlines the class of structural models examined in our empirical analysis. Section 3 presents our econometric method of parameter estimation and specification tests. Section 4 describes the data used in our analysis, and Section 5 reports and discusses our empirical findings. Finally, Section 6 concludes. 2 A Review of Structural Credit Risk Models We consider five representative structural models in our empirical analysis. Specifically, they include the Merton (1974) model, the Black and Cox (1976) model, the Longstaff and Schwartz (1995) model, the Collin-Dufresne and Goldstein (2001) model, and the double exponential jump diffusion model considered in Huang and Huang (2003).2 The Black and Cox model with a flat barrier examined here can be also considered to be a special case of either the exogenous-default version of Leland and Toft (1996) or the one-factor version of 2Examples of other structural models include Geske (1977), Anderson and Sundaresan (1996), Mella- Barral and Perraudin (1997), Leland (1998), Duffie and Lando (2001), and Acharya and Carpenter (2002) etc. 4

Longstaff and Schwartz (1995). Except for the Merton model, all other ones are barrier-type models. Among the five models, Longstaff and Schwartz (1995) and Collin-Dufresne and Goldstein (2001) are two-factor models, and the remaining three are one-factor models. For completeness, below we briefly review the five structural models to be tested in our empirical study. Although thesefivemodels differin certain economic assumptions, theycan be embedded in the same underlying structure that includes specifications of the underlying firm’s asset process, the default boundary, and the recovery rate etc. Let V be the firm’s asset process, K the default boundary, and r the default-free interest rate process. Assume that, under a risk-neutral measure, NQ dV t t = (r δ)dt+σ dWQ +d ZQ 1  λQξQdt, (1) V t− t − v t  X i=1 (cid:16) i − (cid:17)  −   dlnK = κ [ ν φ(r θ ) ln(K /V )]dt (2) t ℓ t ℓ t t − − − − dr = (α βr )dt+σ dZQ (3) t t r t − where δ, σ , κ , θ , ν, φ, α, β, and σ are constants, and WQ and ZQ are both onev ℓ ℓ r dimensional standard Brownian motion under the risk-neutral measure and are assumed to have a constant correlation coefficient of ρ. In Eq. (1), the process NQ is a Poisson process with a constant intensity λQ > 0, the ZQ’s are i.i.d. random variables, and YQ ln(ZQ) i ≡ 1 has a double-exponential distribution with a density given by f YQ (y) = pQ u η u Qe−ηu Qy1 {y≥0} +pQ d η d Qeη d Qy1 {y<0} . (4) In equation (4), parameters ηQ,ηQ > 0 and pQ,pQ 0 are all constants, with pQ +pQ = 1. u d u d ≥ u d The mean percentage jump size ξQ is given by pQηQ pQηQ ξQ = EQ eYQ 1 = u u + d d 1. (5) h − i ηQ 1 ηQ +1 − u − d All five models considered in this analysis are special cases of the general specification in Eqs. (1)-(3). Forinstance, ifthejumpintensityiszero, thentheassetprocessisageometric Brownian motion. This specification is used in the four diffusion models, namely, the models of Merton, Black and Cox (BC), Longstaff and Schwartz (LS), and Collin-Dufresne and Goldstein (CDG). Regarding the specification of the default boundary K, it is a point at the bond maturity in the Merton model. If κ is set to be zero, then the default boundary ℓ 5

is flat (a continuous barrier), an assumption made in Black and Cox (BC), Longstaff and Schwartz (LS), and the jump diffusion (HH) models. The mean-reverting specification in (2) is used in the Collin-Dufresne and Goldstein (CDG) model. The Vasicek model in (3) is used to describe the dynamics of the risk-free rate in the two-factor models of Longstaff and Schwartz (LS) and Collin-Dufresne and Goldstein (CDG) models. If both β and σ are r zero, then the interest rate is constant, an assumption made in the three one-factor models. For simplicity and comparison with other studies, we assume a constant recovery rate. Under each of the five structural models, we can calculate the corresponding risk-neutral default probability and then the CDS spread. Let Q(t,τ) denote the unconditional default probability over (t,t+τ] under the risk-neutral measure (or the forward measure with stochastic interest rates). Then the spread of a τ-year CDS contract is given by (under a one-factor model) (1 R) t+τ e−rsQ′(t,s)ds cds(t,τ) = − t (6) t+τ e− R rs[1 Q(t,s)]ds t − R r(1 R)G(t,τ) = − , (7) 1 e−rτ[1 Q(t,τ)] G(t,τ) − − − where R is the recovery, r is the interest rate, and t+τ G(t,τ) = e−rsQ′(t,s)ds (8) Z t Eq. (7) holds for constant interest rate. As a result, the implementation of the structural models amounts to the calculation of the default probability Q( , ) either analytically or · · numerically. The default probability in the Merton (1974) and the Black and Cox (1976) models is known to have closed form solutions. The default probability in the double exponential jump diffusion model and the two-factor models do not have a known closed form solution but can be calculated using a numerical method (see, e.g., Huang and Huang (2003) for details). 3 A Specification Test of Structural Models Weusethefundamentalpricingrelationshipsimpliedbyvariouscreditriskmodelstoidentify structuralparameterslikeassetvolatility,defaultbarrier,jumpintensity,ordynamicleverage coefficient(s). The intuition is from Merton (1974) — the delta function and pricing equation link equity volatility and credit spread directly to the structural variables and parameters. 6

With these identifying restrictions, we can build an internally consistent GMM estimator (Hansen, 1982), which minimizes the fitted errors of credit spreads and equity volatility, with an appropriate weighting matrix determined by the pricing model and data sample. Along with consistent parameter estimation, we obtain an omnibus specification test, to rank order various credit risk models and to judge their pricing performance in a systematic framework. In addition, we also use the term structure and time series of CDS spreads to evaluate the economic pricing errors, which should by-and-large confirm our GMM specification test results. A structural model would be rejected by the GMM criterion function test, if the pricing errors are relatively large and exhibit systematic variations, assuming that the equity and credit markets are efficient. The implementation of our estimation strategy has several advantages. First, we use high frequency equity returns to construct a more accurate estimate of the equity volatility, therefore minimizing the measurement error imputed into the asset volatility estimate (given any structural model for the underlying asset process), while leaving the main suspect to possible model misspecification which we really care about. Second, we use the CDS spreads as a relative purer measure of the credit risk, therefore sanitizes our approach from the specific pricing error problem associated with bond market iniquity or other non-default characteristics (Longstaff, Mithal, and Neis, 2005). In addition, we use the term structure and time series of CDS spreads in both estimation and pricing exercise, while holding constant the model specification and parameter values, thus avoiding the rolling sample extraction approach that is inconsistent with economic assumption underlying the structural models. More importantly, by bringing in the consistency between observed equity and model implied equity, our approach has the potential to speak directly to the recent finding that time-varying equity volatility has a strong nonlinear forecasting power for credit spreads (Zhang, Zhou, and Zhu, 2006). 3.1 GMM Estimation of Structural Credit Risk Models As described in Section 2, the CDS spread at time t with maturity τ has a general pricing formula for all the structural models under consideration, (1 R) t+τ e−rsQ′(t,s)ds cds(t,τ) = − t , (9) t+τ e− R rs[1 Q(t,s)]ds t − R wherer istheriskfreerate, Ristherecoveryrate,Q(t,s)isthemodel-dependentrisk-neutral default probability at time t for period s, and Q′(t,s) is the risk-neutral default intensity. 7

As pointed out by Merton (1974), the delta function relating the equity volatility and asset volatility is also model-dependent A ∂E t t σ (t) = σ , (10) E A E ∂A t t where the equity volatility σ (t) is generally time-varying while the asset volatility σ may E A be constant. For the jump-diffusion asset value process used by Huang and Huang (2003), the equity volatility of the continuous diffusion component satisfy Eq. (10). With observed CDS spread cds(t,τ) and equity volatility σ (t), we can specify the following overidentifying E restrictions g f cds(t,τ ) cds(t,τ ) 1 1  −  g f(θ,t) =  ··················  (11)  cds(t,τ ) cds(t,τ )   j − j     gσ (t) σ (t)   E − E  where θ is the structural parameter vector fofr various credit risk model under consideration, e.g.,assetvolatility,defaultbarrier,assetjumpintensity,ordynamicleveragecoefficient,etc.. The term structure of CDS spread is represented by four maturities τ = 1, 3, 5, and 10 years. Under the null hypothesis that the model is correctly specified, we have E[f(θ,t)] = 0. (12) Note that both the CDS spread and the equity volatility are allowed to be observed with measurement errors. However, under the appropriately defined GMM metric, these pricing errors must be “small”; or the model specification will be rejected. The corresponding GMM estimator is given by θ ˆ = argming(θ,T)′W(T)g(θ,T), (13) where g(θ,T) refers to the sample mean of the moment conditions, g(θ,T) 1/T T f(θ,t), ≡ t P and W(T) denotes the asymptotic covariance matrix of g(θ,T) (Hansen, 1982). With mild regularity conditions, the estimator of structural parameter θ is √T-consistent and asymptotically normally distributed, under the null hypothesis. Moreover, the minimized value of the objective function multiplied by the sample size J = T ming(θ,T)′W(T)g(θ,T) (14) θ shouldbeasymptoticallydistributedasaChi-squaredistribution, withthedegreeoffreedom equaltothelengthofmomentsf(θ,t)minusthelengthofstructuralparameterθ. Thisallows 8

foranomnibustestoftheoveridentifyingrestrictions. Inaddition,weuseaheteroscedasticity robust estimator for the variance-covariance matrix W(T) (Newey and West, 1987). For the more general models with stochastic interest rate (Longstaff and Schwartz, 1995) or random jumps (Huang and Huang, 2003), one can expand moment restriction vector to include the pricing restrictions of interest rates and/or other maturities of CDS spreads. However, such a joint estimation scheme would be very computationally involved for a twofactor model with stochastic interest rates such as Longstaff and Schwartz (1995) and Collin- Dufresne and Goldstein (2001). This is because the default probability under the forward probability measure, Q(t, ), has to be calculated with discretized numerical approximation. · Tomaketheestimationtractable, weseparatelyestimatethedynamicinterestratemodel and the firm-specific structural parameters. This is a reasonable strategy, since the interest rate parameters are common inputs in those structural credit risk models and those firmspecific parameters do not affect the interest rate process. We use the 3-month LIBOR as an proxy for the short rate and estimate the interest rate volatility σˆ = VAR(r ) accordingly. r t Giventhattheone-factorVasicek(1977)modelisaverycrudeapproximationtotheobserved term structure dynamics, we opt to use a nonlinear least square procedure to estimate the risk-neutral drift parameters α and β month-by-month, τ6 αˆ ,β ˆ = argmin [yˆ y (α,β)]2 t t t,τ t,τ { } τX=τ1 − to best match the term structure of interest swap rates y with maturities of 1, 2, 3, 5, 7, t,τ and 10 years. 4 Data Description 4.1 Credit Default Swap Spreads We choose to use the credit default swap (CDS) premium as a direct measure of credit spreads. CDS is the most popular instrument in the rapidly growing credit derivatives markets. Compared with corporate bond spreads, which were widely used in previous studies in testing structural models, CDS spreads have two important advantages. First, a CDS spread isarelativelypurepricingofdefaultriskoftheunderlyingentity, andthecontractistypically traded on standardized terms. By contrast, bond spreads are more likely to be affected by differences in contractual arrangements, such as seniority, coupon rates, embedded options, 9

and guarantees.3 Second, as shown by Blanco, Brennan, and March (2005) and Zhu (2006), while CDS and bond spreads are quite in line with each other in the long run, in the short run CDS spreads tend to respond more quickly to changes in credit conditions. This means that CDS market may be more efficient than bond market, therefore more appropriate for the specification tests of structural models. Our CDS data are provided by Markit, a comprehensive data source that assembles a network of industry-leading partners who contribute information across several thousand credits on a daily basis. Based on the contributed quotes Markit creates the daily composite quoteforeachCDScontract; whichmustpastthestaledatatest, flatcurvetest, andoutlying data test. Together with the pricing information, the dataset also reports average recovery rates used by data contributors in pricing each CDS contract. In addition, an average of Moody’s and S&P ratings is also included. In this paper we include all CDS quotes written on US entities (sovereign entities excluded) and denominated in US dollars. We eliminate the subordinated class of contracts because of their small relevance in the database and unappealing implication in credit risk pricing. We focus on CDS contracts with modified restructuring (MR) clauses, as they are the most popularly traded in the US market. We require that the CDS time series has at least 36 consecutive monthly observations to be included in the final sample. Another filter is that CDS data have to match equity price (CRSP), equity volatility (TAQ) and accounting variables (COMPUSTAT). We also exclude financial and utility sectors, following previous empirical studies on structural models. After applying these filters, we are left with 93 entities in our study. Our sample period covers January 2002 to December 2004, with maturities of 1, 2, 3, 5, 7, and 10 years.4 For each entity, we create the monthly CDS spread by selecting the latest composite quote in each month, and, similarly, the monthly recovery rates linked to CDS spreads. 4.2 Equity Volatility from High Frequency Data By the theory of quadratic variation, it is possible to construct increasingly accurate measure for the model-free realized volatility or average volatility, during a fixed time interval, say a day or a month, by summing increasingly finer sampled squared high-frequency returns (Andersen, Bollerslev, Diebold, and Labys, 2001b; Barndorff-Nielsen and Shephard, 2002; 3For example, Longstaff, Mithal, and Neis (2005) find that a large proportion of bond spreads are determined by liquidity factors, which do not necessarily reflect the default risk of the underlying asset. 4Additional maturities of 0.5, 15, 20, and 30 years are also available for the CDS data set. Due to the liquidity concern and missing value, we choose to focus on CDS with maturity between 1 and 10 years. 10

Meddahi, 2002). The relative improvement of the high-frequency volatility estimate over the low frequency one is clearly demonstrated by Andersen and Bollerslev (1998) and Andersen, Bollerslev, Diebold, and Ebens (2001a), and its empirical applicability to equity return volatility has been widely accepted (see, Andersen, Bollerslev, and Diebold, 2003, for a survey). In testing structural models, the asset return volatility is unobserved and is usually backed out from the observed equity return volatility (Eom, Helwege, and Huang, 2004), therefore a more accurate measure of equity volatility from high-frequency data is critical in correctly estimating the asset return volatility — the driving force behind behind the firm default risk. Let s logS denote the day t logarithmic price of the firm equity, and the intraday t t ≡ returns are defined as follows: rs s s , (15) t,i ≡ t,i·∆ − t,(i−1)·∆ where rs refers to the ith within-day return on day t and ∆ is the sampling frequency and t,i chosen to be 5-minute. The realized equity volatility (squared) for period t is simply given as 1/∆ σ (t)2 (rs )2 (16) E ≡ X t,i i=1 f which converges to the integrated or average variance during period t. For the doubleexponential jump-diffusion model, the continuous component of equity volatility (squared) can be estimated with the so-called “bi-power variation” π 1/∆ 1/∆ σ (t)2 rs rs . (17) E ≡ 21/∆ 1 X| t,i−1|| t,i| − i=2 f As shown by Barndorff-Nielsen and Shephard (2004), such an estimator of realized equity volatility is robust to the presence of rare and large jumps. The data are provided by the NYSETAQ(TradeandQuote)database, whichincludesintra-day(tick-by-tick)transaction data for all securities listed on NYSE, AMEX, and NASDAQ. The monthly realized variance is the sum of daily realized variances, constructed from the squares of log intra-day 5-minute returns. Then, monthly realized volatility is just the square-root of the annualized monthly realized variance. 11

4.3 Capital Structure and Asset Payout Assets and liabilities are key variables in evaluating structural models of credit risk. The accounting information is obtained from Compustat on a quarterly basis and assigned to each month with the quarter. We calculate the firm asset as the sum of total liability plus market equity, where the market equity is obtained from the monthly CRSP data on shares outstanding and equity prices. Leverage ratio is estimated by the ratio of total liability to the firm asset. The asset payout ratio is proxied by the weighted average of the interest expense and dividend payout. Both ratios are reported as annualized percentages. 4.4 Risk-Free Interest Rates Toproxytherisk-freeinterestratesusedasthebenchmarkinthecalculationofCDSspreads, we use the 3-month LIBOR and the interest rate swaps with maturities of 1, 2, 3, 5, 7, and 10 years. These data are available from the Federal Reserve H.15 Release. 5 Empirical Results In this section we summarize our empirical findings on testing the structural credit risk models, basedontheGMMestimatordefinedinSection3withthetermstructureofCDSspreads and equity volatility. We also provide some diagnostics on various model specifications based on the pricing errors, and discuss some implications for future research. 5.1 Summary Statistics In this paper, we focus on the senior unsecured CDS contracts on U.S. corporations and denominatedinU.S.dollars. Subordinatedclassofcontractsarenotconsideredherefortheir small representations in the fast growing CDS market and their complicated implications in credit risk pricing. We use only the modified restructuring (MR) clauses, as they are the most popularly traded in the U.S. market. After matching with the high frequency equity volatility and firm accounting information, excluding financial and utility firms, we are left with 93 entities spanning from January 2002 to December 2004. Table 1 provides summary statistics on CDS spreads and firm characteristics across both rating categories and sectors. As can be seen from panel A of Table 1, our sample is concentrated in the single-A and triple-B categories, which account for 75 percent of the total 12

sample, reflecting the fact that contracts on investment-grade names dominate the CDS market. In terms of the average over both the time-series and cross-section in our sample, the 5-year CDS spread is 144 basis points, equity volatility is 38.40 percent (annualized), the leverage ratio 48.34 percent, asset payout ratio 2.14 percent, and the quoted recovery rate 40.30 percent. As expected, the CDS spread, equity volatility, and the leverage ratio all increase as rating deteriorates. However, the recovery rate essentially decreases as rating deteriorates but has low variations. Figure 1 plots both the term structure (from 1 year to 10 years) and time evolution (over the period from January 2002 to December 2004) of the average CDS spreads. As can be seen from the figure, the average spreads show large variations and have a peak around late 2002. Figure 2 plots both the 5-year CDS spreads and equity volatility by ratings over the entire sample period. The 5-year CDS spreads clearly have a peak in late 2002 across all three rating groups although the high-yield group has another spike in late 2004. On the other hand, equity volatility is much higher in 2002 than the later part of the sample period and, in particular, has two huge spikes in 2002. 5.2 GMM Specification Test Our econometric method is based on the model implied pricing relationship for CDS spread and equity volatility. There is clear evidence that equity volatility and credit spread are intimately related (Campbell and Taksler, 2003), and the linkage appears to be nonlinear in nature (Zhang, Zhou, and Zhu, 2006). A casual inspection of Figure 2 indicates that CDS spreads and equity volatilities appear to move together sometime during market turmoils but are only loosely related during quiet periods. A structural model with richer time-varying feature in the underlying asset may be called for to account for the observed nonlinear relationship between equity volatility and credit spread.5 The GMM specification test results from each of five structural credit risk models are given in Table 2. In particular, we report the percentage of firms where each of the five models is not rejected, for the whole sample as well as across both ratings and sectors. As can be seen from the table, none of the five models have a rejection rate of 100%. The 5InordertoestimatethestochasticinterestratemodelofLongstaffandSchwartz(1995)andthedynamic leverage model of Collin-Dufresne and Goldstein (2001), we need to first estimate the default-free term structure model of Vasicek (1977) in Eq. (3). Parameter estimates are obtained monthly based on crosssectional data, and not reported here for brevity. The cross-sectional pricing errors that range from 12 to 112 basis points during the sample period. 13

existing empirical studies of the standard structural models based on corporate bond spreads have largely rejected these models as well. Our results indicates that the standard structural models are still missing something even when CDS spreads, presumably a cleaner measure of credit risk than corporate bonds spreads, are used in the empirical analysis. Nonetheless, our empirical results provide new evidence on the relative performance of the five structural models and potential guidance on how to extend the existing models. For instance, notice that the GMM test statistics for the Merton (1974) specification are significantly higher than those for the other four extended models. (Some of the models are not nested so the J-test statistics are not always directly comparable.) Whereas it is known that the Merton model underperforms the richer models, our results are the first in the literature based on a consistent econometric test that takes into account the dynamic behavior of both CDS spreads and equity volatility. Judged by the results reported in the table on the percentage of firms where each of the five models is not rejected, the ranking of the 5 models is as follows Merton < Black-Cox < LS < HH < CDG (This ranking is also consistent with results on the mean test statistic, although as cautioned earlier, J-test statistics are not always directly comparable.) In particular, the double exponential jump-diffusion model considered in Huang and Huang (2003) and especially the CDGstationaryleveragemodeloutperformsignificantlyovertheotherthreemodels, namely, Merton (1974), Black and Cox (1976), and Longstaff and Schwartz (1995). These results imply that both jumps and time varying leverage improve noticeably the model. One finding in Eom, Helwege, and Huang (2004) is that the CDG model improves marginally the fitting of bond spreads over the LS model. Our results here here indicate that the CDG model’s improvement over LS and other models as well is much more significant when CDS spreads are used in the analysis. Another possible reason is that the risk-neutral leverage parameters are estimated directly here, whereas they are estimated indirectly through their counter-parties in the physical measure in Eom, Helwege, and Huang (2004). (ItisactuallymentionedthatinEHHthatdirectestimatingtherisk-neutralleverage parameters may improve the performance of CDG.) Note that our empirical analysis is based on a consistent econometric method that takes thepricingmodelstotheentiretermstructureofCDSspreadsandequityvolatilityestimated using high frequency data. This is in contrast with the prevalent approach of rolling sample 14

estimation and extraction. Of course we are aware that the GMM omnibus test may be biased toward over-rejection of the true model specification (e.g., see, Tauchen, 1986). 5.3 Parameter Estimation In this subsection we report estimates of model parameters. First, we want to mention that we impose additional estimation restrictions to ensure proper identification of model parameters in the Longstaff-Schwartz (1995), CDG, and the jump diffusion models. For the Longstaff-Schwartz (1995) model, if the correlation coefficient ρ is allowed to be free, its estimated value is around -1.2 for almost all firms in the sample. Therefore we restrict ρ to be -1 in the estimation of this model. In the CDG model, the correlation coefficient ρ and sensitivity coefficient φ seem difficult to be simultaneously identifiable and the correlation coefficient is not bounded between -1 and +1. As a result, we impose the restriction that ρ = 0. In the double-exponential jump-diffusion model, the parameters pQ, ηQ, and ηQ enter u u d the solution function multiplicatively with λQ and are very difficult to identify in our GMM estimator. Currently we fix those jump parameters as follows: pQ = 0.5, ηQ = 5, and ηQ = 3, u u d which are similar to the calibration values adopted in Huang and Huang (2003). Table 3 reports estimates of the remaining model parameters and their standard errors across both ratings and sectors. Panel A shows the results for the asset volatility parameter σ , which enters all five models. This parameter is the most accurately estimated one and v significant at all conventional statistical levels. The level of the estimates is reasonable in all models. Panel B of Table 3 reports the estimated default boundary/barrier, a parameter that appears in the three models with a flat default boundary, namely, the Black and Cox (1976) barrier model, the Longstaff-Schwartz (1995) model with stochastic interest rates, and the jump model considered in Huang and Huang (2003). The default barrier scaled by the total debt, V /F, estimated using the BC model ranges from 65% to 103%. The estimates b based on LS are much higher. Results based on the jump model are largely consistent with calibration values used by Huang and Huang (2003) and the empirical estimates by Predescu (2005). Figure 3 plots the relationship between the estimated default boundary V /F and b the observed leverage ratio F/V. As can be seen from the figure, the slope is significantly negative, indicating that a higher default boundary is implied for lower rating names. This finding is also consistent with EHH’s findings based on corporate bond data. 15

Panel C of Table 3 reports the estimate of the jump intensity parameter in HH and the three leverage parameters in CDG. Notice that the estimated jump intensity levels for high-yield names are much higher than those for investment-grade names. In the stationary leverage model (Collin-Dufresne and Goldstein, 2001), parameter κ ℓ is the mean-reverting speed of the risk-neutral log leverage ratio log(K /V ). The mean t t estimated value κˆ ranges from 0.03 for the single CCC-rated name to 17.82 for AA-rated ℓ names, and is much larger than the calibrated value of 0.18 adopted by CDG and also the estimate based on regression in Frank and Goyal (2003). This is perhaps an indication that the model is missing some factor. Parameter ν in CDG is related to θ , the long-run mean of the risk-neutral leverage ℓ ratio, as the following θ = −rt+δt+σ v 2/2 ν. Our choice of estimating a constant ν would ℓ κℓ − imply a time-varying but deterministic θ . The mean estimate νˆ ranges from 0.11 for the ℓ single AAA-rated name to 1.00 for the single CCC-rated name, which is rather close to the calibration value of 0.60 used in CDG. Finally the sensitivity of leverage ratio in interest rate φ in CDG seems to be critical for the model to pass the GMM specification test. More specifically, φ measures the sensitivity of the firm-specific leverage ratio dynamics to the risk-free interest rate process. This is equivalent to the risk factor loading in standard asset pricing models. As can be seen in the table, the estimate of φ varies from a large positive number of the investment grade names to a large negative number of the speculative grade names. This suggests that firms with different credit standing have very different leverage ratio dynamics as the macroeconomic risk changes over time. Such a heterogeneity of dynamics leverage ratio is the key for CDG model pass the GMM omnibus test with more than half of the sample. 5.4 Pricing Performance Evaluation In the literature, the evaluation of structural credit risk models is generally based on comparing their pricing error on corporate bonds, although the models are typically not consistently estimated but rather judged based on ad hoc calibration or rolling sample extractions. Here we connect with the existing literature by looking at the pricing errors of candidate models, after the parameters are consistently estimated and model specification tests are conducted. If our approach is valid, then the specification test result should be consistent with the pric- 16

ing errors evaluations.6 To be more specific, for each month and each maturity, we use the estimated structural parameters and pricing solutions to calculate the model implied CDS spreads and equity volatility. Then we compute the simple difference, absolute difference, and percentage difference between the model implied and observed ones. Finally the mean of the pooled pricing errors is reported for each name. Table4reportsthepricingerrorsonbothCDSspreadsandequityvolatilitybyeachrating group and sector. As can be seen from the table, in terms of average errors, the Merton (1974) model seems to over-estimate the spreads, the barrier and LS models appear to under-estimate the spreads, and the jump and leverage models are more even. The fact that combiningequity price and CDS spreadswould make Merton (1974) overfitis similarly found by Predescu (2005). In terms of absolute pricing performance, the barrier model (Black and Cox, 1976) always outperforms the Merton (1974) model but underperforms the Longstaff- Schwartz (1995) model. The jump model used in Huang and Huang (2003) outperforms all these three models but is dominated by the dynamic leverage model of (Collin-Dufresne and Goldstein, 2001). These results contrast the findings of (Eom, Helwege, and Huang, 2004) based on corporate bond data that richer model specifications do not improve upon the Merton (1974) in terms of pricing errors. It is interesting that, judging from the percentage pricing errors, the jump model performs relatively better for the high rated firms, while the CDG model does better for the low rated firms. The results from equity volatility display similar patterns as those from the CDS spreads. A noticeable difference is that the absolute pricing errors on equity volatility are generally larger than those on CDS spreads, while the percentage pricing errors are about the same order of the magnitude. In order to pass the GMM J-test, a model must perform well on both CDS spreads and equity volatility. Results based on pricing errors indicate that except for the CDG model, the others fail in either one or both dimensions. 5.5 Further Diagnostics on Model Specifications In this subsection, we try to gain further insights on model specification errors, by examining the model-implied term structure and time series of CDS spreads, along with the modelimplied equity volatility. We also discuss some implications of this analysis for improving 6In estimation we use CDS with maturities of 1, 3, 5, 10 years and equity volatility; while 2 and 7 years are too sparse to be included in estimation, they are still useful to be included in pricing error evaluation. 17

the standard structural models. Figure 4 plots the sample average of the CDS term structure from 1 year to 10 years from both the observed data and the five candidate models. A few observations are worth mentioning here: (1) the CDG model almost completely nails the average term structure, especially for the lower rating group; (2) the Merton model clearly misses the CDS spreads, but for high grade (AAA-A) misses mostly the long maturity and for low grade (BBB- CCC) misses mostly the short maturity; (3) the Black-Cox (BC) and LS models seem to fit reasonably the lower grades (BBB-CCC), but underfit the high grades (AAA-A) especially in the short end; (4) the HH model with jumps improves upon the Merton model mostly in the short end, as jumps are sensitive for short term derivatives, although its overall performance is not very satisfying. Overall, the stationary leverage model seems to be the only one to match the curve of average term structure of CDS spread, especially for lower rated names; while the jump model seems to have potential in improving the short end of the term structure, especially for higher rated names. Figure 5 plots the observed 5-year CDS spread against the five model implied ones. For lower ratings BB-CCC, all models seem to match the time-variations of the 5-year CDS spread well, although the CDG model is the best one. For higher ratings AAA-BBB, most models completely miss the CDS dynamics, especially for the first third of the sample, when the risk-free rate remains as low as 1%. Even CDG model can only get the average level right, but not be able to imitate the evolutions. This suggests that for higher rating firms, a time-varying factor in addition to interest rate and leverage ratio — like stochastic asset volatility — may be needed to fully capture the temporal changes in CDS spreads. Figure 6 reports the model implied and fitted equity volatilities. Again, for lower ratings BB-CCC, the CDG model can reasonable capture the time series feature of equity volatility; while other models miss the volatility level, yet produce certain time-variations imitating the volatility dynamics. In contrast, for higher ratings AAA-BBB, all models miss completely thevolatilityspikesduringtheearlysampleperiod. ThepictureforAAA-Aisratherbleak— everymodelgeneratesanearlyconstantequityvolatilitybuttheobservedoneisdramatically changing over time. This evidence indicate that without time varying asset volatility, no existing model can replicate the observed equity volatility dynamics, for top investment grade names. Figure 7 plots the initial spot log leverage ratio log(K /V ) and the long-run mean of t t risk-neutral log leverage ratio. It is clear that for the speculative grade (CCC-BB) these two 18

leverages are very closer to each other. While for low investment grade names (BBB), the observed leverage is significantly lower than the risk-neutral counterpart; and the difference becomes more dramatic for the top investment grade (AAA-A). Such a finding mirrors the recently documented evidence that highly profitable firms may opt to borrow little or no debt (Strebulaev and Yang, 2006; Chen and Zhao, 2006). Such a puzzle is worth further investigation. In summary, dynamic leverage ratio together with stochastic interest rate seem to be crucialforastructuralcreditriskmodeltobettermatchtheCDSspreadandequityvolatility. In addition, incorporating jumps may help to improve the fit of the short end of CDS term structure, especially for the high investment grade names. However, something else needs to be incorporate into the existing models as they all fail to adequately capture the dynamics behavior of CDS spreads and equity volatility, especially for the high investment grade names. This suggests that incorporating a stochastic asset volatility may improve the existing structural models. 6 Conclusions This article provides a consistent econometric specification test of five structural credit risk models using information from both the credit default swap (CDS) market and equity market. In particular, we consider the standard Merton (1974) model, the Black and Cox (1976) barrier model, the Longstaff and Schwartz (1995) model with stochastic interest rates, the stationary leverage model of Collin-Dufresne and Goldstein (2001), and the double exponential jump-diffusion barrier model studied in Huang and Huang (2003). We examine the performance of each model in capturing the behavior of CDS spreads and equity volatility both cross-sectionally and time series wise. Existing empirical studies of structural models mainly based on corporate bond spreads and equity volatility from low frequency daily data. To our best knowledge, this study is the first direct econometric estimation and specification test of structural models using data on the term structure of CDS and equity volatility estimated with high frequency intraday data. This allows us to minimize the effects of measurement error and pricing error, and thus attribute the test results mostly to the specification error. We find that the Merton (1974), Black and Cox (1976), and the Longstaff and Schwartz (1995) models are strongly rejected by our specification test. The jump diffusion model 19

considered in Huang and Huang (2003) improves the performance significantly for the top investment grade names but helps the fit mainly in the short end of the CDS term structure and not much in the long end. Still, the model is rejected for more than half of our sample firms. The best of the five models is the Collin-Dufresne and Goldstein model, that cannot be rejected in more than half of our sample firms. Nonetheless, we show that these structural models still have difficulty predicting credit spreadsaccuratelyevenwhenCDSspreads(apurermeasureofcreditriskthanbondspreads) are used in the analysis. Finally, we document that the five structural models cannot capture the time-series behavior of both CDS spreads and equity volatility. Given that equity volatility in structural models is time-varying, this finding provides a direct evidence that a structural model with stochastic asset volatility (see Huang and Huang, 2003; Huang, 2005; Zhang, Zhou, and Zhu, 2006) may significantly improve the model performance, especially for the investment grade names. 20

References Acharya, Viral and Jennifer Carpenter (2002), “Corporate Bond Valuation and Hedging withStochasticInterestRatesandEndogenousBankruptcy,” Review of Financial Studies, vol. 15, 1355–1383. Andersen, Torben G. and Tim Bollerslev (1998), “Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts,” International Economic Review, vol. 39, 885–905. Andersen, Torben G., Tim Bollerslev, and Francis X. Diebold (2003), Handbook of Financial Econometrics, chap. Parametric and Nonparametric Volatility Measurement, Elsevier Science B.V., Amsterdam, forthcoming. Andersen, Torben G., Tim Bollerslev, Francis X. Diebold, and Heiko Ebens (2001a), “The DistributionofRealizedStockReturnVolatility,” Journal of Financial Economics, vol.61, 43–76. Andersen, Torben G., Tim Bollerslev, Francis X. Diebold, and Paul Labys (2001b), “The Distribution of Realized Exchange Rate Volatility,” Journal of the American Statistical Association, vol. 96, 42–55. Anderson, Ronald W. and Suresh Sundaresan (1996), “Design and valuation of debt contracts,” Review of Financial Studies, vol. 9, 37–68. Arora, N., J.R. Bohn, and F. Zhu (2005), “Reduced Form vs. Structural Models of Credit Risk: A Case Study of Three Models,” Journal of Investment Management, vol. 3, 43–67. Barndorff-Nielsen, Ole and Neil Shephard (2002), “Estimating Quadratic Variation Using Realized Variance,” Journal of Applied Econometrics, vol. 17, 457–478. Barndorff-Nielsen, Ole and Neil Shephard (2004), “Power and Bipower Variation with Stochastic Volatility and Jumps,” Journal of Financial Econometrics, vol. 2, 1–48. Black, Fischer and John Cox (1976), “Valuing Corporate Securities: Some Effects of Bond Indenture Provisions,” Journal of Finance, vol. 31, 351–367. Black, Fischer and Myron S Scholes (1973), “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, vol. 81, 637–54. Blanco, Roberto, Simon Brennan, and Ian W. March (2005), “An Empirical Analysis of the Dynamic Relationship Between Investment-Grade Bonds and Credit Default Swaps,” Journal of Finance, vol. 60, 2255–2281. 21

Campbell, John and Glen Taksler (2003), “Equity Volatility and Corporate Bond Yields,” Journal of Finance, vol. 58, 2321–2349. Chen, Long and Xinlei Zhao (2006), “Why Do more Profitable Firms Have Lower Leverage Ratios?” Working Paper. Chen, R.R., F.J. Fabozzi, G.G. Pan, and R. Sverdlove (2006), “Sources of Credit Risk: Evidence from Credit Default Swaps,” The Journal of Fixed Income, vol. 16, 7–21. Collin-Dufresne, Pierre and Robert Goldstein (2001), “Do Credit Spreads Reflect Stationary Leverage Ratios?” Journal of Finance, vol. 56, 1929–1957. Cossin, Didier and Tomas Hricko (2001), “Exploring for the Determinants of Credit Risk in Credit Default Swap Transaction Data,” Working Paper. Cremers, Martijn, Joost Driessen, Pascal Maenhout, and David Weinbaum (2004), “Explaining the Level of Credit Spreads: Option-Implied Jump Risk Premia in a Firm Value Model,” Working Paper, Cornell University. Delianedis, G. and R. Geske (2001), “The components of corporate credit spreads: Default, recovery, tax, jumps, liquidity, and market factors,” working paper, UCLA. Duan, J.C. (1994), “Maximum Likelihood Estimation Using Price Data Of The Derivative Contract,” Mathematical Finance, vol. 4, 155–167. Duffie, Darrell and David Lando (2001), “Term structures of credit spreads with incomplete accounting information,” Econometrica, vol. 69, 633–664. Eom, YoungHo, JeanHelwege, andJing-ZhiHuang(2004), “StructuralModelsofCorporate Bond Pricing: An Empirical Analysis,” Review of Financial Studies, vol. 17, 499–544. Ericsson, J. and J. Reneby (2005), “Estimating Structural Bond Pricing Models,” Journal of Business, vol. 78, 707–735. Ericsson, Jan, KrisJacobs, andRodolfoOviedo(2005), “TheDeterminantsofCreditDefault Swap Premia,” Working Paper, McGill University. Ericsson, Jan, Joel Reneby, and Hao Wang (2006), “Can Structural Models Price Default Risk? Evidence from Bond and Credit Derivative Markets,” Working Paper, McGill University. Geske, Robert(1977), “TheValuationofCorporateLiabilitiesasCompoundOptions,” Journal of Financial and Quantitative Analysis, vol. 12, 541–552. 22

Hansen, Lars Peter (1982), “Large Sample Properties of Generalized Method of Moments Estimators,” Econometrica, vol. 50, 1029–1054. Houweling, Patrick and Ton Vorst (2005), “Pricing Default Swaps: Empirical Evidence,” Journal of International Money and Finance, vol. 24, 1200–1225. Huang, Jingzhi (2005), “Affine Structural Models of Corporate Bond Pricing,” Working Paper, Penn State University. Huang, JingzhiandMingHuang(2003), “HowMuchoftheCorporate-TreasuryYieldSpread Is Due to Credit Risk?” Working Paper, Penn State and Stanford. Hull, John, Izzy Nelken, and Alan White (2004), “Merton’s Model, Credit Risk, and Volatility Skews,” Working Paper, University of Toronto. Jones, E. Philip, Scott P. Mason, and Eric Rosenfeld (1984), “Contingent Claims Analysis of Corporate Capital Structures: An Empirical Investigation,” Journal of Finance, vol. 39, 611–625. Leland, Hayne E. (1998), “Agency Costs, Risk Management, and Capital Structure,” The Journal of Finance, vol. 53, 1213–1243. Leland, Hayne E. and Klaus B. Toft (1996), “Optimal Capital Structure, Endogenous Bankruptcy, and the Term Structure of Credit Spreads,” Journal of Finance, vol. 51, 987–1019. Longstaff, Francis and Eduardo Schwartz (1995), “A Simple Approach to Valuing Risky Fixed and Floating Rate Debt,” Journal of Finance, vol. 50, 789–820. Longstaff, Francis A., Sanjay Mithal, and Eric Neis (2005), “Corporate Yield Spreads: Default Risk or Liquidity? New Evidence from the Credit-Default-Swap Market,” Journal of Finance, vol. 60, 2213–2253. Lyden, S. and D. Saraniti (2000), “An Empirical Examination of the Classical Theory of Corporate Security Valuation,” working paper, Barclays Global Investors. Meddahi, Nour (2002), “A Theoretical Comparison between Integrated and Realized Volatility,” Journal of Applied Econometrics, vol. 17. Mella-Barral, Pierre and William Perraudin (1997), “Strategic Debt Service,” The Journal of Finance, vol. 52, 531–556. Merton, Robert (1974), “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates,” Journal of Finance, vol. 29, 449–470. 23

Newey, Whitney K. and Kenneth D. West (1987), “A Simple Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix,” Econometrica, vol. 55, 703–708. Predescu, Mirela (2005), “The Performance of Structural Models of Default for Firms with Liquid CDS Spreads,” Working Paper, Rotman School of Management, University of Toronto. Schaefer, Stephen and Ilya A. Strebulaev (2004), “Structural Models of Credit Risk are Useful: Evidence from Hedge Ratios on Corporate Bonds,” Working Paper, London Business School. Strebulaev, Ilya and Baozhong Yang (2006), “The Mystery of Zero-Leverage Firms,” Working Ppaer, Standford GSB. Tauchen,George(1986), “StatisticalPropertiesofGeneralizedMethods-of-MomentsEstimators of Structural Parameters Obtained from Financial Market Data,” Journal of Business and Economic Statistics, vol. 4, 397–416. Vasicek, Oldrich A. (1977), “An Equilibrium Characterization of the Term Structure,” Journal of Financial Economics, vol. 5, 177–188. Zhang, BenjaminYibin, HaoZhou, andHaibinZhu(2006), “ExplainingCreditDefaultSwap Spreads with Equity Volatility and Jump Risks of Individual Firms,” Working Paper, Federal Reserve Board. Zhu, Haibin (2006), “An Empirical Comparison of Credit Spreads between the Bond Market and the Credit Default Swap Market,” Journal of Financial Service Research, vol. 29, 211–235. 24

Table 1: Summary Statistics on CDS Spreads and the Underlying Names This table reports summary statistics on the 93 firms, by ratings (Panel A) and sectors (Panel B), that underly the CDS contracts in the entire sample. Rating is the average of Moody’s and Standard & Poor’s ratings. Equity volatility is estimated using 5-minute intraday returns. Leverage ratio is the total liability divided by the total asset which is equal to total liability plus market equity. Asset payout ratio is the weighted average of dividend payout and interest expense over the total asset. Recovery rate is the quoted recovery rate accompanied with the CDS premium from the dealer-market. CDS spreads have 1-, 2-, 3-, 5-, 7-, and 10-year maturities over the period from January 2002 to December 2004. Panel A: By Ratings Rating Firms % of Equity Leverage Asset Recovery Sample Volatility (%) Ratio (%) Payout (%) Rate (%) AAA 1 1.08% 36.36 63.71 2.22 40.88 AA 6 6.45% 31.50 20.92 1.53 40.92 A 25 26.88% 32.51 38.15 2.02 40.57 BBB 45 48.39% 35.54 51.84 2.26 40.73 BB 11 11.83% 47.19 57.76 2.15 39.51 B 4 4.30% 83.23 72.61 2.28 38.23 CCC 1 1.08% 81.94 93.93 2.89 26.57 Overall 93 100.00% 38.40 48.34 2.14 40.30 Maturity of CDS Rating 1-year 2-year 3-year 5-year 7-year 10-year CDS Spreads Mean (%) AAA 0.23 0.28 0.32 0.43 0.45 0.49 AA 0.12 0.13 0.15 0.20 0.23 0.28 A 0.25 0.29 0.32 0.39 0.43 0.49 BBB 0.74 0.79 0.86 0.94 0.98 1.05 BB 2.62 2.74 2.84 2.90 2.92 2.92 B 7.52 7.20 7.51 7.25 7.01 6.79 CCC 25.26 22.99 20.91 18.81 18.03 17.31 Overall 1.34 1.36 1.40 1.44 1.45 1.49 CDS Spreads Std. Dev. (%) AAA 0.17 0.19 0.21 0.25 0.23 0.24 AA 0.07 0.07 0.07 0.09 0.09 0.10 A 0.23 0.27 0.24 0.25 0.24 0.26 BBB 0.96 0.96 0.96 0.91 0.89 0.84 BB 2.72 2.75 2.59 2.35 2.28 2.14 B 8.67 6.19 7.61 6.12 5.90 5.25 CCC 24.96 19.40 16.48 13.65 12.68 11.81 Overall 4.434 3.775 3.615 3.177 3.036 2.854 25

Table 1: Summary Statistics on CDS Spreads and the Underlying Names Panel B: By Industry Sector Firms % of Equity Leverage Asset Recovery Sample Volatility (%) Ratio (%) Payout (%) Rate (%) Communications 6 6.45% 48.72 42.93 1.99 40.14 Consumer Cyclical 32 34.41% 38.95 48.56 2.01 40.45 Consumer Staple 14 15.05% 33.77 41.68 2.24 40.87 Energy 8 8.60% 39.93 53.89 2.47 40.05 Industrial 18 19.35% 40.24 53.90 2.01 39.90 Materials 11 11.83% 32.85 49.34 2.73 41.35 Technology 4 4.30% 45.22 40.20 1.29 38.95 Overall 93 100.00% 38.68 48.39 2.14 40.39 Maturity of CDS Sector 1-year 2-year 3-year 5-year 7-year 10-year CDS Spreads Mean (%) Communications 2.04 1.99 2.09 2.23 2.16 2.10 Consumer Cyclical 1.57 1.58 1.58 1.61 1.62 1.66 Consumer Staple 0.74 0.81 0.86 0.92 0.94 0.98 Energy 1.58 1.38 1.53 1.43 1.47 1.48 Industrial 1.29 1.38 1.41 1.46 1.48 1.53 Materials 0.92 0.96 1.03 1.10 1.14 1.20 Technology 1.38 1.43 1.48 1.48 1.51 1.52 Overall 1.34 1.36 1.40 1.44 1.45 1.49 CDS Spreads Std. Dev. (%) Communications 4.82 4.13 4.58 4.74 4.33 3.80 Consumer Cyclical 6.19 5.25 4.65 4.06 3.85 3.65 Consumer Staple 2.08 2.21 2.18 2.10 2.02 1.92 Energy 5.60 3.66 4.80 3.32 3.45 3.14 Industrial 2.36 2.54 2.34 2.16 2.09 2.07 Materials 1.46 1.42 1.43 1.39 1.38 1.34 Technology 2.20 2.17 2.12 1.82 1.74 1.59 Overall 4.43 3.78 3.62 3.18 3.04 2.85 26

Table 2: Specification Test of Structural Credit Risk Models This table reports the omnibus GMM test results of overidentifying restrictions under each of 5 structural models. The five moment conditions used in the test are constructed based on the pricing relationship for 1-, 3-, 5- and 10-year CDS spreads and for the equity volatility estimated based on 5-minute intraday data. The five model specifications considered include Merton (1974), Black and Cox (1976), Longstaff and Schwartz (1995), Collin-Dufresne and Goldstein (2001), and the double exponential jump diffusion model (Huang and Huang, 2003). Data used in the test are monthly CDS spreads and equity volatility from January 2002 to December 2004. Merton Model BC Model LS Model HH Model CDG Model Chi-Square d.o.f = 4 d.o.f = 3 d.o.f = 3 d.o.f = 2 d.o.f = 1 Mean 16.70 15.26 14.65 9.95 4.18 Nth 5th 50th 95th 5th 50th 95th 5th 50th 95th 5th 50th 95th 5th 50th 95th Percentile 12.84 17.28 18.04 10.74 15.58 17.69 7.27 15.60 17.79 3.65 10.11 15.66 0.01 2.29 16.52 Sig. level 0.01 0.05 0.10 0.01 0.05 0.10 0.01 0.05 0.10 0.01 0.05 0.10 0.01 0.05 0.10 Proportion Not Rejected 5/93 2/93 0/93 6/93 1/93 0/93 12/93 6/93 3/93 40/93 15/93 11/93 70/93 63/93 51/93 By Ratings AAA 0/1 0/1 0/1 0/1 0/1 0/1 0/1 0/1 0/1 1/1 1/1 1/1 0/1 0/1 0/1 AA 0/6 0/6 0/6 0/6 0/6 0/6 0/6 0/6 0/6 3/6 1/6 1/6 5/6 5/6 5/6 A 0/25 0/25 0/25 0/25 0/25 0/25 0/25 0/25 0/25 15/25 9/25 8/25 22/25 21/25 19/25 BBB 2/45 0/45 0/45 2/45 0/45 0/45 6/45 2/45 2/45 17/45 3/45 1/45 36/45 31/45 23/45 BB 3/11 2/11 0/11 2/11 1/11 0/11 4/11 2/11 1/11 2/11 0/11 0/11 2/11 2/11 2/11 B 0/4 0/4 0/4 2/4 0/4 0/4 1/4 1/4 0/4 1/4 1/4 0/4 4/4 4/4 2/4 CCC 0/1 0/1 0/1 0/1 0/1 0/1 1/1 1/1 0/1 1/1 0/1 0/1 1/1 0/1 0/1 By Sector Communications 1/6 0/6 0/6 2/6 0/6 0/6 1/6 0/6 0/6 4/6 0/6 0/6 5/6 5/6 4/6 Consumer Cyclical 1/32 0/32 0/32 0/32 0/32 0/32 3/32 1/32 0/32 10/32 2/32 2/32 24/32 22/32 18/32 Consumer Staple 0/14 0/14 0/14 0/14 0/14 0/14 0/14 0/14 0/14 9/14 3/14 2/14 11/14 9/14 8/14 Energy 0/8 0/8 0/8 1/8 0/8 0/8 1/8 1/8 0/8 3/8 1/8 0/8 4/8 3/8 2/8 Industrial 1/18 1/18 0/18 2/18 1/18 0/18 4/18 2/18 2/18 9/18 4/18 2/18 15/18 14/18 9/18 Materials 1/11 0/11 0/11 0/11 0/11 0/11 2/11 1/11 0/11 5/11 5/11 5/11 9/11 8/11 8/11 Technology 1/4 1/4 0/4 1/4 0/4 0/4 1/4 1/4 1/4 0/4 0/4 0/4 2/4 2/4 2/4 27

Table 3: Parameter Estimation of Structural Credit Risk Models This table reports the GMM estimation results of the model parameters in each of five structural models. The five moment conditions used in the test are constructed based on the pricing relationship for 1-, 2-, 5- and 10-year CDS spreads and for the equity volatility estimated based on 5-minute intraday data. The five model specifications include Merton (1974), Black and Cox (1976), Longstaff and Schwartz (1995), Collin-Dufresne and Goldstein (2001), and the double exponential jump diffusion model (Huang and Huang, 2003). Panel A reports the asset volatility parameter estimate σ in all five models, Panel B reports the default boundary estimate K in v three barrier type models, and Panel C reports jump intensity estimate λQ in Huang and Huang (2003) model and dynamic leverage parameters κ , ν, φ in Collin-Dufresne and Goldstein (2001) model. ℓ Panel A: Estimate of the Asset Volatility Asset Volatility Merton Model BC Model LS Model HH Model CDG Model Whole Sample N Mean Median Mean Median Mean Median Mean Median Mean Median 93 0.141 0.123 0.178 0.171 0.163 0.151 0.158 0.153 0.186 0.166 (0.007) (0.006) (0.008) (0.007) (0.009) (0.008) (0.006) (0.005) (0.012) (0.011) Percentile N 5th 95th 5th 95th 5th 95th 5th 95th 5th 95th 0.055 0.348 0.087 0.308 0.077 0.299 0.093 0.273 0.081 0.331 (0.002) (0.016) (0.003) (0.016) (0.003) (0.017) (0.003) (0.012) (0.004) (0.027) Ratings N Mean Median Mean Median Mean Median Mean Median Mean Median AAA 1 0.143 0.143 0.087 0.087 0.074 0.074 0.129 0.129 0.108 0.108 (0.004) (0.004) (0.003) (0.003) (0.005) (0.005) (0.007) (0.007) (0.014) (0.014) AA 6 0.054 0.051 0.288 0.302 0.235 0.250 0.180 0.182 0.253 0.249 (0.014) (0.014) (0.015) (0.015) (0.015) (0.015) (0.007) (0.006) (0.020) (0.020) A 25 0.096 0.089 0.176 0.179 0.157 0.160 0.162 0.163 0.198 0.190 (0.009) (0.009) (0.007) (0.007) (0.009) (0.009) (0.005) (0.005) (0.012) (0.011) BBB 45 0.136 0.126 0.151 0.141 0.138 0.132 0.148 0.150 0.167 0.157 (0.005) (0.004) (0.007) (0.006) (0.007) (0.007) (0.006) (0.005) (0.011) (0.010) BB 11 0.210 0.212 0.220 0.211 0.221 0.195 0.183 0.141 0.212 0.165 (0.006) (0.005) (0.008) (0.006) (0.011) (0.008) (0.008) (0.008) (0.013) (0.009) B 4 0.363 0.378 0.245 0.250 0.252 0.246 0.179 0.183 0.200 0.180 (0.010) (0.010) (0.009) (0.009) (0.010) (0.011) (0.010) (0.009) (0.011) (0.010) CCC 1 0.381 0.381 0.178 0.178 0.054 0.054 0.038 0.038 0.046 0.046 (0.011) (0.011) (0.010) (0.010) (0.002) (0.002) (0.004) (0.004) (0.002) (0.002) 28

Table 3: Parameter Estimation of Structural Credit Risk Models Panel A (count.): Estimate of the Asset Volatility Asset Volatility Merton Model BC Model LS Model HH Model CDG Model Whole Sample N Mean Median Mean Median Mean Median Mean Median Mean Median 93 0.141 0.123 0.178 0.171 0.163 0.151 0.158 0.153 0.186 0.166 (0.007) (0.006) (0.008) (0.007) (0.009) (0.008) (0.006) (0.005) (0.012) (0.011) Percentile N 5th 95th 5th 95th 5th 95th 5th 95th 5th 95th 0.055 0.348 0.087 0.308 0.077 0.299 0.093 0.273 0.081 0.331 (0.002) (0.016) (0.003) (0.016) (0.003) (0.017) (0.003) (0.012) (0.004) (0.027) Sector N Mean Median Mean Median Mean Median Mean Median Mean Median Communications 6 0.199 0.176 0.183 0.168 0.164 0.140 0.187 0.173 0.239 0.269 (0.006) (0.004) (0.007) (0.007) (0.010) (0.011) (0.007) (0.006) (0.020) (0.021) Consumer Cyclical 32 0.145 0.133 0.179 0.183 0.151 0.152 0.158 0.151 0.186 0.167 (0.007) (0.005) (0.008) (0.007) (0.008) (0.007) (0.006) (0.005) (0.012) (0.011) Consumer Staple 14 0.103 0.079 0.193 0.188 0.178 0.165 0.148 0.148 0.175 0.161 (0.008) (0.008) (0.009) (0.008) (0.010) (0.009) (0.005) (0.005) (0.012) (0.011) Energy 8 0.150 0.113 0.160 0.147 0.151 0.145 0.144 0.134 0.163 0.139 (0.006) (0.004) (0.007) (0.006) (0.007) (0.006) (0.006) (0.005) (0.008) (0.007) Industrial 18 0.151 0.124 0.164 0.140 0.163 0.130 0.160 0.151 0.175 0.159 (0.007) (0.006) (0.006) (0.006) (0.008) (0.007) (0.006) (0.005) (0.010) (0.009) Materials 11 0.119 0.112 0.163 0.159 0.144 0.120 0.144 0.145 0.167 0.185 (0.007) (0.004) (0.007) (0.006) (0.008) (0.009) (0.005) (0.005) (0.010) (0.010) Technology 4 0.164 0.148 0.256 0.250 0.268 0.249 0.208 0.183 0.280 0.243 (0.007) (0.008) (0.009) (0.009) (0.017) (0.017) (0.009) (0.008) (0.024) (0.017) 29

Table 3: Parameter Estimation of Structural Credit Risk Models Panel B: Estimate of the Default Boundary Default Barrier BC Model LS Model HH Model Whole Sample N Mean Median Mean Median Mean Median 93 0.911 0.827 1.163 1.093 0.824 0.761 (0.039) (0.033) (0.050) (0.039) (0.087) (0.064) Percentile N 5th 95th 5th 95th 5th 95th 0.606 1.444 0.660 1.881 0.492 1.505 (0.009) (0.086) (0.007) (0.117) (0.018) (0.259) Ratings N Mean Median Mean Median Mean Median AAA 1 0.959 0.959 1.115 1.115 0.759 0.759 (0.018) (0.018) (0.022) (0.022) (0.034) (0.034) AA 6 0.760 0.738 1.217 1.052 1.279 1.378 (0.081) (0.080) (0.115) (0.110) (0.166) (0.152) A 25 1.028 0.958 1.319 1.206 0.901 0.869 (0.050) (0.046) (0.064) (0.060) (0.094) (0.067) BBB 45 0.911 0.866 1.130 1.091 0.758 0.752 (0.035) (0.030) (0.041) (0.033) (0.085) (0.060) BB 11 0.834 0.751 1.041 1.019 0.821 0.711 (0.025) (0.022) (0.031) (0.036) (0.043) (0.032) B 4 0.655 0.630 0.857 0.864 0.480 0.504 (0.017) (0.016) (0.046) (0.029) (0.074) (0.079) CCC 1 0.736 0.736 1.011 1.011 0.603 0.603 (0.011) (0.011) (0.001) (0.001) (0.105) (0.105) Sector N Mean Median Mean Median Mean Median Communications 6 1.059 1.114 1.351 1.497 0.764 0.729 (0.037) (0.039) (0.072) (0.070) (0.128) (0.115) Consumer Cyclical 32 0.924 0.826 1.208 1.145 0.843 0.773 (0.043) (0.037) (0.050) (0.038) (0.099) (0.061) Consumer Staple 14 0.888 0.766 1.123 0.950 0.959 0.839 (0.053) (0.051) (0.064) (0.058) (0.078) (0.057) Energy 8 0.870 0.777 1.055 0.962 0.756 0.702 (0.031) (0.024) (0.042) (0.035) (0.062) (0.065) Industrial 18 0.902 0.904 1.136 1.129 0.743 0.744 (0.030) (0.026) (0.033) (0.030) (0.070) (0.047) Materials 11 0.904 0.855 1.137 1.197 0.760 0.792 (0.038) (0.028) (0.053) (0.043) (0.097) (0.066) Technology 4 0.815 0.640 1.067 0.935 0.965 0.734 (0.027) (0.027) (0.054) (0.048) (0.062) (0.057) 30

Table 3: Parameter Estimation of Structural Credit Risk Models Panel C: Estimates of Other Parameters in HH and CDG Model HH Model CDG Model Parameter λQ κ ν φ ℓ Whole Sample N Mean Median Mean Median Mean Median Mean Median 93 0.224 0.130 13.215 14.784 0.304 0.169 2.254 1.817 (0.061) (0.037) (0.095) (0.048) (0.085) (0.011) (0.418) (0.145) Percentile N 5th 95th 5th 95th 5th 95th 5th 95th 0.042 0.878 0.289 20.273 0.093 1.139 -2.609 5.008 (0.009) (0.176) (0.008) (0.463) (0.005) (0.198) (0.057) (1.130) Ratings N Mean Median Mean Median Mean Median Mean Median AAA 1 0.051 0.051 15.044 15.044 0.106 0.106 1.183 1.183 (0.011) (0.011) (0.023) (0.023) (0.012) (0.012) (0.151) (0.151) AA 6 0.095 0.091 17.818 19.147 0.855 0.349 13.383 3.304 (0.029) (0.024) (0.056) (0.036) (0.118) (0.022) (2.598) (0.506) A 25 0.125 0.129 16.471 16.015 0.185 0.173 2.225 1.981 (0.038) (0.027) (0.043) (0.026) (0.013) (0.011) (0.203) (0.157) BBB 45 0.173 0.134 14.407 14.816 0.218 0.148 1.491 1.724 (0.056) (0.037) (0.087) (0.045) (0.125) (0.009) (0.248) (0.121) BB 11 0.329 0.191 4.057 1.456 0.508 0.320 1.977 1.385 (0.072) (0.059) (0.157) (0.100) (0.045) (0.024) (0.475) (0.312) B 4 0.986 0.954 0.568 0.511 0.500 0.377 -3.438 -4.038 (0.248) (0.237) (0.436) (0.354) (0.111) (0.119) (0.417) (0.235) CCC 1 1.788 1.788 0.026 0.026 1.001 1.001 -2.609 -2.609 (0.286) (0.286) (0.007) (0.007) (0.252) (0.252) (0.065) (0.065) Sector N Mean Median Mean Median Mean Median Mean Median Communications 6 0.285 0.164 11.802 13.594 0.278 0.239 0.333 2.377 (0.121) (0.080) (0.205) (0.162) (0.052) (0.028) (0.518) (0.253) Consumer Cyclical 32 0.230 0.146 13.502 15.280 0.341 0.173 2.483 1.871 (0.064) (0.039) (0.098) (0.042) (0.176) (0.011) (0.336) (0.145) Consumer Staple 14 0.222 0.115 15.230 15.416 0.307 0.158 3.820 1.859 (0.033) (0.022) (0.052) (0.028) (0.052) (0.009) (1.128) (0.135) Energy 8 0.265 0.136 9.848 13.758 0.256 0.206 0.667 1.338 (0.051) (0.049) (0.185) (0.072) (0.046) (0.011) (0.303) (0.164) Industrial 18 0.201 0.117 14.145 15.175 0.251 0.145 0.478 1.613 (0.065) (0.031) (0.054) (0.042) (0.025) (0.008) (0.117) (0.101) Materials 11 0.228 0.115 13.105 14.125 0.245 0.168 4.473 1.899 (0.062) (0.039) (0.048) (0.052) (0.020) (0.011) (0.179) (0.182) Technology 4 0.119 0.138 8.836 8.960 0.523 0.380 2.871 2.564 (0.056) (0.047) (0.188) (0.087) (0.044) (0.023) (0.694) (0.618) 31

Table 4: Pricing Errors of CDS Spreads and Equity Volatility This table reports the pricing errors CDS Spreads and Equity Volatility under each of five structural models. The pricing errors of the CDS spreads are calculated as the average, absolute, average percentage, and absolute percentage differences between the model implied and observed spreads, across six maturities, 1, 2, 3, 5, 7, and 10 years, and monthly observations from January 2002 to December 2004. The fitted errors of equity volatility are calculated in a similar fashion. ThefivemodelspecificationsincludeMerton(1974), BlackandCox(1976), LongstaffandSchwartz (1995), Collin-Dufresne and Goldstein (2001), and the double exponential jump diffusion model (Huang and Huang, 2003). Panel A: By Ratings CDS Spreads Average Pricing Error Absolute Pricing Error Rating N Merton BC LS HH CDG Merton BC LS HH CDG Overall 93 0.35 -0.96 -0.52 -0.40 -0.06 1.58 1.01 1.06 0.70 0.67 AAA 1 0.23 -0.32 -0.25 -0.01 -0.11 0.39 0.32 0.26 0.19 0.20 AA 6 -0.19 -0.11 -0.16 -0.05 -0.06 0.19 0.13 0.16 0.09 0.12 A 25 -0.30 -0.24 -0.23 -0.07 -0.03 0.37 0.29 0.28 0.15 0.18 BBB 45 0.13 -0.70 -0.57 -0.27 -0.05 1.29 0.73 0.70 0.42 0.51 BB 11 -0.14 -1.43 -0.56 -0.20 -0.21 2.59 1.68 2.46 1.38 1.70 B 4 6.08 -5.18 -4.21 -3.57 -0.22 7.94 5.21 4.77 4.15 2.32 CCC 1 12.10 -13.81 7.07 -6.84 1.10 17.47 13.81 12.25 9.37 6.27 Equity Volatility Average Pricing Error Absolute Pricing Error Rating N Merton BC LS HH CDG Merton BC LS HH CDG Overall 93 -0.13 -3.26 -0.42 1.33 1.02 26.53 13.36 15.05 10.89 11.37 AAA 1 -3.02 -13.89 -16.65 0.40 -7.19 12.19 14.94 16.82 11.27 11.87 AA 6 -25.15 4.22 -2.38 -6.14 0.11 25.15 11.47 11.08 8.00 8.84 A 25 -17.60 -4.58 -7.17 -2.95 -0.58 18.17 10.07 10.53 7.49 8.45 BBB 45 -3.16 -3.81 -3.90 1.69 0.22 19.08 10.85 11.98 9.53 9.82 BB 11 3.66 4.16 13.11 10.78 8.55 22.18 20.24 22.69 19.59 19.60 B 4 63.02 -10.88 41.06 6.12 -2.33 79.04 39.07 54.34 22.63 21.98 CCC 1 431.35 -31.04 38.30 14.73 21.56 431.35 39.42 47.10 31.12 36.28 CDS Spreads Average Percentage Pricing Error Absolute Percentage Pricing Error Rating N Merton BC LS HH CDG Merton BC LS HH CDG Overall 93 24.36 -66.72 -36.36 -27.99 -4.11 109.94 70.70 73.77 48.50 46.91 AAA 1 62.12 -85.15 -67.79 -2.04 -30.18 106.02 85.15 70.98 50.94 53.87 AA 6 -99.91 -58.88 -84.84 -28.49 -32.19 99.91 71.03 84.89 45.87 64.31 A 25 -80.22 -65.34 -62.02 -18.14 -7.04 99.23 77.23 75.92 39.66 48.14 BBB 45 14.45 -76.20 -61.68 -28.90 -5.43 140.22 79.11 76.44 46.12 55.23 BB 11 -4.93 -49.71 -19.38 -6.96 -7.28 89.64 58.31 85.19 47.82 58.93 B 4 83.20 -70.83 -57.68 -48.80 -3.07 108.73 71.25 65.26 56.75 31.73 CCC 1 60.29 -68.80 35.22 -34.08 5.50 87.03 68.80 61.03 46.68 31.22 Equity Volatility Average Percentage Pricing Error Absolute Percentage Pricing Error Rating N Merton BC LS HH CDG Merton BC LS HH CDG Overall 93 -0.34 -8.43 -1.08 3.44 2.65 68.59 34.53 38.91 28.15 29.40 AAA 1 -8.17 -37.58 -45.04 1.07 -19.46 32.98 40.42 45.50 30.50 32.13 AA 6 -79.35 13.31 -7.51 -19.39 0.36 79.35 36.20 34.96 25.24 27.89 A 25 -53.53 -13.94 -21.81 -8.96 -1.76 55.26 30.62 32.01 22.78 25.70 BBB 45 -8.84 -10.66 -10.91 4.74 0.61 53.43 30.38 33.55 26.69 27.48 BB 11 7.66 8.70 27.45 22.58 17.90 46.45 42.38 47.50 41.02 41.04 B 4 75.25 -13.00 49.04 7.31 -2.78 94.38 46.66 64.89 27.02 26.24 CCC 1 533.76 -38.41 47.40 18.22 26.68 533.76 48.78 58.28 38.51 44.89 32

Table 4: Pricing Errors of CDS Spreads and Equity Volatility Panel B: By Sectors CDS Spreads Average Pricing Error Absolute Pricing Error Sector N Merton BC LS HH CDG Merton BC LS HH CDG Overall 93 0.35 -0.96 -0.52 -0.40 -0.06 1.58 1.01 1.06 0.70 0.67 Communications 6 -0.57 -1.59 -1.76 -1.49 -0.14 1.28 1.60 1.77 1.52 0.91 ConsumerCyclical 32 1.07 -1.13 -0.48 -0.46 -0.04 2.34 1.20 1.14 0.71 0.76 ConsumerStaple 14 0.47 -0.68 -0.73 -0.11 -0.01 1.09 0.70 0.78 0.30 0.30 Energy 8 1.45 -1.14 -0.68 -0.68 -0.33 2.34 1.14 0.76 0.84 0.62 Industrial 18 -0.45 -0.78 -0.06 -0.34 -0.08 0.89 0.89 1.32 0.60 0.65 Materials 11 -0.34 -0.77 -0.42 -0.37 0.25 0.84 0.77 0.65 0.44 0.73 Technology 4 -1.19 -0.52 -0.34 0.92 -0.49 1.19 0.71 0.78 1.52 1.01 Equity Volatility Average Pricing Error Absolute Pricing Error Sector N Merton BC LS HH CDG Merton BC LS HH CDG Overall 93 -0.13 -3.26 -0.42 1.33 1.02 26.53 13.36 15.05 10.89 11.37 Communications 6 -12.65 -15.45 -14.26 -5.74 -2.34 19.25 17.48 18.38 13.12 16.17 ConsumerCyclical 32 13.24 -2.01 -0.71 3.59 2.40 40.10 13.64 15.97 13.11 12.17 ConsumerStaple 14 -5.31 2.67 6.87 -0.97 -0.60 26.05 13.03 18.68 8.39 9.16 Energy 8 6.94 -6.12 2.81 3.46 0.21 28.17 16.13 14.34 11.74 10.19 Industrial 18 -8.76 -5.81 -1.84 1.29 -0.47 13.68 11.60 12.86 9.09 9.53 Materials 11 -9.67 -1.34 -1.82 1.58 2.60 13.52 10.04 10.03 7.18 9.72 Technology 4 -19.27 -3.84 0.98 -2.76 4.72 21.01 17.53 15.12 15.12 20.73 CDS Spreads Average Percentage Pricing Error Absolute Percentage Pricing Error Sector N Merton BC LS HH CDG Merton BC LS HH CDG Overall 93 24.36 -66.72 -36.36 -27.99 -4.11 109.94 70.70 73.77 48.50 46.91 Communications 6 -26.37 -74.30 -81.99 -69.65 -6.53 59.83 74.65 82.81 71.09 42.34 ConsumerCyclical 32 66.34 -69.96 -29.66 -28.68 -2.55 144.94 74.16 70.80 44.26 46.95 ConsumerStaple 14 52.73 -76.98 -82.34 -12.78 -1.04 122.89 78.94 87.97 34.36 33.93 Energy 8 96.35 -75.45 -45.42 -45.09 -21.70 155.62 75.97 50.58 55.90 40.96 Industrial 18 -30.75 -53.32 -3.78 -23.40 -5.19 60.41 60.82 90.31 40.75 44.15 Materials 11 -30.80 -70.36 -38.20 -33.74 22.90 77.16 70.67 59.74 40.31 66.60 Technology 4 -80.20 -35.09 -22.83 62.14 -32.95 80.20 48.03 52.30 102.10 68.18 Equity Volatility Average Percentage Pricing Error Absolute Percentage Pricing Error Sector N Merton BC LS HH CDG Merton BC LS HH CDG Overall 93 -0.34 -8.43 -1.08 3.44 2.65 68.59 34.53 38.91 28.15 29.40 Communications 6 -25.96 -31.72 -29.26 -11.78 -4.81 39.51 35.87 37.72 26.92 33.19 ConsumerCyclical 32 33.98 -5.17 -1.82 9.21 6.17 102.94 35.01 41.00 33.67 31.25 ConsumerStaple 14 -15.73 7.90 20.33 -2.88 -1.76 77.14 38.59 55.29 24.83 27.14 Energy 8 17.37 -15.34 7.04 8.67 0.53 70.55 40.40 35.91 29.40 25.52 Industrial 18 -21.77 -14.43 -4.56 3.19 -1.17 33.99 28.83 31.96 22.58 23.68 Materials 11 -29.45 -4.08 -5.55 4.82 7.92 41.16 30.56 30.52 21.85 29.60 Technology 4 -42.62 -8.48 2.16 -6.11 10.43 46.46 38.77 33.44 33.43 45.83 33

Table A1: Summary Statistics of Individual Names This appendix table reports ratings, 5-year CDS spread, equity volatility, leverage ratio, asset payout, and recovery rate, for each of the 93 firms similar as those by ratings and sectors in Table 1. Last Five Yr Equity Leverage Asset Recovery Company Rating CDS (%) Volatility (%) Ratio (%) Payout (%) Rate (%) AirProds&ChemsInc A 0.238 28.358 33.067 2.086 40.863 AlbertsonsInc BBB 0.692 35.540 54.662 3.650 41.008 AmeradaHessCorp BB 0.817 28.458 61.871 2.929 40.081 AnadarkoPeteCorp BBB 0.427 31.244 47.816 1.688 39.439 ArrowElectrsInc BBB 2.175 44.325 62.279 2.259 39.269 AutozoneInc BBB 0.708 33.269 30.222 0.827 41.977 AvonProdsInc A 0.230 27.128 17.924 0.998 41.353 BakerHughesInc A 0.298 39.469 20.584 1.764 40.833 BaxterIntlInc BBB 0.493 39.739 33.159 1.739 40.526 BellSouthCorp A 0.550 43.254 39.213 3.308 41.848 Black&DeckerCorp BBB 0.389 29.569 45.897 1.566 42.200 BoeingCo A 0.517 36.815 56.877 1.744 39.336 BorgWarnerInc BBB 0.572 29.766 48.270 1.285 40.623 BowaterInc BB 2.751 30.755 62.578 3.583 41.287 CSXCorp BBB 0.607 29.651 69.128 2.305 40.486 CampbellSoupCo A 0.319 27.171 36.114 2.699 40.063 CaterpillarInc A 0.350 32.081 57.902 1.992 40.122 CendantCorp BBB 1.595 42.626 59.864 1.291 39.440 CentexCorp BBB 0.895 41.148 69.613 2.543 40.670 ClearChannelCommsInc BBB 1.413 45.192 35.378 1.487 40.789 CocaColaEntpersInc A 0.327 34.774 68.903 2.281 40.019 ComputerAssocIntlInc BB 2.889 54.727 35.045 1.044 35.840 ComputerSciencesCorp A 0.565 41.122 43.578 1.182 39.763 ConAgraFoodsInc BBB 0.470 27.510 43.829 3.516 39.320 CorningInc BB 5.412 80.739 41.995 1.138 36.807 DelphiCorp BBB 1.470 40.828 77.164 1.535 40.539 DeltaAirLinesInc CCC 18.806 81.939 93.931 2.885 26.566 DevonEngyCorp BBB 0.732 31.487 56.495 2.281 40.513 DiamondOffshoreDrillingInc BBB 0.488 39.213 32.696 1.701 40.833 DowChemCo A 0.817 35.536 48.723 3.166 39.775 EIduPontdeNemours&Co AA 0.241 30.318 37.916 2.574 41.409 EastmanKodakCo BBB 1.317 37.618 56.431 2.550 38.839 EatonCorp A 0.335 27.783 42.526 1.527 40.815 ElectrDataSysCorp BB 2.087 51.554 50.321 2.332 40.349 EliLilly&Co AA 0.219 35.486 13.956 1.898 40.494 FedtDeptStoresInc BBB 0.675 38.303 54.236 1.966 41.664 FordMtrCo BBB 2.977 47.060 92.612 2.769 41.849 GAPacCorp BB 3.824 48.523 74.892 3.547 42.054 GenElecCoInc AAA 0.427 36.356 63.713 2.223 40.883 GenMlsInc BBB 0.539 24.225 44.680 3.095 41.508 GenMtrsCorp BBB 2.434 35.537 94.017 2.595 41.278 GilletteCo AA 0.147 28.421 17.574 1.672 40.977 GoodrichCorp BBB 1.230 35.427 61.064 3.187 39.736 GoodyearTire&RubrCo B 7.671 65.509 88.106 2.245 39.840 HJHeinzCo A 0.310 23.404 39.061 3.199 41.748 HiltonHotelsCorp BBB 2.141 36.860 51.553 2.754 41.065 HomeDepotInc AA 0.222 39.170 14.502 0.741 42.223 34

Table A1: Summary Statistics of Individual Names (continued) Last Five Yr Equity Leverage Asset Recovery Company Rating CDS (%) Volatility (%) Ratio (%) Payout (%) Rate (%) IKONOfficeSolutionsInc BB 3.460 48.604 73.673 1.337 38.221 IntlBusinessMachsCorp A 0.381 31.166 32.683 0.578 39.991 IntlPaperCo BBB 0.740 30.566 58.274 2.944 39.674 JCPenneyCoInc BB 2.949 45.576 61.984 2.343 37.818 JonesApparelGpInc BBB 0.634 32.547 26.906 1.353 41.338 KerrMcgeeCorp BBB 0.745 26.472 59.613 3.398 41.242 LockheedMartinCorp BBB 0.501 32.241 44.982 1.815 41.173 LowesCosInc A 0.356 36.642 19.222 0.587 41.788 LtdBrandsInc BBB 0.584 44.878 21.283 3.854 41.529 LucentTechInc B 9.525 96.827 63.895 1.255 37.988 MGMMIRAGE BB 2.167 33.197 57.910 2.675 39.764 MascoCorp BBB 0.612 33.101 35.400 2.758 42.234 MattelInc BBB 0.534 35.721 21.203 2.269 40.322 MayDeptStoresCo BBB 0.608 36.953 52.074 3.923 41.765 MaytagCorp BBB 0.773 38.307 58.938 2.213 41.476 McDonaldsCorp A 0.322 38.651 30.956 2.107 40.051 NordstromInc BBB 0.609 40.304 43.145 1.555 41.820 NorfolkSthnCorp BBB 0.471 36.021 61.054 2.704 39.724 NorthropGrummanCorp BBB 0.675 26.992 51.679 1.844 40.890 OmnicomGpInc BBB 0.906 36.220 42.475 0.887 40.262 PPGIndsInc A 0.360 27.727 37.415 2.667 42.133 PhelpsDodgeCorp BBB 1.780 38.034 48.840 1.877 41.547 PitneyBowesInc A 0.211 27.063 46.124 2.645 41.674 PraxairInc A 0.291 28.048 33.167 1.730 42.060 Procter&GambleCo AA 0.163 23.275 21.002 1.289 40.450 Rohm&HaasCo BBB 0.353 29.283 43.281 2.241 42.235 RyderSysInc BBB 0.590 29.285 65.616 2.294 39.827 SBCCommsInc A 0.598 43.723 42.509 3.587 38.423 SafewayInc BBB 0.724 39.373 52.084 1.893 41.592 SaraLeeCorp A 0.281 28.465 42.474 2.900 39.904 SealedAirCorpUS BBB 2.349 35.792 44.043 1.820 37.390 SherwinWilliamsCo A 0.396 29.004 32.345 1.896 41.694 SolectronCorp B 4.976 86.414 54.483 1.908 39.241 SouthwestAirlsCo A 0.723 43.900 29.447 0.624 40.323 TheGapInc BB 2.889 50.769 27.086 1.429 41.034 TheKrogerCo. BBB 0.754 39.574 55.452 1.960 41.729 TribuneCo A 0.413 25.200 34.934 1.500 41.228 UtdTechCorp A 0.260 30.856 37.047 1.116 39.475 VFCorp A 0.323 25.458 31.046 2.687 38.877 ValeroEngyCorp BBB 1.075 36.741 65.574 2.174 40.715 VisteonCorp BB 2.671 46.160 87.957 1.297 41.348 WalMartStoresInc AA 0.193 32.359 20.540 0.991 39.991 WaltDisneyCo BBB 0.714 43.767 38.906 1.644 39.191 WeyerhaeuserCo BBB 0.753 29.759 62.255 3.509 41.164 WhirlpoolCorp BBB 0.477 31.043 58.506 2.305 40.512 WilliamsCosInc B 6.836 84.181 83.953 3.724 35.851 35

Time−Series and Term Structure of Average CDS Spreads (%) 3 2.5 2 1.5 1 0.5 10 7 5 0 3 Maturity in Years Jan2002 Jan2003 2 Jan2004 Dec2004 Figure 1: Average CDS Spreads over the Entire Sample This figure plots the average CDS spreads of 93 firms with maturities ranging from 1 year to 10 years from January 2002 to December 2004. CDS spreads are in annualized percentage. 36

Average 5−Year CDS Spreads by Rating Groups (%) 11 Rating A to AAA 10 Rating BBB Rating CCC to BB 9 8 7 6 5 4 3 2 1 0 Jan2002 Jan2003 Jan2004 Dec2004 Average Equity Volatility by Rating Groups (%) 130 Rating A to AAA 120 Rating BBB Rating CCC to BB 110 100 90 80 70 60 50 40 30 20 Jan2002 Jan2003 Jan2004 Dec2004 Figure 2: CDS Spreads and Equity Volatility This figure plots the time series of average 5-year CDS spreads and equity volatility by the rating groups (A-AAA, BBB, CCC-BB). Equity volatility is estimated based on 5-minute intraday stock return data. 37

Barrier Model 2 1.5 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Debt/Asset tbeD/reirraB y = −.7x + 1.2 Corr = −0.48 LS Model 3 2.5 2 1.5 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Debt/Asset tbeD/reirraB y = −1.2x + 1.74 Corr = −0.56 HH Model 3 2.5 2 1.5 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Debt/Asset tbeD/reirraB y = −x + 1.31 Corr = −0.66 Figure 3: Leverage Ratio and Default Boundary These are scatter plots between the oberserved debt/asset ratios and the estiamted barrier/debt ratios of all the 93 firms for constant barrier models—Black and Cox (1976), Longstaff and Schwartz (1995), and the double exponential jump diffusion model (Huang and Huang, 2003). 38

Rating A to AAA 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5 6 7 8 9 10 Rating BBB 2 1.5 1 0.5 0 1 2 3 4 5 6 7 8 9 10 Rating CCC to BB 15 10 5 0 1 2 3 4 5 6 7 8 9 10 Merton Barrier LS HH CDG Observed CDS Spread Figure 4: Observed and Model Implied CDS Term Structure The figure plots time-series average CDS term structure by three rating groups. The five model specifications considered include Merton (1974), Black and Cox (1976), Longstaff and Schwartz (1995), Collin-Dufresne and Goldstein (2001), and the double exponential jump diffusion model (Huang and Huang, 2003). 39

Rating A to AAA 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Jan2002 Jan2003 Jan2004 Dec2004 Rating BBB 2 1.5 1 0.5 0 Jan2002 Jan2003 Jan2004 Dec2004 Rating CCC to BB 14 12 10 8 6 4 2 0 Jan2002 Jan2003 Jan2004 Dec2004 Merton Barrier LS HH CDG Observed 5−Yr CDS Spread Figure 5: Observed and Model 5-Year CDS Spreads This figure plots the time series of observed 5-year CDS spreads and model implied ones estimated from five structural models — Merton (1974), Black and Cox (1976), Longstaff and Schwartz (1995), Collin-Dufresne and Goldstein (2001), and the double exponential jump diffusion model (Huang and Huang, 2003). 40

Rating A to AAA 80 70 60 50 40 30 20 10 Jan2002 Jan2003 Jan2004 Dec2004 Rating BBB 80 70 60 50 40 30 20 Jan2002 Jan2003 Jan2004 Dec2004 Rating CCC to BB 160 140 120 100 80 60 40 20 Jan2002 Jan2003 Jan2004 Dec2004 Merton Barrier LS HH CDG Observed Equity Volatility Figure 6: Observed and Model Implied Equity Volatility This figure plots the realized volatility—estimated based 5-minute intraday stock returns— and model implied equity volatility extracted from five structural models—Merton (1974), Black and Cox (1976), Longstaff and Schwartz (1995), Collin-Dufresne and Goldstein (2001), and the double exponential jump diffusion model (Huang and Huang, 2003). 41

Rating A to AAA 1 0.8 0.6 0.4 0.2 0 Jan2002 Jan2003 Jan2004 Dec2004 Rating BBB 1 0.8 0.6 0.4 0.2 0 Jan2002 Jan2003 Jan2004 Dec2004 CCC to BB 1 0.8 0.6 0.4 0.2 0 Jan2002 Jan2003 Jan2004 Dec2004 Observed Spot Leverage Mean Risk−Neutral Leverage Figure 7: Observed Spot Leverage and the Long-Run Mean of Risk-Neutral Leverage This figure plots the time series of leverage ratio (debt/asset) across three rating groups. The long-run mean of the risk-neutral leverage is estimated using the Collin-Dufresne and Goldstein (2001) model. 42