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A. Valuation Framework

Defaultable coupon bonds, like other debt contracts, are defined by their promised stream of cash flows through time. Typically these consist of a promised face value, F, to be paid at maturity T and a stream of coupon payments to be paid in the interim. To accommodate both discrete and continuous coupon payments, we denote by the nondecreasing function of cumulated coupon payments until time t. In addition to specifying the promised payments, defaultable debt recognizes that there is a random time 𝒯 at which default occurs. At this time, a payment is made in fulfillment of the debt obligation. The recovery, , if any, is generally far below the value of remaining promised payments. We associate with 𝒯 the unit step function : The defaultable debt contract can now be defined by the entities (F, T, , ) that are presumed adapted to a Brownian subfiltration of the information filtration of a probability space (Ω, ℐ, P) satisfying the usual technical conditions. The filtration extends by the knowledge of the default time process .

Suppose that is the accumulation of the money market account, where the spot interest rate is . According to Duffie (1996), the absence of arbitrage is ensured by the existence of a probability measure Q equivalent to P under which the money market discounted gains processes for all assets are martingales. It follows that the time t price of the defaultable coupon bond with maturity τ periods from time t, denoted , is where an is expectation operator under the probability measure Q. Henceforth, we will abbreviate the left‐hand side of equation (2) as with the understanding that this price is observed only if a default has not yet occurred. The first integral in equation (2) accounts for the stream of coupon payments received as long as there is no default and stopped at the default time. The second term accounts for the promised face value given no default, and the last integral accounts for the single recovery at the default time (note that equals the Kronecker delta function at and is zero at times other than the default time when it is one). The formulation of default in (2) is consistent with the surprise stopping time approach (Lando 1998; Duffie and Singleton 1999; Collin‐Dufresne, Goldstein, and Hugonnier 2004).

For a surprise default time that is a stopping time, there exists a positive process , called the hazard rate process, such that is a martingale. When we have a martingale under probability Q, we refer to as the risk‐neutral hazard rate process. Heuristically speaking, models the probability of default in the interval (t, ), the instantaneous likelihood of default. The process is adapted to . In this case we may write When we use (3) and iterated expectations, first computing expectations with respect to and then again with respect to , it follows that Unlike equation (2), the pricing equation (4) eliminates reference to the discontinuous process and reduces the problem of pricing defaultable debt to that of pricing nondefaultable debt with an altered discount rate and cash flow claim.

B. Duffie‐Singleton (1999) Model

If we now follow Duffie and Singleton (1999) and define recovery as a proportion, , of the predefault value of the defaultable debt so that then the defaultable debt price is (see Duffie and Singleton [1999] for intermediate steps) Because it is not possible in the Duffie‐Singleton approach to separate the effects of the hazard rate process from that of the loss process , the aggregate defaultable discount rate can be defined as This discount rate consolidates time value, default arrival rates, and recovery. With deterministic and continuous coupon rate , the debt equation can be simplified to where is the price of the unit face defaultable zero‐coupon bond with maturity .

C. Specification of the Defaultable Discount Rate

In this subsection we specify that lead to empirically testable closed‐form models for the price of defaultable coupon debt. Consider the family of aggregate defaultable discount rate models shown below: where surrogates firm‐specific distress indexed by n and postulated either as the level or as the logarithm of the variable. When we incorporate both an economywide and a firm‐specific variable, the linearity of in and affords analytical tractability. To keep a parsimonious factor structure, we assume that distress is driven by a single factor . However, our characterization of is versatile to accommodate a 𝒦‐factor model of . Such an angle is pursued further in the empirical section.

Equation (10) indicates that cross‐sectional variations in credit risk are due to cross‐sectional variations in Λ_0,n, Λ_r,n, Λ_s,n, and . The term Λ₀ measures the unconditional instantaneous credit yield. If , defaultable yields are positively related to interest rates; existing evidence on comovements between Treasury and corporate yield curves is indicative of (see, e.g., Duffee 1998; Collin‐Dufresne, Goldstein, and Martin 2001; Driessen 2005). Under the assumption that is positively associated with firm‐specific distress, the credit quality of the firm deteriorates when distress rises (provided that ).

The parameters of can vary systematically with credit rating as modeled by Jarrow et al. (1997). Two special cases of (10) are of relevance. Case 1: Setting and gives the term structure of default‐free bonds. Case 2: If , one obtains a class of credit risk models driven solely by systematic factors (Duffie and Singleton 1997).

D. Term Structure of Default‐Free Bonds

Let represent the long‐run mean of the short rate and The following two‐factor model is adopted on empirical and theoretical grounds: where and are standard Brownian motions under the risk‐neutral measure with correlation . The term ( ) is the mean reversion rate for ( ) and is the long‐run mean for . The risk‐neutral dynamics (11) results by assuming that the risk premium associated with is and the dynamics under the physical measure is as specified in (24) below. As in Duffee (2002), this four‐parameter risk premium specification permits the risk‐neutral long‐term mean and the coefficient of mean reversion to depart from their statistical counterparts.

When we solve a standard valuation equation, the price of default‐free discount bond is where and The yield to maturity, to be used in the Kalman filtering estimation, is Buhler et al. (1999) and Dai and Singleton (2000) make the point that the one‐factor model is statistically rejected in favor of a two‐factor model. Moreover, Litterman and Scheinkman (1991), Chen and Scott (1993, 2003), and Duffee (1998) show that movements in the level and slope of the yield curve capture a large fraction of the Treasury term structure variations.

E. Term Structure of Single‐Name Defaultable Bonds

For its analytical tractability, we assume, under the risk‐neutral measure, that the distress factor obeys where is standard Brownian motion under the risk‐neutral measure and is the risk premium for . Let and . The form of (14) is a robust four‐parameter specification for the distress factor.

Consider the price of a unit face defaultable discount bond with τ periods to maturity. Using the risk‐neutral dynamics of and and solving (9), we have where and

We first observe that is exponential affine in three state variables: the short interest rate, the stochastic long‐run mean interest rate, and the firm‐specific distress factor. The bond price is driven by 18 structural parameters: there are three parameters in the defaultable discount rate specification, 10 in the interest rate process, and five associated with the dynamics of firm‐specific distress. Second, under the positivity of Λ_r, the defaultable discount bond price is negatively related to and : The bond price is also negatively associated with the distress factor, as seen by provided that . The expressions for the local risk exposures can be employed to develop hedges for marked‐to‐market risks.

Third, the yield‐to‐maturity of the defaultable discount bond, for maturity τ, is and it decomposes single‐name credit yield into a systematic component and a firm‐specific component. When , the asymptotic defaultable yield is and The three‐factor model offers the flexibility to produce various yield curve shapes including double‐humped. The yield curve of the defaultable coupon bond, , inherits the same structure as displayed by the zeros. Compared to yield dynamics (23), the hazard rate specification in Duffee (1999) is based on two latent factors for the interest rate process plus a single factor that is independent of the interest rate process (see also Jacobs and Li 2004; Driessen 2005).

Five candidates for are selected for their empirical plausibility. As discussed below, each choice leads to a distinct testable model of credit risk.

Depending on theoretical plausibility and analytical convenience, equations (10) and (14) offer the flexibility to model the observable distress variable in levels or in logs. Guided by Collin‐Dufresne and Goldstein (2001), we expand on the set of default risk models by including a logarithmic transformation for leverage, B/M, and volatility (these variables are strictly positive). Our later empirical exercises also present the results of two additional tests. First, we investigate whether the chosen dynamics in level or in logs is more consistent with the Gaussian specification for the physical process (31) below. Second, we assess model fitting errors with both level and log transformations.

In what follows, the model will be used to benchmark performance, and we will focus on reduced‐form models with identifiable factors that may or may not be tradable. As default is triggered only at maturity, the Merton (1974) model cannot be easily adapted to price defaultable coupon bonds. Unless jumps are added, these models imply counterfactually low credit spreads for short‐maturity bonds; Collin‐Dufresne and Goldstein (2001) establish that one‐factor structural models have undesirable long‐run yield properties. In addition, Jones, Mason, and Rosenfeld (1984), Wei and Guo (1997), Eom, Helwege, and Huang (2004), and Ericsson and Reneby (2005) present evidence rejecting such models.

A. Kalman Filtering Estimation Approach

To make proper connections, we require an assumption on the evolution of under the physical measure, that is, where and ω_r and ω_z are standard Brownian motions with . This assumption has the implication that risk‐neutral density and physical density of share the same parametric form. Integrating the stochastic differential equation (24), we get where I is a identity matrix. It follows that is a normally distributed vector with zero mean and variance‐covariance matrix , where The conditional moments of given are and where Δt is the time interval between and t. The exact discrete‐time state space formulation is then based on the assumption that , …, is a Markov process with initial state and .

Let be the time t observed Treasury yields, where K denotes the number of yields employed in the estimation. As in Babbs and Nowman (1999), Duffee (1999), and Chen and Scott (2003), the measurement equation for observed yields can be expressed as where , and is an diagonal matrix with element . The term is an vector with ith element , and is an matrix with ith row . The transition equation for the unobserved state variable is where , and is a matrix given by (27). The term is a vector and is a matrix .

The normality assumption on and allows us to implement a Kalman filter recursion based on the results of Harvey (1991). Briefly, let denote the optimal estimate of based on information and denote the covariance matrix of the estimation error. In this setting the conditional distribution of is normal with mean , and the prediction errors , are independently and normally distributed with mean zero and variance‐covariance .1 The log likelihood function is accordingly which can be maximized to generate the state space system parameters. In our optimization routine we set the initial value for to its unconditional moment and the initial value for to

B. Interpretation of Results

For the Kalman filter estimation, we select a broad cross section of Treasury yields with maturity of six months, two years, five years, and 10 years. In the discussion to follow, we impose the restriction since the maximum likelihood procedure failed to converge when is freely determined as in Duffee (2002). Table 1 reports restricted estimation results with (labeled set 1) as well as for the specification that accommodates nonzero (labeled set 2). The key observation to make from the two estimations is that the log likelihood, ℒ, barely changes from 2,180.99 to 2,181.05. Thus the three‐parameter risk premium specification for fares no better than the two‐parameter counterpart. Given the insensitivity of ℒ to , the remainder of the paper focuses on set 1, which maintains (the defaultable yields and credit spreads were also insensitive to estimated with no impact on performance statistics).

Table 1 Kalman Filtering Estimation of the Interest Rate Process: Interest Rate Parameters

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The parameter estimates with set 1 are reasonable and in line with economic intuition. Take the long‐run interest rate, μ_z, as an example, which is estimated at 8.7%; this measure of central tendency compares with 7.28% estimated by Babbs and Nowman (1999). Our results indicate strong mean reversion for the process but a near random walk behavior for the process. Specifically, the estimated and imply a half‐life of 1.31 and 13.38 years for the and process, respectively.

As would be expected, and comove positively as reflected in the estimated value of 0.356 (see also Jegadeesh and Pennacchi 1996). Because the risk‐neutral drift of the short rate is , a negative market price of interest rate risk, , has the impact of increasing the risk‐neutral drift relative to the physical drift. To the contrary, a positive λ_z decreases the risk‐neutral drift of . In general, the estimated parameters are several times larger than their standard errors, suggesting statistical significance.

The goodness of fit of the interest rate model is formally assessed in table 2. First, we report the standard deviations of the measurement errors, which compare favorably with those in existing studies (e.g., Babbs and Nowman 1999; Duffee 1999; Chen and Scott 2003). For instance, the estimated standard deviation of the measurement error for six‐month, two‐year, five‐year, and 10‐year Treasury yields is, respectively, 0, 13, 0, and 14 basis points (bp). Second, the model provides desirable fitting errors (actual minus model yield) across the yield curve: the median in‐sample absolute errors for six‐month, two‐year, five‐year, and 10‐year yields is 14 bp, 23 bp, 21 bp, and 19 bp.

Table 2 Kalman Filtering Estimation of the Interest Rate Process: Fitting Errors and Standard Deviation of Measurement Errors

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The log likelihood and the p‐value from a ‐distributed likelihood ratio statistic validate a two‐factor specification over a restricted specification with nonstochastic. Therefore, allowing for stochastic enhances the ability of the three‐factor credit risk models to describe credit curves more realistically. When we demonstrate the plausibility of the two‐factor interest rate model from a different perspective, (i) the correlation between the Kalman filter predicted ( ) and the six‐month (10‐year) market yield is 0.997 (0.872), and (ii) the slope of the yield curve (10‐year market yield minus six‐month yield) has a correlation of 0.993 with the model‐generated slope. Overall the two factors reasonably explain the cross section of yields as well as yield curve movements through time.

A. Determinants of Credit Risk

Because credit risk models are often employed in marking‐to‐market other illiquid securities, evaluating model performance is of broad interest (Dai and Singleton 2003). Does a three‐factor credit risk model with leverage improve on a two‐factor counterpart with firm‐specific distress considerations absent? Is a credit risk model based on leverage less misspecified compared to the alternatives of book‐to‐market, profitability, volatility, and distance‐to‐default? To address these questions and to model credit risk determinants, table 4 presents four measures of performance for each model: (i) the absolute yield valuation errors (AYE) (in basis points), (ii) the absolute percentage valuation errors (APE) (in %), (iii) the Schwarz‐Bayes information criterion (SIC), and (iv) the absolute credit spread errors (ACE) (in basis points) for a maturity‐matched spread of corporate over Treasury. Each measure provides a perspective on the effectiveness of a model to explain the term structure of defaultable yields and credit spreads.

Table 4 Valuation Errors for Defaultable Coupon Bond Models

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On the basis of the results for yield errors on BBB‐rated bonds, a three‐factor model with leverage (in level) is attractive relative to the two‐factor interest rate model. In the row labeled “all,” note that incorporating leverage in the defaultable discount rate specification leads to yield errors of 36 bp with the leverage (level) model compared to 45 bp with the interest rate model. This strong improvement over the base model is, in fact, observed at all maturities, with each model performing slightly worse at the short end of the defaultable yield curve. From a different angle, the documented fitting errors suggest that systematic variables have a first‐order impact on defaultable term structures, with leverage providing a sizeable next‐order explanatory ability.

Compared with the three‐factor setting of Duffee (1999) and the single‐factor intensity process specification of Longstaff, Mittal, and Neis (forthcoming), the average yield error is −7 bp (not reported in the table to save on space), making the leverage model reasonably accurate for marking‐to‐market credit risks. Although academic work has mostly focused on default factors and our results are encouraging, realize that omitted nondefault attributes such as illiquidity of corporate bonds and actual spread volatility can cause model prices to systematically deviate from observed market prices even in the presence of properly specified default factors (see the practitioner perspective outlined in Kao [2000] and Dignan [2003]).

Table 4 imparts the insight that the interest rate model and the leverage (in level or logs) model are virtually indistinguishable for high‐grade bonds at all maturities. This seemingly counterintuitive result can be understood to mean that high‐rated bonds are mostly determined by systematic factors and are rather insensitive to variations in firm‐specific leverage. Although Λ_s has the theoretically correct positive sign in the leverage model that raises default probability for firms with higher leverage, the estimated magnitude is insufficiently strong to affect the pricing performance for high‐grade bonds. Consistent with Λ₀ and Λ_r of the interest rate model, we may write the defaultable discount rate as , implying that fixed‐income markets value high‐grade bonds using a translated default‐free process. Thus a more complex model that accounts for firm‐specific distress need not necessarily perform better for high‐grade bonds. As would be expected, each credit risk model (in particular the interest rate model) is less misspecified for high‐grade bonds than its low‐grade counterpart. Shifts in interest rate risk play a fundamental role in the valuation of single‐name high‐grade bonds (see, e.g., Kwan 1996; Collin‐Dufresne, Goldstein, and Martin 2001).

According to empirical results for BBB‐rated bonds, the leverage (in level) model performs the best, followed in turn by the B/M (in level) model, the profitability model, the stock volatility model, and the distance‐to‐default model. The interest rate model is the last‐place performer. To fix main ideas, consider long‐term bonds and the level specification: the percentage pricing errors (yield errors) are 2.66% (33 bp), 3.07% (39 bp), 3.24% (41 bp), 3.18% (40 bp), 3.20% (39 bp), and 3.37% (42 bp), respectively, for the leverage, B/M, profitability, volatility, distance‐to‐default, and interest rate models. It must be noted that this model rank ordering does not perfectly coincide with the average log likelihood ratio based ordering presented in table 3. The reason is that the objective function in the estimation is not to minimize the sum of squared pricing errors; instead it is derived from the conditional density of and the assumed cross‐correlation structure of the valuation errors to give realistic single‐name empirical moments. Given the large notional value of fixed‐income securities, the documented superiority of the leverage model is economically meaningful.

Another way to judge performance is to compare the SIC statistic, which rewards goodness of fit while penalizing the dimensionality of the model. Across most maturities, the SIC of the interest rate model exceeds the SIC of the leverage model for BBB‐rated bonds; the reverse is true for high‐grade bonds. This empirical yardstick thus favors the more heavily parameterized leverage model over the interest rate model for BBB‐rated bonds. However, for high‐grade bonds, simplicity is a more desirable modeling element. It is worth noting that in the case of BBB‐rated firms, the SICs of the B/M and distance‐to‐default models exceed the SIC of the leverage model and therefore perform no better. Issues related to model differentiation based on statistical significance are formalized in the next subsection.

Relevant to our findings on high‐grade bonds, Collin‐Dufresne, Goldstein, and Martin (2001) argue that while variation in corporate yields is predominantly due to Treasury yields, the credit spread is a much harder entity to predict. Are credit spreads mainly driven by the short interest rate or firm‐specific distress variables? To answer this question, we provide additional information on the absolute difference between the observed and model generated credit spreads and report them as ACE in table 4. The chief result that emerges from this exercise is that ACEs have a large interest rate component, and accounting for leverage and other firm‐specific distress is important. Consider BBB‐rated bonds in which the credit spread errors can be reduced from 40 bp in the case of the interest rate model to 33 bp in the case of the leverage model (in level). When this feature is shared with AYE, adding a firm‐specific distress variable to the credit risk model does not help to lessen ACEs for high‐grade bonds. Thus the performance metrics that rely on yields (i.e., AYE) and credit spreads (i.e., ACE) provide similar conclusions.

To gauge the robustness of the findings, table 5 displays valuation results based on a holdout sample of one year. The standard argument favoring such an exercise is that model performance need not improve out of sample for a more complex valuation model: Extra parameters have identification problems and may penalize accuracy. In implementation, we compute theoretical prices evaluated at the Kalman filter forecast of and and the optimal forecast of based on the parametric model (31). The results are more striking in the out‐of‐sample context for BBB‐rated bonds. First, when we compare the entries in table 4 and table 5, there is a general deterioration in the performance of credit risk models and particularly the interest rate model. Thus the interest rate model is most misspecified out of sample, especially at the short end of the low‐grade yield curve. Second, the leverage model enhances the overall effectiveness of the interest rate model by 16 bp and the B/M (distance‐to‐default) model by 8 (9) bp. Third, the results tell us that the relative ranking between credit risk models is unaltered in sample versus out of sample.

Table 5 Out‐of‐Sample Valuation Errors Based on a Holdout Sample

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On the basis of in‐sample and out‐of‐sample absolute yield errors and the SIC statistic, we generally observe that the model relying on the distress variable specified in level does slightly better than the log counterpart. For instance, the absolute yield error in table 5 for BBB firms with leverage (in level) is 40 bp versus 49 bp with leverage (in logs). Similar improvement is documented for SIC in table 4, which is −7.53 for leverage (level) and −7.32 for leverage (logs).

To confirm whether the level or the log of leverage is more consistent with the assumed Gaussian physical dynamics, we also computed the Shapiro‐Wilk normality statistic applied to firm‐level leverage series. This test shows that moving from level of leverage to the logarithm of leverage improves the Shapiro‐Wilk statistic for 31% of the firms but worsens for 69% of the firms. In summary, the level of leverage conforms to the normality hypothesis better than the log of leverage. However, the log B/M appears more consistent with the Gaussian dynamics for 63% of the firms.

B. GMM Test of Model Comparisons and Specification Tests

The ask price is not reported in the Lehman database, so we are unable to assess the extent of the improvement relative to the bid‐ask spreads. However, for any two defaultable coupon bond models, define the disturbance terms and where μ^int and μ^lev represent the mean monthly absolute yield errors (averaged across short, medium, and long) for the interest rate model and the leverage model (in level).

Statistics presented in table 6 are helpful for judging whether the incremental explanatory power of a model is statistically significant relative to other models. First, we report the rejection rate for the null hypothesis versus the alternative . When the weighting matrix is taken from the unconstrained estimation, this test is done by performing a restricted estimation with . Then, T times the difference in the restricted and the unrestricted criterion functions is asymptotically distributed. Second, we present the rejection rate for the one‐sided test using the Newey‐West method to adjust for heteroskedasticity and autocorrelation. Finally, we compute (or ), which reflects the average absolute yield error (credit spread error) reduction in moving from the interest rate model (the base model) to the leverage model.

Table 6 Generalized Method of Moments Tests of Statistical Significance

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Consider low‐grade bonds: valuation improvement is statistically large by adding leverage (in level) to the defaultable discount rate specification of the interest rate model. Overall, leverage results in a 30.1%, 14.3%, 25.2%, 33.8%, and 15.2% relative improvement over the interest rate, the B/M, profitability, volatility, and distance‐to‐default models. Between leverage and interest rate models, the two‐sided (one‐sided) rejection rate is 70% (67%), implying statistical improvement for the majority of the single‐names (and likewise for credit spreads).

The B/M (log) generalization to the interest rate model also delivers economically and statistically significant performance enhancement, albeit by a lesser extent relative to the leverage model (we focus on log since it is more consistent with the Gaussian hypothesis, although the results are similar with level). The relative improvement is 18.4% over the interest rate model with a one‐sided rejection rate of 46%. With high‐grade bonds, however, there is no discernible improvement over the interest rate model with either leverage or B/M. In this regard, Agrawal et al. (2001) reason that differences in default are insufficient to explain yield spreads observed across rating classes. Such a result also parallels a calibration‐based finding of Huang and Huang (2003) that default risk accounts for only a small fraction of aggregate credit spreads, and this result can now be extended to high‐grade single‐name credit spreads. Allowing for a richer stochastic structure beyond the interest rate model in the firm‐specific dimension is of greater relevance for yield errors and credit spread error of low‐grade bonds.

To further study properties of model mispricing, we appeal to Collin‐Dufresne, Goldstein, and Martin (2001) and investigate whether pricing errors from the leverage model are contemporaneously correlated. Specifically, we focus on four systematic factors: (i) the default spread between low‐rated and high‐rated corporate bonds (denoted DEFAULT), (ii) the spread between the 10‐year and two‐year Treasury yield (denoted SLOPE), (iii) the size premium (SMB), and (iv) the VIX volatility index from the Chicago Board Options Exchange. The following regression is performed for each firm (letting ): Three findings are relevant to our modeling approach. First, we find that the average adjusted across our sample of firms is 9%, implying that single‐name pricing errors show little time covariation with the chosen systematic factors (50% of the firms had an less than 5%). Second, the sensitivity coefficients are typically positive. In economic terms, this means that model mispricing tends to be greater during periods of widening default spreads or steepening Treasury curves (or equivalently when VIX/size premium are high). Third, the t‐statistics (corrected for heteroskedasticity and autocovariance) are greater than two for at most 19% of the firms. For the majority of the firms, the pricing errors are thus statistically unrelated to the systematic factors. Extant research has found that corporate bond markets are subject to high transaction cost and low volume. Therefore, it appears likely that bond market illiquidity has contributed to model mispricing. More empirical work is needed to study the role of bond market liquidity along the lines of Collin‐Dufresne, Goldstein, and Martin (2001). If there are marketwide default and liquidity factors, they are perhaps not adequately captured by the set of included systematic variables.

Additionally, we performed a principal component analysis of absolute yield errors and credit spread errors of each model by binning the data into nine groups (by short, medium, and long and by AA, A, and BBB credit ratings). The results establish that there is a dominant component for each of the models, with the first two components explaining the bulk of the variation. Take the leverage model as an example, where the first component explains 83.6% of the total variation. Together, the first two components capture 92.7% of the variation. As shown by the regression analysis, none of the systematic variables appear to be a good proxy for the predominant factors.

Tables 4 and 5 have the unfortunate implication that credit risk models are misspecified, albeit to various degrees. One key issue that remains to be addressed is whether cross‐sectional variations in model misspecification are related to excluded firm‐specific distress risk. Indeed, there is no theoretical basis for picking a firm‐specific distress variable successively in a default model. In doing so, we were guided by parsimony and ease of implementation: a four‐factor generalization driven by (31) has four additional parameters and two correlation terms requiring the estimation of 22 parameters (plus five Cholesky decomposition constants). How can a four‐factor credit risk model with two firm‐specific factors be expected to fare relative to a three‐factor credit risk model? To informally attend to this concern, two regression specifications are explored using the entire cross section. First, we regress the absolute yield errors of the leverage model on B/M ( ) and obtain where the reported coefficients are time‐series pooled averages and the t‐statistics are based on GMM. Second, we regress B/M mispricing on leverage: On the basis of the sensitivity coefficients, one may conjecture that it is possible to lower the pricing errors of defaultable bonds: the valuation errors of the B/M model are statistically higher for firms with high leverage, even after one accounts for B/M (and vice versa). Owing to burdensome MLE implementation of higher‐dimensional models and the rather low explanatory power of and in the aforementioned cross‐sectional estimations, we leave a formal analysis of four‐factor credit risk models to a later study.

C. Hedging Dynamic Credit Exposures

This subsection concentrates on hedging a portfolio consisting of a short position in a corporate bond and a long position in a Treasury bond with matched maturity. Notice that there is an arbitrage link between the corporate bond market, the Treasury market, and the credit default swap market. According to Duffie (1999), Duffie and Liu (2001), and Houweling and Vorst (2005), holding a τ‐period par corporate bond and buying credit protection through a credit default swap is approximately risk‐free (adjusting for floating‐rate feature and coupon equivalence). For this reason the price of single‐name credit protection is the same as reference entity credit spread. Our approach, as shown, can be useful in understanding which traded fixed‐income instruments can most effectively hedge dynamic credit exposures and, to first order, potentially hedge the risks of writing credit protection. This connection arises since sellers of credit default swaps can hedge their exposure by shorting corporate bonds (see Chen, Cheng, and Wu 2005; Longstaff, Mittal, and Neis 2005) or vice versa.

Fix the credit risk model as the interest rate model. In this case the combined position can be written as which hedges credit exposure using two risk‐free discount bonds with short maturity τ₁ and long maturity τ₂. Taking the target as the log relative of the corporate coupon bond price to the Treasury bond price and normalizing it by τ imparts it a yield interpretation. To dynamically hedge and risks, we take the required partial derivatives and derive the positioning as with and determined from (12). The hedge portfolio is made self‐financing by setting Local risk exposures for the defaultable coupon bond are the aggregated face and coupon exposures: where and are displayed in (20) and (21). The combined position in the defaultable bond and the replicating portfolio is liquidated at time . Dollar‐hedging errors are defined as When τ is selected as two years and seven years, the hedging strategy is executed each month and for each single‐name bond. We compute absolute dollar hedging errors each month as and absolute percentage hedging errors as

Implementing a dynamic hedging strategy based on leverage and B/M models involves nontraded leverage and book values, and only a partial hedge is feasible. To hedge , risks, we employ two Treasury discount bonds and a single‐name equity to dynamically hedge the market component of leverage. However, this leaves the long‐term debt component of leverage unhedged. Notice that we can write leverage as , where is the number of shares outstanding and is debt value. Assuming that the stock price follows geometric Brownian motion, we derive and as shown in (40) and (41) and where

The results reported in panels A and B of table 7 reveal that leaving the debt (book) component of leverage (B/M) unaccounted in the hedging strategy can impair dynamic ability of three‐factor credit risk models. Despite the flexibility of an additional instrument, the absolute hedging errors from the leverage model and the B/M model are no better than that of the two‐factor interest rate model. From the perspective of absolute hedging errors, interest rate movements are dominant. When averaged over BBB‐rated single‐name bonds, the absolute hedging error with the leverage (book‐to‐market) model is $0.107 ($0.107) compared to $0.106 with the interest rate model (assuming $100 face value and for years). Credit risk models relying on purely traded factors tend to produce better hedging effectiveness. Regardless of maturity, the two Treasury instruments do a fairly reasonable job of neutralizing dynamic movements in single‐name credit exposures for firms in our sample. We also investigated the relation of single‐name hedging errors using (38) and found them to be characterized by a lack of covariation.

Table 7 Hedging Credit Exposures

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Although not reported, a robustness check shows that the same conclusion applies when the target is a short position in a corporate bond. Our results open up the wider implication that short positions in credit default swaps could be equivalently hedged using Treasury instruments up to BBB‐rated single‐name reference entities.