The Cross-Section of Credit Risk Premia and Equity Returns

Transcription

1 The Cross-Section of Credit Risk Premia and Equity Returns Nils Friewald Christian Wagner Josef Zechner May 14, 2011 Abstract We analyze whether distress risk is priced in equity returns by exploring the joint cross-section of credit default swaps (CDS) and stocks for US firms from 2004 to While previous research uses either real-world or risk-neutral default probabilities, we argue that credit risk premia priced in stock returns depend on both. We decompose risk premia extracted from the term structure of CDS spreads using a single-factor model à la Cochrane and Piazzesi (2005). Consistent with predictions from structural models, our empirical results reveal a strong positive relation between stock returns and expected risk premia and an inverse relation to unexpected credit news. We find that risk premia embedded in CDS spreads contain information beyond size and book-to-market but also that equity excess returns of credit-sorted portfolios are highest for small firms and value stocks. Our results are robust across the pre-crisis and crisis subsamples. JEL classification: G12, G13 Keywords: asset pricing equity returns, default risk, risk premia, credit default swaps, cross-sectional We thank Rui Albuquerque, Michael Brandt, Pierre Collin-Dufresne, Andrea Gamba, Miriam Marra, Lucio Sarno, Clemens Sialm, Paul Schneider as well as seminar participants at Cass Business School and at Warwick Business School for helpful comments. Institute for Finance, Banking and Insurance; WU Vienna. nils.friewald@wu.ac.at. Institute for Finance, Banking and Insurance; WU Vienna. christian.wagner@wu.ac.at. Institute for Finance, Banking and Insurance; WU Vienna. josef.zechner@wu.ac.at. 1

2 1 Introduction While financial theory generally suggests a positive relationship between distress risk and equity returns the corresponding empirical evidence is far from unanimous. Most papers use measures for real-world default probabilities to sort firms into portfolios ranging from low to high distress risk. The results of e.g. Vassalou and Xing (2004) and Chava and Purnanandam (2010) suggest that there is a positive relation between default risk and equity returns, however, the majority of authors present evidence indicating that the relation is negative. For instance, Dichev (1998) and Campbell et al. (2008) document a distress anomaly because firms with high default risk earn abnormally low returns. Geroge and Hwang (2010) explain this puzzle showing that firms facing high distress costs choose low leverage to reduce their default probability but they retain greater exposure to systematic default risk than high leverage firms. Anginer and Yildizhan (2010) sort firms into portfolios using risk-neutral default probabilities to measure exposure to systematic default risk. Their results suggest that default risk is not priced in the cross-section of equity returns. In this paper, we argue that there is a positive relation between equity returns and credit risk premia which are a function of, both, risk-neutral default probabilities and real-world default probabilities. We show that this claim is consistent with the structural framework of Merton (1974) and we present supportive empirical evidence by exploring the joint crosssection of credit default swaps (CDS) and stocks for US firms from 2004 to In the Merton (1974) model, asset dynamics imply, both, a real-world and a risk-neutral default probability. Asset and hence equity excess returns (because equity is a call option on the underlying assets) are linked to both default probabilities, in other words, excess returns are a function of credit risk premia. We generalize asset dynamics to allow for time-variation in the market price of risk driven by time-varying credit risk premia and show that equity excess returns are negatively related to changes in the risk-neutral default probability and positively to credit risk premia. We describe how approaches established in the fixed-income literature can be used to extract credit risk premia from the term structure of CDS spreads by exploiting the direct link between CDS spreads and risk-neutral default probabilities. In particular, we use a one- 2

3 factor model à la Cochrane and Piazzesi (2005) to decompose CDS-implied credit risk premia into an expected risk premium and an unexpected news component. We show that firms with higher credit risk premia should earn higher equity excess returns while the relationship between unanticipated credit news and stock returns should be negative; for instance, a reduction in credit risk due to unexpectedly high earnings should be associated with higher stock returns. In our empirical analysis, we provide evidence consistent with these arguments. Neither sorting stocks into portfolios using the distance-to-default (which is monotonically related to real-world default probabilities) nor using the level of CDS spreads (monotonically related to risk-neutral default probabilities) does provide insights on whether default risk is priced in equity returns. It also appears that the link between default probabilities and equity returns is different in the pre-crisis (01/2004 to 06/2007) and in the crisis period (07/2007 to 06/2010). However, consistent with our arguments, equity excess returns strongly co-move with changes in CDS spreads, foremost with the risk premium component that drives CDS returns in excess of the risk-neutral expectation. Using the one-factor model, we decompose CDS excess returns to obtain estimates for expected risk premia and unexpected credit news. Consistent with our priors, we find that the higher firms credit risk premia, the higher their excess returns. The higher the unanticipated increase in default risk, the lower excess returns of firms are. Throughout our empirical analysis, we account for traditional risk factors by regressing excess returns on the market return, the three factors of Fama and French (1993), and the four factors of Carhart (1997). We find that expected risk premia are priced in equity returns even after controlling for these factors and that expected risk premia convey information beyond size and book-to-market: the factor model alphas are highly significant while the factor loadings are generally not different from zero. Similarly, unexpected news are reflected in stock returns but appear largely unrelated to other risk factors. These findings apply to the full sample, as well as to the pre-crisis and crisis sub-periods. To take a closer look at the relationship between portfolio returns and firm characteristics, we double sort portfolios, first using either size or book-to-market and subsequently by either 3

4 expected risk premia or residual news. Expected risk premia are significantly priced in all size portfolios but the excess returns are highest for small firms. We find a similar pattern when we use book-to-market as a control variable: expected risk premium sorted portfolios earn highest returns for value firms (high book-to-market) while the effect is not significant for growth firms (low book-to-market). Finally, to control for liquidity issues, we perform sequential sorts where we rank firms based on their (relative) CDS bid-ask spreads in the first stage. We find that the link between equity returns to expected risk premia and unanticipated news gets stronger as liquidity increases. This result suggests that risk premia estimated from CDS excess returns do not reflect compensation for illiquidity and that the pricing effect is stronger for companies with low transaction costs in the CDS market. Relation to Literature Our work can be related to various streams of research. We motivate our theoretical prediction that there is a link between asset dynamics and credit risk premium dynamics using insights from the structural model of Merton (1974). The static link between (time-invariant) real-world and risk-neutral default probabilities as well as the resulting relation to the (average) market price of risk are discussed e.g. in Duffie and Singleton (2003). Berg (2010) shows that the relation between real-world and risk-neutral default probabilities is hardly affected when moving away from the Merton framework to a first-passage time framework, to strategic default models or to models with unobservable asset values; there are only minor differences across different structural models of default such as those of Black and Cox (1976), Leland (1994), Leland and Toft (1996), and Duffie and Lando (2001) The empirical evidence on whether default risk is priced in stock returns is mixed. Some papers do find that there is a positive relation between default risk and equity returns. Vassalou and Xing (2004) construct a market-based measure of the default probability using the Merton (1974) model and find that distressed stocks earn higher returns. Chava and Purnanandam (2010) argue that ex post realized returns are too noisy to estimate expected returns and that using estimates based on implied cost of capital reveals a positive relation between expected stock returns and default risk. However, there is a wealth of papers documenting a negative relation between the real-world default probability and stock returns. 4

5 For instance, Dichev (1998) uses the Altman (1968) Z-score and the Ohlson (1980) O score to measure default risk and reports a negative relation to equity returns. More recently, Campbell et al. (2008) use a dynamic panel regression approach that incorporates accounting data and market data (such as past stock returns and standard deviations as well as returns in excess of the market). They find that firms with high distress risk deliver abnormally low returns. Also Garlappi et al. (2008) find that the Expected Default Frequency (EDF) measure of Moody s KMV is in general not positively related to expected stock returns. Related, Avramvov et al. (2009) find that the distress puzzle is more pronounced for worst-rated stocks around rating downgrades. Geroge and Hwang (2010) present an explanation of the puzzle based on optimal capital structure choice and costs of financial distress. 1 Firms facing high distress costs choose low leverage to reduce their default probability but they retain greater exposure to systematic risk than high leverage firms. Building on this argument, Anginer and Yildizhan (2010) use corporate yield spreads to measure risk-neutral default probabilities thereby allowing them to rank firms based on their exposure to systematic default risk. They neither find that firm s default risk is priced in equity markets nor that firms with high distress risk earn anomalous low returns. Our finding that there is a positive relation between equity returns and (expected) credit risk premia (a function of real-world and risk-neutral default probabilities) complements the mixed evidence in the literature. To measure credit risk premia we extract information from the term-structure of CDS spreads. Using CDS data offers several advantages as compared to using corporate yield spreads because contracts are standardized and comparable across reference companies. Furthermore, issuing a CDS on a particular firm does not change the firm s capital structure and CDS maturities can be chosen independent of the firm s debt maturity structure. The CDS market is much more liquid than the corporate bond market 1 Other attempts to explain the distress anomaly build on long-run risk models. Avramov et al. (2010) show that the negative cross-sectional relations between expected stock returns and forecast dispersion, idiosyncratic volatility, and credit risk arises out of a long-run risk economy where the cross-section of expected returns is determined by a firms s cash flow duration. They argue that, while firms with high cash flow durations are strongly exposed to systematic shocks, low duration firms are more sensitive to firm-specific shocks. It follows that firms with high measures of idiosyncratic risk (such as high default risk) tend to have low systematic risk and, hence, low expected returns. Related, Radwanski (2010) argues that distressed firms have short expected lifetimes and consequently earn lower returns because they are not exposed to long-run risk factors. 5

6 and CDS spreads are less contaminated by non-default components; see e.g. Longstaff et al. (2005) and Ericsson et al. (2007). There is also evidence that information from the CDS market is fresher, for instance, Blanco et al. (2005) show that the CDS market leads the bond market in determining the price of credit risk. With the growing time-series and cross-section available for CDS data it has become possible to explore the link between CDS and stock markets. Some papers study lead-lag relations between CDS, equity, and also bond markets, see for instance Blanco et al. (2005) and Norden and Weber (2009). Acharya and Johnson (2007) find that there is an information flow from CDS to equity markets: under circumstances consistent with the use of non-public information by informed banks, recent increases in CDS spreads predict negative stock returns. This negative relation is also found by Ni and Pan (2010) in their study of the consequences of short sale bans in stock markets. In the presence of such bans, it takes more time for negative information in CDS markets to get incorporated into stock prices and returns become predictable. In an empirical study, Han and Zhou (2011) find that the slope of the term structure of CDS spreads negatively predicts stock returns. Similar to the aforementioned papers, they argue that this predictability emerges form slow information diffusion but that it cannot be explained by standard risk factors or default risk. The authors stress that their findings are thus distinct from the literature on the cross-sectional relationship between expected stock returns and default or distress risk (see Han and Zhou, 2011, p. 5). Hence, all of these papers investigate the (informational) linkages between CDS and equity markets in a rather general way. In contrast, we directly exploit the suitability of CDS data for analyzing the link between credit risk and stock returns by extracting credit risk premia from the CDS term structure. We also discuss that our risk premium results are consistent with the negative relation between the slope of the CDS term structure and equity returns documented by Han and Zhou (2011). Most authors interested in default risk premia embedded in CDS spreads construct a measure by linking risk-neutral default probabilities implied from CDS spreads to real-world default probabilities implicit in EDFs of Moody s KMV; see e.g. Berndt et al. (2008). Thus, these measures draw on different data sources and typically require some modeling assump- 6

7 tions for default probabilities or intensities. Relying on approaches established in the bond market literature, we generate estimates of credit risk premia that are essentially model-free and only require the use of CDS data. Analogously to the case for interest rates, e.g. Fama and Bliss (1987) and Campbell and Shiller (1991), forward CDS spreads should be unbiased predictors of future spot CDS spreads if there is no time-variation in risk premia. We document that there is time-variation in CDS excess returns and use a single-factor model based on the idea of Cochrane and Piazzesi (2005) to disentangle expected credit risk premia and unanticipated changes in companies credit quality. The remainder of the paper is organized as follows: In Section 2 we discuss how stock returns are related to default risk in the Merton (1974) model and extend the model to allow the market price of risk and default probabilities to vary over time. We then use these findings in Section 3 to show how we extract credit risk premia from the term structure of CDS spreads. In Section 4 we describe the data and report our empirical findings which we underpin by several robustness checks. Section 5 concludes. The appendix describes technical details. 2 Default Risk, Asset Dynamics and Stock Returns In this section, we start from noting that asset dynamics in the Merton (1974) model imply credit risk premia which are a function of, both, real-world and risk-neutral default probabilities. We use this insight as a motivation to generalize asset dynamics to accommodate time-variation in asset excess returns driven by time-varying credit risk premia. Finally, we decompose realized asset excess returns per unit of risk into expected risk premia and unexpected credit news. 2.1 Credit Risk Premia in the Merton Model In the model of Merton (1974), the asset process is modeled as a log-normal diffusion and the real-world measure (P) dynamics follow dv t = µv t dt + σv t dw P t (1) 7

8 with µ being the drift, V denotes the asset value, σ is the volatility, and W P denotes a standard P-Brownian motion. In this framework two types of claims exist: debt and equity. Debt is a zero-coupon bond with face value D and time-to-maturity T. Default occurs if the value of assets at maturity is below the face value of debt. The default probability, given the asset dynamics in (1), is P D P t = Φ ( log(v ) t/d) + (µ 1 2 σ2 )T σ T }{{} DD (2) where Φ is the standard normal distribution function and DD defines a measure for the distance-to-default (which is just another way of stating the default probability). Pricing of claims on assets has to be done under the risk-neutral measure Q. Assuming a constant riskless rate r, in the drift of the asset dynamics µ is replaced by r, and the risk-neutral probability of default is given by P D Q t = Φ ( log(v t/d) + (r 1 2 σ2 )T σ T ). (3) The firm s shareholders have a European call option on assets struck at the face value of debt which will thus be in the money if default does not occur. Hence, equity E t can be valued by applying the Black and Scholes (1973) formula, i.e. ( ) E t =V t Φ Φ 1 (1 P D Q t ) + σ T D exp( rt )(1 P D Q t ) (4) where Φ 1 denotes the inverse of the standard normal distribution function. Equation (4) shows that the value of equity increases with the asset value and is inversely related to the risk-neutral default probability. The value of any claim that serves as default protection, such as a protective put or a credit default swap, is positively related to the risk-neutral default probability and it increases as the asset value falls. Hence, changes in equity prices should be inversely related to changes in prices of credit instruments. Asset dynamics are governed by the market price of risk, which in the Merton framework is a function of the difference between risk-neutral and real-world default probabilities (see 8

9 e.g. Duffie and Singleton, 2003, p. 119f). Combining Equations (2) and (3) yields P D Q t ( = Φ Φ 1 (P D P t ) + µ r σ T ), (5) and we define the asset excess return per unit of volatility λ by λ µ r T σ = Φ 1 (P D Q ) Φ 1 (P D P ). (6) Note that we have omitted the time t-subscripts because the market price of risk is timeinvariant, due to the Merton (1974) model assumptions of constant drift and constant riskless rate. Furthermore, the functional form of the link between real-world and risk-neutral default probabilities as the difference of the respective inverses of the standard normal distribution implies that risk-adjusted asset excess returns are largely driven by the difference between P D P and P D Q while the level of the default probability is typically less important. The difference between real-world and risk-neutral default probabilities can be interpreted as a (constant) credit risk premium. In the literature, various definitions of credit risk premia exist, two common measures are the difference and the ratio of risk-neutral and real-world default probabilities; see e.g. Amato (2005) and Berndt et al. (2008). Berg (2010) shows that these risk premia are hardly affected when considering other structural models such as Black and Cox (1976), Leland (1994), Leland and Toft (1996), and Duffie and Lando (2001). Higher asset excess returns are thus associated with higher credit risk premia and because equity is a European call option on assets the analogous argument applies for stock returns. 2.2 Time-Variation in Market Price of Risk and Default Probabilities Building on this link between the asset excess returns and default probabilities, we now motivate asset dynamics which allow for time-variation in the market price of risk when there is time-variation in the real-world and/or risk-neutral probabilities of default. 2 As a 2 We implicitly allow the drift to vary over time but similarly we could (additionally) allow asset volatility and/or interest rates to vary over time. 9

10 consequence, changes in expected default probabilities, credit risk premia, and unexpected credit news have an impact on market price of risk. We consider the log asset return from time t to t + τ, with τ < T. At t + τ the time to maturity for debt is T T τ. 3 We use Equation (3) to calculate log asset values for time t and t + τ respectively, log(v t ) and log(v t+τ ). The realized asset return is a function of the risk-neutral default probabilities at time t and t + τ and is given by 4 log(v t+τ /V t ) = ( r 1 [ ] 2 σ2) τ σ Φ 1 (P D Q t+τ ) T Φ 1 (P D Q t ) T. (7) We define the realized asset return in excess of the risk-neutral drift, rx t+τ, ( rx t+τ log(v t+τ /V t ) r 1 2 σ2) τ, (8) and the excess return per-unit of volatility, λ t+τ rx t+τ /σ, is [ ] λ t+τ = Φ 1 (P D Q t+τ ) T Φ 1 (P D Q t ) T. (9) Equation (9) represents a dynamic analogue to Equation (6). It shows that risk-adjusted excess returns are negatively related to innovations in the risk-neutral default probability which is consistent with the arguments and empirical findings of e.g. Kwan (1996) and Collin- Dufresne and Goldstein (2001). In particular, Equation (9) shows that the excess return is zero if the inverse of the risk-neutral probability realizes exactly at its Q-expectation, i.e. if ] Φ 1 (P D Q t+τ ) = EQ t [Φ 1 (P D Q t+τ ), because E Q t ] [Φ 1 (P D Q t+τ ) = Φ 1 (P D Q t ) T/T. (10) As a result, positive (negative) excess returns are realized if P D Q t+τ is less (greater) than its risk-neutral expectation at time t. 5 3 We hence maintain the assumption that debt has a fixed maturity date, however, having a constant term to maturity over time would even simplify our results as T τ would be equal to T below. 4 Note that in the standard Merton (1974) framework, the second term on the right hand side would just be a Brownian increment. 5 See Appendix A.1 for a derivation of the Q-expectation of Φ 1 (P D Q t+τ ). 10

11 Asset excess returns are thus driven by expected credit risk premia and/or unexpected news about the company s default risk. To disentangle these components of realized returns, we decompose the right hand side of Equation (9). Assuming that asset dynamics obey rational expectations, we have that λ t+τ = E P t [λ t+τ ] + η t+τ [ ] [ ] = Φ 1 (P D Q t+τ ) T Φ 1 (P D Q t ) T + η t+τ. (11) E P t We add and subtract E Q t [ Φ 1 (P D Q t+τ ) ] T and get λ t+τ = = [ ( [ ] E Q t Φ 1 (P D Q t+τ ) ) T Φ 1 (P D Q t ) T }{{} =0 by Eq. (10) ( + E P t [Φ 1 (P D Q t+τ ) ] E Q t ] [ ]) Φ 1 (P D Q t+τ ) T + η t+τ [ ] ( ] [ ] ) E Q t [Φ 1 (P D Q t+τ ) E P t Φ 1 (P D Q t+τ ) T + η t+τ. (12) Equation (12) shows that asset excess returns compensate for credit risk premia given by the difference between P and Q expectations about the future inverse of the standard normal distribution of the risk-neutral default probabilities. In other words, it shows that asset excess returns inversely co-move with changes in risk-neutral default probabilities in (9) because these changes reveal information about credit risk premia and unexpected credit news (η t+τ ). In what follows, we show how we extract P- and Q-measure information about a company s default risk from the term structure of CDS spreads. 3 Using CDS spreads to extract credit risk information In our empirical analysis, we use CDS market information to explore whether credit risk is priced in equity returns. This section lays out how we extract credit risk premia from the term structure of CDS spreads. Without loss of generality, we illustrate the link between CDS spreads and default prob- 11

12 abilities using a deterministic hazard rate model. The T -year CDS spread at time t is given by S T t ( ) log 1 P D Q = κ T t t (1 R) = (1 R) (13) T where κ T t is the hazard rate and R denotes the constant recovery rate. In this example, St T is monotonically increasing in κ T t and, hence, in P D Q t. Changes in risk-neutral default probabilities (which drive asset excess returns) directly translate into changes in CDS spreads. Assuming rational expectations, changes in CDS spreads from t to t + τ can be written and decomposed as follows St+τ T St+τ T St T = E P [ ] t S T t+τ S T t + ξ t+τ ( = E Q [ ] ) ( t S T t+τ S T t E Q t [ S T t+τ ] E P t [ S T t+τ ] ) + ξ t+τ (14) We start from noting that, for any horizon τ and maturity T, we can extract E Q t [ S T t+τ ] from the term structure of CDS spreads because E Q t [ ] S T t+τ = F τ T t, where Ft τ T denotes the forward CDS spread contracted at time t and being effective from time t + τ for T periods. We refer to F τ T t S T t = E Q t [ S T t+τ ] S T t (15) as the CDS forward premium which represents the risk-neutral expectation of the change in the CDS spread. We define, in analogy to the bond market, the CDS excess return RX T t+τ S T t+τ (F τ T t S T t ) (16) which is the change in the CDS spread in excess to the Q-expected change. We define Λ T t+τ RX T t+τ (17) because minus the CDS excess return has the same informational content as λ t+τ in Equation 12

13 (12). To see this, we combine Equations (14) to (17) and get Λ T t+τ = ( E Q t [ S T t+τ ] E P t [ S T t+τ ] ) ξ t+τ. (18) Given the link between CDS spreads and risk-neutral default probabilities, constructing portfolios by ranking firms using either Λ T t+τ or λ t+τ results in identical portfolios. For the purpose of a cross-sectional asset pricing exercise exploring the link between credit risk and equity returns one can therefore use CDS data to test the predictions of structural models (as described in Section 2) because Λ T t+τ shares the properties of λ t+τ. Realized asset and, hence, also equity excess returns are positively related to Λ T t+τ and thus inversely related to RXt+τ T. Decomposing Λ T t+τ into the two terms in Equation (18), equity excess returns should increase with expected risk premia (first term) and decrease with unanticipated credit news ξ t+τ (for instance, unexpected high earnings implying a reduction in credit risk should be associated with higher stock returns). Note that the above predictions only rely on the monotonically increasing relation between S T t and P D Q t, a property that is common to all CDS valuation models. For the purpose of our paper, it is therefore sufficient to use available CDS market data without additionally specifying a model for CDS spreads. It is not necessary to explore the determinants of levels, changes, or excess returns of CDS spreads to answer the question whether credit risk premia extracted from CDS spreads are priced in equity returns in a manner consistent with the Merton (1974) model. 6 In the following subsections, we show how we use data on the term structure of CDS spreads to estimate expected risk premia and unexpected credit news. Details with respect to the valuation of spot and forward CDS contracts are delegated to Appendix A.3. 6 There is a separate stream of research on the determinants of credit spreads using yield spreads and/or CDS spreads. Recent empirical and theoretical work investigates both the role of firm characteristics (e.g. leverage, growth options) as well as market-wide factors (e.g. catastrophe risk, macroeconomic conditions), see for instance Collin-Dufresne and Goldstein (2001), Berndt and Obreja (2010), Collin-Dufresne et al. (2010), and Arnold et al. (2011). The term structure of credit spreads is driven by company specific and common factors, however, firms may also be specific in their link to common factors. For instance, accounting for firms heterogeneity in asset composition, firms react differently to common macroeconomic risk, thereby causing cross-sectional differences in credit risk (see e.g. Arnold et al., 2011). 13

14 3.1 The Term Structure of CDS Spreads and CDS Excess Returns We estimate risk premia by analyzing the link between spot and forward CDS spreads building on approaches used in research on interest rates and exchange rates. We start from the case of risk-neutrality E P t [ ] S T t+τ = E Q [ ] t S T t+τ = F τ T t (19) implying that forward CDS spreads should be unbiased predictors of future CDS spreads because risk premia are assumed to be zero. An empirical test of this forward unbiasedness hypothesis (FUH) can be done using regressions analogue to those used in the bond market (see, e.g., Fama (1984) and Fama and Bliss (1987)) and currency market literature (Fama (1984)). Assuming rational expectations and setting τ = 1, running the regression S T t+1 = α 0 + β 0 (F 1 T t S T t ) + ɛ 0,t+1. (20) should result in an estimate of β 0 that equals one if the FUH holds. While a non-zero α 0 would be indicative for a constant (i.e. time-invariant) risk premium, any deviation of β 0 from its FUH implied value of one would be suggestive for the existence of time-varying risk premia. This can be seen from the complementary regression that is obtained from subtracting the CDS forward premium, i.e. (F 1 T t S T t ), from both sides of (20) RX T t+1 = α 1 + β 1 (F 1 T t S T t ) + ɛ 1,t+1 (21) where RX T t+1 is the CDS excess return as in Equation (16). Since Equation (21) contains the same information as Equation (20), the intercepts and the residuals are the same, i.e. α 1 = α 0 and ɛ 1,t+1 = ɛ 0,t+1, and the estimate of the slope, β 1 = β 0 1, should equal zero if the FUH holds. If the estimate is non-zero, this implies predictable time-varying risk premia. Note that F 1 T t and S T t on the right-hand side contain the same information about (expected) CDS spreads for the period from t+1 to t+t. In other words, the disjunct information embedded in the spot and forward CDS spreads relate to the periods from t to t + 1 (which is only reflected in the spot spread) and from t + T to t + T + 1 (which is only reflected in 14

15 the forward spread). We thus consider a slightly modified version of Equation (21) using the forward premium between the 1-year forward CDS spread starting in T years and the current 1-year CDS spread: RX T t+1 = α 2 + β 2 (F T 1 t S 1 t ) + ɛ 2,t+1. (22) In other words, we investigate whether the one-period CDS excess return in the T -period CDS contract is related to the Q-expected change in the one-period CDS spread over the next T years. 7 Finding a non-zero β 2 would imply the presence of time-varying risk premia that are predictable. In the next subsection, we show how one can estimate expected CDS excess returns using a single-factor model exploiting the information embedded in the term structure of 1-year forward CDS spreads. 3.2 Single-Factor Model for CDS Excess Returns We now relate risk premia to the full term structure of forward CDS spreads rather than only to one particular T -period forward premium. Our approach is guided by the ideas of Cochrane and Piazzesi (2005) who extract a single factor from forward interest rates to predict bond risk premia. The term structure of forward CDS spreads is represented by the current 1-year CDS spread and 1-year forwards for T = 1, 3, 5, 7. As described in detail in the data section, we choose these maturities because they correspond to the canonical CDS spreads of 1, 3, 5, 7, and 10 years. The starting point for the single-factor model is given by regressing CDS excess returns of T -year CDS contracts (with T = 1, 3, 5, 7) on all forward rates RX T t+1 = δ T 0 + δ T 1 S 1 t + δ T 2 F 1 1 t + δ T 3 F 3 1 t + δ T 4 F 5 1 t + δ T 5 F 7 1 t + ε T t+1. (23) In the single-factor model, all T -excess returns are driven by the the same linear combination of CDS spreads, parameterized with γ = (γ 0, γ 1, γ 2, γ 3, γ 4, γ 5 ). With d T denoting the T - 7 Note that in the case of interest rates, the regressions (21) and (22) are fully equivalent: since both rates on the right hand side of Equation (21) can be expressed as sums of one-period forward rates and taking the difference results in rates being effective between t + 1 and t + T canceling out; see also Fama and Bliss (1987). In the context of valuing CDS contracts this relation is more involved as also the risk-adjusted present values have to be accounted for; see Appendix A.3. Empirically, using these two specifications leads to the same conclusions, though. 15

16 specific loading on the single factor we have RXt+1 T = d T ( γ 0 + γ 1 St 1 + γ 2 Ft γ 3 Ft γ 4 Ft γ 5 Ft 7 1 ) + ε T t+1. (24) To estimate the parameters of Equation (24), we use a two step approach. We first estimate the single factor by regressing average excess returns across maturities on all forward rates. With RX t+1 1/4 T ={1,3,5,7} RXT t+1, we identify γ through RX t+1 = γ 0 + γ 1 S 1 t + γ 2 F 1 1 t + γ 3 F 3 1 t + γ 4 F 5 1 t + γ 5 F 7 1 t + ε t+1. = γ F t + ε t+1. (25) In the second step, one can estimate the T -specific loadings d T on the single factor, γ F t, by regressing T -excess returns on the factor RX T t+1 = d T ( γ F t ) + ε T t+1. (26) 3.3 Estimates of Expected Risk Premia and Credit News Analogous to the T -specific definition in Equation (17), we define Λ t+1 RX t+1 = γ F t ε t+1. (27) From Equation (27) we obtain the estimates for expected risk premia, ÊRP t+1, and unanticipated credit news, NEW S t+1, ÊRP t+1 = γ F t, (28) NEW S t+1 = ε t+1. (29) Based on the above arguments, stock returns should be related positively to risk premia and negatively to news (e.g. lower credit risk and hence lower CDS spreads due to unexpected high earnings should be associated with higher stock returns). 16

17 4 Empirical Analysis 4.1 Data We obtain daily CDS spreads for 677 USD denominated contracts of US based obligors from Datastream for the period between January 2, 2004 and June 30, We use only the five canonical CDS maturities of 1, 3, 5, 7, and 10 years since these are most frequently quoted and traded. The protection payment may be triggered by several different restructuring events, ranging from no-restructuring to full-restructuring. For our analysis of the US market we include contracts that adopt the modified-restructuring (MR) clause, which was the market convention before the introduction of the CDS Big Bang protocol in April 2009, and contracts that adopt the no-restructuring (NR) clause, which has been the market standard since the changes of the protocol took place. 8 Moreover, we remove non-corporate obligors (such as states and counties) and daily observations where the 5-year CDS spread is not available which is typically an indication for contracts on reference entities being traded infrequently. This leaves us with 906,936 observations across 599 firms. To compute discount factors for the purpose of fitting survival curves and calculating forward CDS spreads, we obtain US Libor rates for maturities of 1 week, 1, 2, 3, 6, 9, and 12 months and swap rates for 2, 3, 4, 5, 7, and 10 years from Datastream. The bootstrap procedure follows standard industry practice; Feldhütter and Lando (2008) show that swap rates are the best parsimonious proxy for riskless rates. For our analysis of the link between equity and CDS markets, we require stock prices, which we also obtain from Datastream at a daily frequency for the same sample period. We exclude firms for which stock data is not available (in most cases these are privately-held firms or non-list subsidiaries). We also apply a filter to remove stale price observations, where we define prices to be stale if we observe equal prices on at least five consecutive days. In such a case we only consider the first of these observations and classify the subsequent observations as not available. We obtain data on several company characteristics from Datastream and Compustat. In 8 We still find observations in the CDS data set where the restructuring field is empty. Note that we do not remove these observations because according to Datastream the field of the restructuring clause was added not before

18 order to calculate various proxies for distress risk we obtain each firm s book-to-market ratio and market value from Datastream. To compute firms distance-to-default we obtain book values of liabilities using the Compustat annual files. To estimate the firm s notional debt value we assume that it to consists of short-term and long-term debt. For short-term debt we use Compustat data item Debt Due in 1st Year (DD1) which represents the current portion of long-term debt. For long-term debt we use the Compustat data item Long-Term Debt - Total (DLTT). We finally end up with a merged data set consisting of 620,336 observations of 404 firms in the period between January 2, 2004 and June 30, 2010 where CDS spreads, stock prices and firm characteristics are available. To correct for standard risk factors in our asset pricing tests, we use the CAPM market factor, the three factors proposed by Fama and French (1993), and the four factors proposed by Carhart (1997). We obtain all these factors from Kenneth French s website Descriptive Statistics and Predictability of CDS Returns We present various descriptive statistics for the CDS data in Table 1, with the left part presenting results for the full sample (01/ /2010), the middle for the pre-crisis period (01/ /2007), and the right for the crisis (07/ /2010). All statistics are based on monthly data for all companies and presented in basis points. Forward CDS spreads are calculated as described in Appendix A.3. The descriptives show that CDS markets behave differently before and during the crisis. The average level of CDS spreads has been approximately 2% higher during the crisis as compared to before. Prior to the the crisis the standard deviation of CDS spreads increases with maturity of the contract while the opposite is true later. On average the term structure of CDS spreads is flatter during the crisis as judged by the slopes; this result is driven by the fact that while the term structure is almost always upward sloping before the crisis, one frequently observes inverted shapes during the crisis. Analogous arguments apply for forward CDS premia. Changes in CDS spreads are on average negative prior to July 2007 while after

19 the start of the crisis changes have a positive mean, are larger in absolute terms, and more volatile. Furthermore, CDS excess returns tend to be negative prior to the crisis but positive during the crisis. This suggests that forward CDS spreads overestimated future CDS spreads in the first part of our sample but underestimated subsequent spreads in the latter part. The statistics for CDS excess returns potentially provide a first indication for the presence of risk premia. To assess whether CDS spreads comprise time-varying risk premia that are predictable, we run the regressions described in Section 3 on company-by-company basis and present estimates in Table 2. The results for the unbiasedness regression, Equation (20), in Panel A reveal that the pre-crisis estimates of β 0 are typically negative and in most cases significantly different from one. This finding resembles evidence from bond markets (see e.g. Fama and Bliss, 1987) and currency markets (see e.g. Fama, 1984) and is suggestive for the presence of time-varying risk premia. There is also some evidence for predictability with the average R 2 being around During the crisis the β 0 estimates are on average positive but, on average, one would not reject that the estimate equals one. However, taking a closer look reveals that estimates are much more dispersed and that one frequently observes, both, estimates that are significantly lower than unity and even more often estimates that are significantly larger than one. This again provides an indication for time-varying risk premia, however, there appear to be substantial cross-sectional differences. During the crisis, shortmaturity CDS spreads seem to be more predictable than prior to the crisis while we observe to opposite for long maturities. For both sub-samples, the predictability of CDS excess returns using the complementary regression in Equation (21) is very similary; see the R 2 s in Panel B. Following the ideas of Cochrane and Piazzesi (2005), we relate CDS excess returns to the full term structure of 1-period CDS spreads and present R 2 s of the unrestricted estimation, Equation (23), in Panel C and for the single-factor model, Equation (26), in Panel D. Our findings suggest that CDS excess returns are indeed predictable in both sub-samples but also in the full sample. Overall, the single-factor model captures most of the variation that is explained by the unrestricted estimation. We only observe a large difference in R 2 s (mean of 0.40 as compared to 0.28) for the 1-year maturity in the pre-crisis period suggesting that 19

20 there is predictable variation in these excess returns not captured by the single factor. For longer maturities, our results suggest that variation in excess returns across maturities is by and large driven by the single factor. Moreover, the R 2 s for the unrestricted estimation range from 0.32 to 0.36, for the single-factor model from 0.28 to During the crisis, risk premia appear to be somewhat less predictable with average R 2 s ranging from 0.28 to 0.32 for the unrestricted model and from 0.26 to 0.28 for the single factor model. Since the difference in results between the two estimation strategies is small, the term structure of CDS excess returns appears to be largely driven by the single factor during the crisis. 10 Overall, our findings suggest that the term structure of CDS spreads contains information about time-varying risk premia. CDS excess returns are predictable and, on average, the single factor captures around 28% of the variation in the pre-crisis as well as the crisis sub-sample and almost 20% when considering the two (substantially different) periods jointly. 4.3 Asset Pricing Results In this section, we present empirical evidence that distress risk is priced in the cross-section of equity returns. We compute monthly equity excess returns of value-weighted quintile portfolios constructed by ranking firms based on their default risk. First, we report our core results that stock returns increase with (expected) risk premia and decrease with unfavorable credit news, both prior and during the crisis. Second, we present results from sequential portfolio sorts to further control for firms size, book-to-market, and liquidity of CDS contracts. Finally, we summarize additional robustness checks Equity Returns and Default Risk Portfolios Sorted by Default Probabilities For comparison with prior research, we first sort companies into portfolios based on their real-world default probability using the 10 Other results not reported, include diagnostic checks of the residuals of the firm-by-firm estimations of the single-factor model. In particular, we test for serial correlation using the Durbin-Watson, Box-Pierce, and Ljung-Box statistics. Average (bootstrapped) p-values are around 0.40 across statistics and samples. We only detect significant auto-correlation for a few firms, typically with shorter time series when using monthly data. Furthermore, as a benchmark for the predictability results, we also estimate AR(1) models for CDS excess returns. On average, the R 2 s are somewhat lower than those of the unbiasedness regressions and substantially lower than those of the single-factor model. 20

21 distance-to-default (DD) and based on their risk-neutral default probability using the firm s 5-year CDS spread (S5). 11 We summarize our findings in Table 3, with the left part presenting results for the full sample (01/ /2010), the middle for the pre-crisis period (01/ /2007), and the right for the crisis period (07/ /2010). The results for the DD-sorted portfolios in Panel A suggest that, over the full sample period, companies with highest default probability and, hence, lowest DD (portfolio P1) earn lowest value-weighted excess returns and those with highest DD (P5) earn the highest. These results resemble the findings in the literature that firms with higher PDs earn abnormally low returns, see e.g. Campbell et al. (2008). In our sample, however, the excess return from buying P1 and selling P5 is not significant. While one may attribute the lack of significance to the comparably smaller number of companies that we use (because of the availability of CDS data) and/or the shorter sample period, it is interesting to see that the pattern as such is driven by the crisis period. In the pre-crisis period, companies with highest DDs earn lower excess returns than those with lowest DDs. This finding is more in line with work documenting that equity returns compensate for higher default risk, see e.g. Vassalou and Xing (2004); however, there is no monotonic pattern across portfolios and the P1 P5 return is not significant. The sub-panel labeled Portfolio Characteristics presents averages of other risk measures for the DD-sorted portfolios. Companies with lowest DD, i.e. highest default risk, have highest CDS spreads, are the smallest firms in our sample, and have the highest book-tomarket ratio. These patterns are monotonic from P1 to P5 and consistent with standard arguments that these measures proxy for distress risk. To control for traditional risk factors, we also perform factor model regressions and report the alphas of the CAPM, the Fama and French (1993) 3-factor model, and the Carhart (1997) 4-factor extension. None of the P1 P5 α estimates is significant in any of the three sample periods that we consider. We also present the loadings on the Fama-French factors RM, SMB, and HML. For the full sample we find that loadings on RM monotonically decrease from P1 to P5 and that the loading of the P1 P5 11 While Merton-model implied default probabilities may be biased as compared to empirical default frequencies, the distance-to-default nevertheless serves well for the purpose of ranking companies. Using KMV EDFs instead would not change our results because the transformation applied basically only changes the level of default probabilities but not the ranking across companies (see Lando, 2004, p. 49) 21

22 return is close to one and significant; this finding is driven by the crisis. We do not find any particular patterns for the other factors. The results for the CDS-sorted portfolios in Panel B are qualitatively similar. Firms with higher CDS spreads appear to earn lower excess returns, however, this finding is again driven by the crisis period and the P1 P5 return is not significant. Also, the results with respect to alphas and factor loadings are similar. Overall, neither sorting portfolios based on firms real-world default probabilities nor on their risk-neutral ones provides a clear picture whether distress risk is priced in equity returns. Furthermore, it appears that the relationship between probabilities of default and stock returns may be different prior as compared to during the crisis. Portfolios Sorted by Changes in CDS Spreads The dynamics motivated by structural models in Section 2.2 suggest that equity excess returns are negatively related to changes in risk-neutral default probabilities, see Equation (9). Table 4 presents supporting evidence by sorting firms into portfolios based on their contemporaneous CDS spread changes. We find that equity excess returns monotonically increase from P1 to P5, i.e. firms with largest increases in CDS spreads earn lowest equity excess returns over that month and vice versa. The long-short strategy reveals significant return differentials in the full sample as well as in the subperiods. The portfolio characteristics suggest that largest CDS spreads changes in absolute terms are associated with lower DDs and higher S5. 12 Furthermore, absolute spread changes appear to be largest for firms that are small in size and have high book-to-market ratios. The longshort portfolio returns remain highly significant even after controlling for risk factors as judged by the t-statistics of the factor model alphas. There are no significant factor loadings of the long-short portfolios except on SMB in the full sample. Note that these results are consistent with the recent finding of Han and Zhou (2011) that the slope of the CDS term structure negatively predicts stock returns. Very similar to their 12 Given this finding, we also consider relative changes in CDS spreads in our robustness checks. Using log or percentage changes in spreads is relatively unusually, though. Analogue to the bond market literature and work on corporate yield spreads it is common to use differences. However, using relative changes produces very similar findings. We do not report these results here to conserve space but they are available from the authors upon request. 22

23 results, we find (but do not report) that the slope positively predicts changes in CDS spreads and negatively predicts stock returns. However, building on the insights from the structural framework of Merton (1974), it is not surprising to find a negative relation between slope and equity returns. In fact, our work provides a rationale for the finding of Han and Zhou (2011). The slope of the CDS term structure negatively predicts equity returns because it positively predicts changes in risk-neutral default probabilities underlying CDS spreads. This is consistent with the notion of priced default risk premia since, as described in Sections 2.2 and 3, changes in risk-neutral default probabilities reveal information about (expected) risk premia embedded in the term structure of CDS spreads. 13 The next step therefore is to decompose CDS spread changes into a risk-neutral expectation and a risk premium component. Portfolios Sorted by CDS Forward Premia and Realized Risk Premia We now sort portfolios using the two components of changes in CDS spreads as in Equation (14): the CDS forward premium representing the risk-neutral expected change in the CDS spread E Q [ ] t S T t+τ = F τ T t St T and the realized risk premium Λ T t+τ RXt+τ T. From Equation (12), we expect that the realized risk premium, that we later decompose into an expected risk premium as well as unexpected news, is priced in equity returns, while the forward implied change is not. The results in Table 5 confirm our priors empirically. Panel A shows that firms with higher forward-implied CDS returns are associated with higher stock returns in the full sample but the P1 P5 return is not significant and largely driven by the crisis period. Also, for the crisis period the P1 P5 return is not significant once we control for other risk factors by estimating factor model alphas. Thus, consistent with our prior, we do not find that the risk-neutral expected change in the default probability is priced in equity returns. Panel B presents evidence that there is a strong link between stock returns and realized risk premia extracted from CDS spreads. When we sort portfolios based on Λ T t+τ, we find 13 Also note that the CDS predictability results of Han and Zhou (2011) and in our paper can be viewed in analogy to those reported for bond markets. Finding that slope predicts future spread changes appears consistent with the expectations hypothesis (EH) while predictability of excess returns appears to be inconsistent with the EH. This is very similar to EH paradox uncovered by Campbell and Shiller (1991) and the EH failure documented in Fama and Bliss (1987). Recent research shows that these puzzling empirical results are caused by the omission of risk premia. For instance, Dai and Singleton (2002) and Backus et al. (2001) show that the EH-implied coefficients in predictive regressions can be recovered when accounting for risk premia. 23