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The Determinants of Credit Spread Changes.
 Journal of Finance
, 2001
"... ABSTRACT Using dealer's quotes and transactions prices on straight industrial bonds, we investigate the determinants of credit spread changes. Variables that should in theory determine credit spread changes have rather limited explanatory power. Further, the residuals from this regression are ..."
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Cited by 422 (2 self)
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ABSTRACT Using dealer's quotes and transactions prices on straight industrial bonds, we investigate the determinants of credit spread changes. Variables that should in theory determine credit spread changes have rather limited explanatory power. Further, the residuals from this regression are highly crosscorrelated, and principal components analysis implies they are mostly driven by a single common factor. Although we consider several macroeconomic and financial variables as candidate proxies, we cannot explain this common systematic component. Our results suggest that monthly credit spread changes are principally driven by local supply/demand shocks that are independent of both creditrisk factors and standard proxies for liquidity. * CollinDufresne is at Carnegie Mellon University. Goldstein is at Washington University in St. Louis. Martin is at Arizona State University. A significant portion of this paper was written while Goldstein and Martin were at The Ohio State University. We thank Rui Albuquerque, Gurdip Bakshi, Greg Bauer, Dave Brown, Francesca Carrieri, Peter Christoffersen, Susan Christoffersen, Greg Duffee, Darrell Duffie, Vihang Errunza, Gifford Fong, Mike Gallmeyer, Laurent Gauthier, Rick Green, John Griffin, Jean Helwege, Kris Jacobs, Chris Jones, Andrew Karolyi, Dilip Madan, David Mauer, Erwan Morellec, Federico Nardari, NR Prabhala, Tony Sanders, Sergei Sarkissian, Bill Schwert, Ken Singleton, Chester Spatt, René Stulz (the editor), Suresh Sundaresan, Haluk Unal, Karen Wruck, and an anonymous referee for helpful comments. We thank Ahsan Aijaz, John Puleo, and Laura Tuttle for research assistance. We are also grateful to seminar participants at Arizona State University, University of Maryland, McGill University, The Ohio State University, University of Rochester, and Southern Methodist University. The relation between stock and bond returns has been widely studied at the aggregate level (see, for example, Campbell and Ammer (1993), Keim and Stambaugh (1986), Fama and French (1989), and Fama and French (1993)). Recently, a few studies have investigated that relation at both the individual firm level (see, for example, Kwan (1996)) and portfolio level (see, for example, Blume, Keim and Patel (1991), and Cornell and Green (1991)). These studies focus on corporate bond returns, or yield changes. The main conclusions of these papers are: (1) highgrade bonds behave like Treasury bonds, and (2) lowgrade bonds are more sensitive to stock returns. The implications of these studies may be limited in many situations of interest, however. For example, hedge funds often take highly levered positions in corporate bonds while hedging away interest rate risk by shorting treasuries. As a consequence, their portfolios become extremely sensitive to changes in credit spreads rather than changes in bond yields. The distinction between changes in credit spreads and changes in corporate yields is significant: while an adjusted R 2 of 60 percent is obtained when regressing highgrade bond yield changes on Treasury yield changes and stock returns (see Kwan (1996)) we find that the R 2 falls to five percent when the dependent variable is credit spread changes. Hence, while much is known about yield changes, we have very limited knowledge about the determinants of credit spread changes. Below, we investigate the determinants of credit spread changes. From a contingentclaims, or noarbitrage standpoint, credit spreads obtain for two fundamental reasons: 1) there is a risk of default, and 2) in the event of default, the bondholder receives only a portion of the promised payments. Thus, we examine how changes in credit spreads respond to proxies for both changes in the probability of future default and for changes in the recovery rate. Separately, recent empirical studies find that the corporate bond market tends to have relatively high transactions costs and low volume. 1 These findings suggest looking beyond the pure contingentclaims viewpoint when searching for the determinants of credit spread changes, since one might expect to observe a liquidity premium. Thus, we also examine the extent to which credit spread changes can be explained by proxies for liquidity changes. Our results are, in summary: although we consider numerous proxies that should measure both changes in default probability and changes in recovery rate, regression analysis can only explain about 25 percent of the observed credit spread changes. We find, however, that the residuals from these regressions are highly crosscorrelated, and principal components analysis implies that they are mostly driven by a single common factor. An important implication of this finding is that if any explanatory variables have been omitted, they are likely not firmspecific. We therefore rerun the regression, but 1 this time include several liquidity, macroeconomic, and financial variables as candidate proxies for this factor. We cannot, however, find any set of variables that can explain the bulk of this common systematic factor. Our findings suggest that the dominant component of monthly credit spread changes in the corporate bond market is driven by local supply/demand shocks that are independent of both changes in creditrisk and typical measures of liquidity. We note that a similar, but significantly smaller effect has been documented in the mortgage backed (Ginnie Mae) securities market by Boudoukh, Richardson, Stanton, and Whitelaw (1997), who find that a 3factor model explains over 90 percent of Ginnie Mae yields, but that the remaining variation apparently cannot be explained by the changes in the yield curve. 2 In contrast, our multiplefactor model explains only about onequarter of the variation in credit spreads, with most of the remainder attributable to a single systematic factor. Similarly, Duffie and Singleton (1999) find that both creditrisk and liquidity factors are necessary to explain innovations in U.S. swap rates. However, when analyzing the residuals they are unable to find explanatory factors. They conclude that swap marketspecific supply/demand shocks drive the unexplained changes in swap rates. Existing literature on credit spread changes is limited. 3 Pedrosa and Roll (1998) document considerable comovement of credit spread changes among index portfolios of bonds from various industry, quality, and maturity groups. Note that this result by itself is not surprising, since theory predicts that all credit spreads should be affected by aggregate variables such as changes in the interest rate, changes in business climate, changes in market volatility, etc. The particularly surprising aspect of our results is that, after controlling for these aggregate determinants, the systematic movement of credit spread changes still remains, and indeed, is the dominant factor. Brown The rest of the paper is organized as follows. In Section I, we examine the theoretical determinants of credit spread changes from a contingentclaims framework. In Section II, we discuss the data and define the proxies used. In Section III, we analyze our results. In Section IV, we provide evidence for the robustness of our results on several fronts. First, we repeat the analysis using transactions (rather than quotes) data to obtain credit spread changes. Second, we consider a host of new explanatory variables that proxy for changes in liquidity and other macroeconomic effects. Finally, we perform a regression analysis on simulated data to demonstrate that our empirical findings are not being driven by the econometric techniques used. We conclude in Section V. 2 I. Theoretical Determinants of Credit Spread Changes Socalled structural models of default provide an intuitive framework for identifying the determinants of credit spread changes. 4 These models build on the original insights of Black and Scholes (1973), who demonstrate that equity and debt can be valued using contingentclaims analysis. Introduced by Merton (1974) and further investigated by, among others, Black and Cox (1976), Leland (1994), Longstaff and Schwartz (1995), Bryis and de Varenne (1997), and CollinDufresne and Goldstein Mathematically, contingentclaims pricing is most readily accomplished by pricing derivatives under the socalled riskneutral measure, where all traded securities have an expected return equal to the riskfree rate (see Cox and Ross (1976) and Harrison and Kreps (1979)). In particular, the value of the debt claim is determined by computing its expected (under the riskneutral measure) future cash flows discounted at the riskfree rate. As the credit spread CS(t) is uniquely defined through: (1) the price of a debt claim, (2) this debt claim's contractual cash flows, and (3) the (appropriate) riskfree rate, we can write CS(t) = CS(V t , r t , {X t }), where V is firm value, r is the spot rate, and {X t } represents all of the other "state variables" needed to specify the model. 6 Since credit spreads are uniquely determined given the current values of the state variables, it follows that credit spread changes are determined by changes in these state variables. Hence, structural models generate predictions for what the theoretical determinants of credit spread changes should be, and moreover offer a prediction for whether changes in these variables should be positively or negatively correlated with changes in credit spreads. We discuss these proposed determinants individually. Changes in the Spot Rate As pointed out by Longstaff and Schwartz (1995), the static effect of a higher spot rate is to increase the riskneutral drift of the firm value process. A higher drift reduces the incidence of default, and in turn, reduces the credit spreads. This prediction is borne out in their data. Further evidence is provided by Duffee (1998), who uses a sample restricted to noncallable bonds and 3 finds a significant, albeit weaker, negative relationship between changes in credit spreads and interest rates. Changes in Slope of Yield Curve Although the spot rate is the only interestratesensitive factor that appears in the firm value process, the spot rate process itself may depend upon other factors as well. 7 For example, Litterman and Scheinkman (1991) find that the two most important factors driving the term structure of interest rates are the level and slope of the term structure. If an increase in the slope of the Treasury curve increases the expected future short rate, then by the same argument as above, it should also lead to a decrease in credit spreads. From a different perspective, a decrease in yield curve slope may imply a weakening economy. It is reasonable to believe that the expected recovery rate might decrease in times of recession. 8 Once again, theory predicts that an increase in the Treasury yield curve slope will create a decrease in credit spreads. Changes in Leverage Within the structural framework, default is triggered when the leverage ratio approaches unity. Hence, it is clear that credit spreads are expected to increase with leverage. Likewise, credit spreads should be a decreasing function of the firm's return on equity, all else equal. Changes in Volatility The contingentclaims approach implies that the debt claim has features similar to a short position in a put option. Since option values increase with volatility, it follows that this model predicts credit spreads should increase with volatility. This prediction is intuitive: increased volatility increases the probability of default. Changes in the Probability or Magnitude of a Downward Jump in Firm Value Implied volatility smiles in observed option prices suggest that markets account for the probability of large negative jumps in firm value. Thus, increases in either the probability or the magnitude of a negative jump should increase credit spreads. Changes in the Business Climate Even if the probability of default remains constant for a firm, changes in credit spreads can occur due to changes in the expected recovery rate. The expected recovery rate in turn should be 4 a function of the overall business climate. 9 II. Data Our first objective is to investigate how well the variables identified above explain observed changes in credit spreads. Here, we discuss the data used for estimating both credit spreads and proxies for the explanatory variables. Credit Spreads The corporate bond data are obtained from Lehman Brothers via the Fixed Income (or Warga) Database. We use only quotes on noncallable, nonputtable debt of industrial firms; quotes are discarded whenever a bond has less than four years to maturity. Monthly observations are used for the period July 1988 through December 1997. Only observations with actual quotes are used, since it has been shown by Sarig and Warga (1989) that matrix prices are problematic. 10 To determine the credit spread, CS i t , for bond i at month t, we use the Benchmark Treasury rates from Datastream for maturities of 3, 5, 7, 10, and 30 years, and then use a linear interpolation scheme to estimate the entire yield curve. Credit spreads are then defined as the difference between the yield of bondi and the associated yield of the Treasury curve at the same maturity. Treasury Rate Level We use Datastream's monthly series of 10year Benchmark Treasury rates, r 10 t . To capture potential nonlinear effects due to convexity, we also include the squared level of the term structure, (r 10 t ) 2 . Slope of Yield Curve We define the slope of the yield curve as the difference between Datastream's 10year and 2year Benchmark Treasury yields, slope t ≡ r 10 t − r 2 t . We interpret this proxy as both an indication of expectations of future short rates, and as an indication of overall economic health. Firm Leverage For each bond i, market values of firm equity from CRSP and book values of firm debt from COMPUSTAT are used to obtain leverage ratios, lev 5 Since debt levels are reported quarterly, linear interpolation is used to estimate monthly debt figures. We note that previous studies of yield changes have often used the firm's equity return to proxy for changes in the firm's health, rather than changes in leverage. For robustness, we also use each firm's monthly equity return, ret i t , obtained from CRSP, as an explanatory variable. Volatility In theory, changes in a firm's future volatility can be extracted from changes in implied volatilities of its publicly traded options. Unfortunately, most of the firms we investigate lack publicly traded options. 11 Thus, we are forced to use the best available substitute: changes in the VIX index, VIX t , which corresponds to a weighted average of eight implied volatilities of nearthemoney options on the OEX (S&P 100) index. 12 These data are provided by the Chicago Board Options Exchange. While use of VIX in place of firmspecific volatility assumes a strong positive correlation between the two, this assumption does not seem to affect our results significantly. Indeed, one of our main findings is that most of the credit spread innovation is unexplained, and that the residuals are highly correlated crosssectionally. Note that if changes in individual firm volatility and market volatility are not highly correlated, then our proxy should bias our results away from finding residuals which are so systematic. Jump Magnitudes and Probabilities Changes in the probability and magnitude of a large negative jump in firm value should have a significant effect on credit spreads. This factor is rather difficult to proxy because historical occurrences of such jumps are rare enough to be of little value in predicting future probabilities and magnitude of such jumps. Therefore, we approach the problem using a forwardlooking measure. In particular, we employ changes in the slope of the "smirk" of implied volatilities of options on S&P 500 futures to determine perceived changes in the probability of such jumps. Options and futures prices were obtained from Bridge. Our proxy is constructed from atand outofthe money puts, and atand inthemoney calls with the shortest maturity on the nearby S&P 500 futures contract. We first compute implied volatilities for each strike K using the standard Black and Scholes (1973) model. We then fit the linearquadratic regression σ(K) = a + bK + cK 2 , where K is the strike price. Our estimate of this slope, jump t , is defined via where F is the atthe money strike price, which equals the current futures price. We choose to look at the implied volatility at K = .9F because we do not want 6 to extrapolate the quadratic regression beyond the region where actual option prices are most typically observed. Note that if there is a nonnegligible probability of large negative jumps in firm value, then the appropriate hedging tool for corporate debt may not be the firm's equity, but rather deep outofthemoney puts on the firm's equity. Assuming large negative jumps in firm value are highly correlated with market crashes, we hope to capture systematic changes in the market's expectation of such events with this proxy. We expect that a steepening in the slope of the smirk will trigger an increase in credit spreads. Changes in Business Climate We use monthly S&P 500 returns, S&P t , as a proxy for the overall state of the economy. The data are obtained from CRSP. For ease of analysis, each bond is assigned to a leverage group based on the firm's average leverage ratio for those months where the bond has quotes available. In Panels II and III of INSERT In Maturity subsample results are also presented in Panels II and III of
TimeChanged Lévy Processes and Option Pricing
, 2002
"... As is well known, the classic BlackScholes option pricing model assumes that returns follow Brownian motion. It is widely recognized that return processes differ from this benchmark in at least three important ways. First, asset prices jump, leading to nonnormal return innovations. Second, return ..."
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Cited by 189 (23 self)
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As is well known, the classic BlackScholes option pricing model assumes that returns follow Brownian motion. It is widely recognized that return processes differ from this benchmark in at least three important ways. First, asset prices jump, leading to nonnormal return innovations. Second, return volatilities vary stochastically over time. Third, returns and their volatilities are correlated, often negatively for equities. We propose that timechanged Lévy processes be used to simultaneously address these three facets of the underlying asset return process. We show that our framework encompasses almost all of the models proposed in the option pricing literature. Despite the generality of our approach, we show that it is straightforward to select and test a particular option pricing model through the use of characteristic function technology.
Asset pricing at the millennium
 Journal of Finance
"... This paper surveys the field of asset pricing. The emphasis is on the interplay between theory and empirical work and on the tradeoff between risk and return. Modern research seeks to understand the behavior of the stochastic discount factor ~SDF! that prices all assets in the economy. The behavior ..."
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Cited by 189 (0 self)
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This paper surveys the field of asset pricing. The emphasis is on the interplay between theory and empirical work and on the tradeoff between risk and return. Modern research seeks to understand the behavior of the stochastic discount factor ~SDF! that prices all assets in the economy. The behavior of the term structure of real interest rates restricts the conditional mean of the SDF, whereas patterns of risk premia restrict its conditional volatility and factor structure. Stylized facts about interest rates, aggregate stock prices, and crosssectional patterns in stock returns have stimulated new research on optimal portfolio choice, intertemporal equilibrium models, and behavioral finance. This paper surveys the field of asset pricing. The emphasis is on the interplay between theory and empirical work. Theorists develop models with testable predictions; empirical researchers document “puzzles”—stylized facts that fail to fit established theories—and this stimulates the development of new theories. Such a process is part of the normal development of any science. Asset pricing, like the rest of economics, faces the special challenge that data are generated naturally rather than experimentally, and so researchers cannot control the quantity of data or the random shocks that affect the data. A particularly interesting characteristic of the asset pricing field is that these random shocks are also the subject matter of the theory. As Campbell, Lo, and MacKinlay ~1997, Chap. 1, p. 3! put it: What distinguishes financial economics is the central role that uncertainty plays in both financial theory and its empirical implementation. The starting point for every financial model is the uncertainty facing investors, and the substance of every financial model involves the impact of uncertainty on the behavior of investors and, ultimately, on mar* Department of Economics, Harvard University, Cambridge, Massachusetts
Counterparty Risk and the Pricing of Defaultable Securities
 THE JOURNAL OF FINANCE
, 2001
"... Motivated by recent financial crises in East Asia and the United States where the downfall of a small number of firms had an economywide impact, this paper generalizes existing reducedform models to include default intensities dependent on the default of a counterparty. In this model, firms have c ..."
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Cited by 184 (11 self)
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Motivated by recent financial crises in East Asia and the United States where the downfall of a small number of firms had an economywide impact, this paper generalizes existing reducedform models to include default intensities dependent on the default of a counterparty. In this model, firms have correlated defaults due not only to an exposure to common risk factors, but also to firmspecific risks that are termed “counterparty risks.” Numerical examples illustrate the effect of counterparty risk on the pricing of defaultable bonds and credit derivatives such as default swaps.
Pricing the risks of default
 Review of Derivatives Research
, 1998
"... the problems and opportunities facing the financial services industry in its search for competitive excellence. The Center's research focuses on the issues related to managing risk at the firm level as well as ways to improve productivity and performance. The Center fosters the development of a ..."
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Cited by 179 (7 self)
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the problems and opportunities facing the financial services industry in its search for competitive excellence. The Center's research focuses on the issues related to managing risk at the firm level as well as ways to improve productivity and performance. The Center fosters the development of a community of faculty, visiting scholars and Ph.D. candidates whose research interests complement and support the mission of the Center. The Center works closely with industry executives and practitioners to ensure that its research is informed by the operating realities and competitive demands facing industry participants as they pursue competitive excellence. Copies of the working papers summarized here are available from the Center. If you would like to learn more about the Center or become a member of our research community, please let us know of your interest.
Pricing And Hedging Derivative Securities In Markets With Uncertain Volatilities
 Applied Mathematical Finance
, 1995
"... We present a model for pricing and hedging derivative securities and option portfolios in an environment where the volatility is not known precisely, but is assumed instead to lie between two extreme values oe min and oe max . These bounds could be inferred from extreme values of the implied volatil ..."
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Cited by 165 (3 self)
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We present a model for pricing and hedging derivative securities and option portfolios in an environment where the volatility is not known precisely, but is assumed instead to lie between two extreme values oe min and oe max . These bounds could be inferred from extreme values of the implied volatilities of liquid options, or from highlow peaks in historical stock or optionimplied volatilities. They can be viewed as defining a confidence interval for future volatility values. We show that the extremal nonarbitrageable prices for the derivative asset which arise as the volatility paths vary in such a band can be described by a nonlinear PDE, which we call the BlackScholesBarenblatt equation. In this equation, the "pricing" volatility is selected dynamically from the two extreme values oe min ,oe max , according to the convexity of the valuefunction. A simple algorithm for solving the equation by finitedifferencing or a trinomial tree is presented. We show that this model capture...
Pricing American options: a duality approach.
 Operation Research
, 2004
"... Abstract We develop a new method for pricing American options. The main practical contribution of this paper is a general algorithm for constructing upper and lower bounds on the true price of the option using any approximation to the option price. We show that our bounds are tight, so that if the ..."
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Cited by 147 (6 self)
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Abstract We develop a new method for pricing American options. The main practical contribution of this paper is a general algorithm for constructing upper and lower bounds on the true price of the option using any approximation to the option price. We show that our bounds are tight, so that if the initial approximation is close to the true price of the option, the bounds are also guaranteed to be close. We also explicitly characterize the worstcase performance of the pricing bounds. The computation of the lower bound is straightforward and relies on simulating the suboptimal exercise strategy implied by the approximate option price. The upper bound is also computed using Monte Carlo simulation. This is made feasible by the representation of the American option price as a solution of a properly defined dual minimization problem, which is the main theoretical result of this paper. Our algorithm proves to be accurate on a set of sample problems where we price call options on the maximum and the geometric mean of a collection of stocks. These numerical results suggest that our pricing method can be successfully applied to problems of practical interest.
A Macroeconomic Model with a Financial Sector," 41
 Journal of Monetary Economics
, 2009
"... This paper studies the full equilibrium dynamics of an economy with financial frictions. Due to highly nonlinear amplification effects, the economy is prone to instability and occasionally enters volatile episodes. Risk is endogenous and asset price correlations are high in down turns. In an enviro ..."
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Cited by 145 (8 self)
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This paper studies the full equilibrium dynamics of an economy with financial frictions. Due to highly nonlinear amplification effects, the economy is prone to instability and occasionally enters volatile episodes. Risk is endogenous and asset price correlations are high in down turns. In an environment of low exogenous risk experts assume higher leverage making the system more prone to systemic volatility spikes a volatility paradox. Securitization and derivatives contracts leads to better sharing of exogenous risk but to higher endogenous systemic risk. Financial experts may impose a negative externality on each other by not maintaining adequate capital cushion.
Portfolio and consumption decisions under meanreverting returns: An exact solution for complete markets
 Journal of Financial and Quantitative Analysis 37, 63–91. The Journal of Finance Wachter, Jessica A
, 2003
"... This paper solves, in closed form, the optimal portfolio choice problem for an investor with utility over consumption under meanreverting returns. Previous solutions either require approximations, numerical methods, or the assumption that the investor does not consume over his lifetime. This paper ..."
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Cited by 135 (7 self)
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This paper solves, in closed form, the optimal portfolio choice problem for an investor with utility over consumption under meanreverting returns. Previous solutions either require approximations, numerical methods, or the assumption that the investor does not consume over his lifetime. This paper breaks the impasse by assuming that markets are complete. The solution leads to a new understanding of hedging demand and of the behavior of the approximate loglinear solution. The portfolio allocation takes the form of a weighted average and is shown to be analogous to duration for coupon bonds. Through this analogy, the notion of investment horizon is extended to that of an investor who consumes at multiple points in time.