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Transform Analysis and Asset Pricing for Affine JumpDiffusions
 Econometrica
, 2000
"... In the setting of ‘‘affine’ ’ jumpdiffusion state processes, this paper provides an analytical treatment of a class of transforms, including various Laplace and Fourier transforms as special cases, that allow an analytical treatment of a range of valuation and econometric problems. Example applicat ..."
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Cited by 710 (38 self)
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In the setting of ‘‘affine’ ’ jumpdiffusion state processes, this paper provides an analytical treatment of a class of transforms, including various Laplace and Fourier transforms as special cases, that allow an analytical treatment of a range of valuation and econometric problems. Example applications include fixedincome pricing models, with a role for intensitybased models of default, as well as a wide range of optionpricing applications. An illustrative example examines the implications of stochastic volatility and jumps for option valuation. This example highlights the impact on option ‘smirks ’ of the joint distribution of jumps in volatility and jumps in the underlying asset price, through both jump amplitude as well as jump timing.
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
The JumpRisk Premia Implicit in Options: Evidence from an Integrated TimeSeries Study
 Journal of Financial Economics
"... Abstract: This paper examines the joint time series of the S&P 500 index and nearthemoney shortdated option prices with an arbitragefree model, capturing both stochastic volatility and jumps. Jumprisk premia uncovered from the joint data respond quickly to market volatility, becoming more p ..."
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Cited by 419 (3 self)
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Abstract: This paper examines the joint time series of the S&P 500 index and nearthemoney shortdated option prices with an arbitragefree model, capturing both stochastic volatility and jumps. Jumprisk premia uncovered from the joint data respond quickly to market volatility, becoming more prominent during volatile markets. This form of jumprisk premia is important not only in reconciling the dynamics implied by the joint data, but also in explaining the volatility “smirks” of crosssectional options data.
Explaining the rate spread on corporate bonds
 Journal of Finance
, 2001
"... The purpose of this article is to explain the spread between spot rates on corporate and government bonds. We find that the spread can be explained in terms of three elements: (1) compensation for expected default of corporate bonds (2) compensation for state taxes since holders of corporate bonds p ..."
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Cited by 383 (3 self)
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The purpose of this article is to explain the spread between spot rates on corporate and government bonds. We find that the spread can be explained in terms of three elements: (1) compensation for expected default of corporate bonds (2) compensation for state taxes since holders of corporate bonds pay state taxes while holders of government bonds do not, and (3) compensation for the additional systematic risk in corporate bond returns relative to government bond returns. The systematic nature of corporate bond return is shown by relating that part of the spread which is not due to expected default or taxes to a set of variables which have been shown to effect risk premiums in stock markets Empirical estimates of the size of each of these three components are provided in the paper. We stress the tax effects because it has been ignored in all previous studies of corporate bonds. 1
Corporate Yield Spreads: Default Risk or Liquidity? New Evidence from the Credit Default Swap Market
 Journal of Finance
, 2005
"... Copyright c○2004 by the authors. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. ..."
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Cited by 359 (8 self)
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Copyright c○2004 by the authors. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.
Term structures of credit spreads with incomplete accounting information
 ECONOMETRICA
, 2001
"... We study the implications of imperfect information for term structures of credit spreads on corporate bonds. We suppose that bond investors cannot observe the issuer’s assets directly, and receive instead only periodic and imperfect accounting reports. For a setting in which the assets of the firm ..."
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Cited by 317 (19 self)
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We study the implications of imperfect information for term structures of credit spreads on corporate bonds. We suppose that bond investors cannot observe the issuer’s assets directly, and receive instead only periodic and imperfect accounting reports. For a setting in which the assets of the firm are a geometric Brownian motion until informed equityholders optimally liquidate, we derive the conditional distribution of the assets, given accounting data and survivorship. Contrary to the perfectinformation case, there exists a defaultarrival intensity process. That intensity is calculated in terms of the conditional distribution of assets. Credit yield spreads are characterized in terms of accounting information. Generalizations are provided.
Structural Models of Corporate Bond Pricing: An Empirical Analysis
, 2003
"... This paper empirically tests five structural models of corporate bond pricing: those of Merton (1974), Geske (1977), Leland and Toft (1996), Longsta# and Schwartz (1995), and CollinDufresne and Goldstein (2001). We implement the models using a sample of 182 bond prices from firms with simple capita ..."
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Cited by 245 (6 self)
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This paper empirically tests five structural models of corporate bond pricing: those of Merton (1974), Geske (1977), Leland and Toft (1996), Longsta# and Schwartz (1995), and CollinDufresne and Goldstein (2001). We implement the models using a sample of 182 bond prices from firms with simple capital structures during the period 19861997. The conventional wisdom is that structural models do not generate spreads as high as those seen in the bond market, and true to expectations we find that the predicted spreads in our implementation of the Merton model are too low. However, most of the other structural models predict spreads that are too high on average. Nevertheless, accuracy is a problem, as the newer models tend to severely overstate the credit risk of firms with high leverage or volatility and yet su#er from a spread underprediction problem with safer bonds. The Leland and Toft model is an exception in that it overpredicts spreads on most bonds, particularly those with high coupons. More accurate structural models must avoid features that increase the credit risk on the riskier bonds while scarcely a#ecting the spreads of the safest bonds.
The Link between Default and Recovery Rates: Effects on the Procyclicality of Regulatory Capital Ratios, BIS Working Papers, No 113.
, 2002
"... ..."
An Empirical Analysis of the Dynamic Relationship between InvestmentGrade Bonds and Credit Default Swaps
, 2004
"... ..."
Equity volatility and corporate bond yields
 Journal of Finance
, 2003
"... This paper explores the e¡ect of equity volatility on corporate bond yields. Panel data for the late 1990s show that idiosyncratic ¢rmlevel volatility can explain as much crosssectional variation in yields as can credit ratings. This ¢nding, together with the upward trend in idiosyncratic equity v ..."
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Cited by 190 (1 self)
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This paper explores the e¡ect of equity volatility on corporate bond yields. Panel data for the late 1990s show that idiosyncratic ¢rmlevel volatility can explain as much crosssectional variation in yields as can credit ratings. This ¢nding, together with the upward trend in idiosyncratic equity volatility documented by Campbell, Lettau, Malkiel, and Xu (2001), helps to explain recent increases in corporate bond yields. DURING THE LATE 1990s, THE U.S. EQUITY and corporate bond markets behaved very di¡erently. As displayed in Figure 1, stock prices rose strongly, while at the same time, corporate bonds performed poorly. The proximate cause of the low returns on corporate bonds was a tendency for the yields on both seasoned and newly issued corporate bonds to increase relative to the yields of U.S.Treasury securities. These increases in corporate^Treasury yield spreads are striking because they occurred at a time when stock prices were rising; the optimism of stock market investors did not seem to be shared by investors in the corporate bond market.