On the Term Structure of Default Premia in the Swap and LIBOR Markets

Existing theories of the term structure of swap rates provide an analysis of the Treasury–swap spread based on either a liquidity convenience yield in the Treasury market, or default risk in the swap market. Although these models do not focus on the relation between corporate yields and swap rates (the LIBOR–swap spread), they imply that the term structure of corporate yields and swap rates should be identical. As documented previously (e.g., in Sun, Sundaresan, and Wang (1993)) this is counterfactual. Here, we propose a model of the default risk imbedded in the swap term structure that is able to explain the LIBOR–swap spread. Whereas corporate bonds carry default risk, we argue that swap contracts are free of default risk. Because swaps are indexed on "refreshed"-credit-quality LIBOR rates, the spread between corporate yields and swap rates should capture the market's expectations of the probability of deterioration in credit quality of a corporate bond issuer. We model this feature and use our model to estimate the likelihood of future deterioration in credit quality from the LIBOR–swap spread. The analysis is important because it shows that the term structure of swap rates does not reflect the borrowing cost of a standard LIBOR credit quality issuer. It also has implications for modeling the dynamics of the swap term structure.     

Affine Models for Credit Risk Analysis

Continuous-time affine models have been recently introducedin the theoretical financial literature on credit risk. Theyprovide a coherent modeling, rather easy to implement, but havenot yet encountered the expected success among practitionersand regulators. This is likely due to a lack of flexibilityof these models, which often implied poor fit, especially comparedto more ad hoc approaches proposed by the industry. The aimof this article is to explain that this lack of flexibilityis mainly due to the continuous-time assumption. We developa discrete-time affine analysis of credit risk, explain howdifferent types of factors can be introduced to capture separatelythe term structure of default correlation, default heterogeneity,correlation between default, and loss-given-default; we alsoexplain why the factor dynamics are less constrained in discretetime and are able to reproduce complicated cycle effects. Thesemodels are finally used to derive a credit-VaR and various decompositionsof the spreads for corporate bonds or first-to-default basket.     

Modelling bonds and credit default swaps using a structural model with contagion

This paper develops a two-dimensional structural framework for valuing credit default swaps and corporate bonds in the presence of default contagion. Modelling the values of related firms as correlated geometric Brownian motions with exponential default barriers, analytical formulae are obtained for both credit default swap spreads and corporate bond yields. The credit dependence structure is influenced by both a longer-term correlation structure as well as by the possibility of default contagion. In this way, the model is able to generate a diverse range of shapes for the term structure of credit spreads using realistic values for input parameters.     

The systematic risk of corporate bonds: default risk,term risk,and index choice

In this paper, we empirically examine the systematic risk of corporate bonds in the Euro area. Based on a unique sample of 784 bonds from 1999 to 2010, we show that the systematic risk of constructed bond portfolios and individual bonds—measured against three different market indices—depends on credit quality, term risk, and index choice. A significant increase in systematic risk for lower-rated bonds is observed following the start of the financial crisis. In multi-factor models, bond portfolios load significantly on default and term risk, which are included as additional factors. Conducting Fama and MacBeth cross-sectional tests, we find that default and term risk are priced with economically relevant premiums that range from 0.35 to 0.62 % per month. Our results are robust to the inclusion of characteristics such as rating and time to maturity.     

中小企业集合债券总体信用风险度量研究

中小企业集合债券总体信用风险既包括系统风险产生的周期性违约风险,又包括相互关联关系导致的传染性违约风险。首先通过对因素模型的改进构建模型Ⅰ,研究集合债券的周期性违约风险;在此基础上引入违约传染建立模型Ⅱ,分析违约传染对违约概率及违约相关性的影响,研究集合债券的总体信用风险。最后基于模型Ⅱ进行算例研究,得出结论：企业间的相互关联关系降低了其1次违约概率,增加了其多次违约概率即违约相关性。     

Understanding the term structure of credit default swap spreads

The shape of the term structure of credit default swap (CDS) spreads displays large variations over time and across firms. Consistent with the predictions of structural models of credit risk, we find that the slope of CDS spread term structure increases with firm leverage and volatility, but decreases with the level and the slope of the Treasury yield curve. However, these variables together have rather limited explanatory power for CDS slope and there is a significant common component in the regression residuals. In addition, we find that CDS slope predicts future changes in the CDS spreads, even after controlling for the contemporaneous variables that determine changes in the CDS spreads according to the structural models. Our results suggest that while structural models are qualitatively useful for understanding the shape of credit term structure, there are missing factors that importantly affect the term structure of CDS spreads.     

Modeling CDS spreads: A comparison of some hybrid approaches

According to the credit risk model proposed by Cathcart and El-Jahel (2006), default can occur either expectedly, when a certain signaling variable breaches a lower barrier, or unexpectedly, as the first jump of a Poisson process, whose intensity depends on the signaling variable itself and on the interest rate. In the present paper we test the performances of such a model and of other three models generalized by it in fitting the term structure of credit default swap (CDS) spreads. In order to do so, we derive a semi-analytical formula for pricing CDSs and we use it to fit the observed term structures of 65 different CDSs. The analysis reveals that all the model parameters yield a relevant contribution to credit spreads. Moreover, if the dependence of the default intensity on both the signaling variable and the interest rate is removed, the pricing of CDSs becomes very simple, from both the analytical and the computational standpoint, while the goodness-of-fit is reduced by only a few percentage points. Therefore, when using the credit risk model proposed by Cathcart and El-Jahel (2006), assuming a constant default intensity provides an interesting and efficient compromise between parsimony and goodness-of-fit. Furthermore, by fitting the term structure of CDS spreads on a period of about twelve years, we find that the parameters of the model with constant default are rather stable over time, and the goodness-of-fit is maintained high.     

Financial statement comparability and credit risk

Prior research shows that firms’ financial statement comparability improves the accuracy of market participants’ valuation judgments and thus may reduce firms’ costs of capital. Distinct from prior research focusing on the equity market, we develop measures of comparability relevant to debt market participants based on the within-industry variability of Moody’s adjustments to reported accounting numbers for the purposes of credit rating. We examine two sets of adjustments: (1) to the interest coverage ratio and (2) to non-recurring income items. We validate these comparability measures by providing evidence that greater comparability is associated with lower frequency and magnitude of split ratings by credit rating agencies. We predict and find that greater comparability is associated with (1) lower estimated bid-ask spreads for traded bonds, (2) lower credit spreads for both bonds and five-year credit default swaps, and (3) a steeper one- to five-year credit default swap term structure. Our results are consistent with financial statement comparability reducing debt market participants’ uncertainty about and pricing of firms’ credit risk.     

Pricing defaultable bonds: a middle-way approach between structural and reduced-form models

In this paper we present a valuation model that combines features of both the structural and reduced-form approaches for modelling default risk. We maintain the cause and effect or ‘structural’ definition of default and assume that default is triggered when a state variable reaches a default boundary. However, in our model, the state variable is not interpreted as the assets of the firm, but as a latent variable signalling the credit quality of the firm. Default in our model can also occur according to a doubly stochastic hazard rate. The hazard rate is a linear function of the state variable and the interest rate. We use the Cox et al. (A theory of the term structure of interest rates. Econometrica, 1985, 53(2), 385–407) term structure model to preclude the possibility of negative probabilities of default. We also horse race the proposed valuation model against structural and reduced-form default risky bond pricing models and find that term structures of credit spreads generated using the middle-way approach are more in line with empirical observations.     

Empirical credit cycles and capital buffer formation

We model 1927–1997 US business failure rates using an unobserved components time series model. Clear evidence is found of cyclical behavior in default rates. We also detect significant longer term movements in default rates and default correlations. In a multi-year backtest experiment we show that accommodation of default rate dynamics has important consequences for credit risk capitalization requirements. Static or myopic variants of credit portfolio models miss significant periods of credit risk accumulation. Empirically congruent dynamic models by contrast provide more timely warning signals of credit risk build-up. In this way they may mitigate some of the pro-cyclicality concerns.     

Capital Structure and Debt Priority

We study a defaultable firm's debt priority structure in a simple structural model where the firm issues senior and junior bonds and is subject to both liquidity and solvency risks. Assuming that the absolute priority rule prevails and that liquidation is immediate upon default, we determine the firm's interior optimal priority structure along with its optimal capital structure. We also obtain closed‐form solutions for the market values of the firm's debt and equity. We find that the magnitude of the spread differential between junior and senior bond yields is positively, but not linearly related to the total debt level and the riskiness of assets. Finally, we provide an in‐depth analysis of probabilities of default and the term structure of credit spreads.     

A reduced form model of default spreads with Markov-switching macroeconomic factors

We examine the ability of observed macroeconomic factors and the possibility of changes in regime to explain the proportion of yield spreads caused by the risk of default in the context of a reduced form model. For this purpose, we extend the Markov-switching risk-free term structure model of Bansal and Zhou (2002) to the corporate bond setting and develop recursive formulas for default probabilities, risk-free and risky zero-coupon bond yields as well as credit default swap premia. The model is calibrated with consumption, inflation, risk-free yields and default data for Aa, A and Baa bonds from the 1987 to 2008 period. We find that our macroeconomic factors are linked with two out of three sharp increases in the spreads during this sample period, indicating that the spread variations can be related to macroeconomic undiversifiable risk.     

Investment‐Based Corporate Bond Pricing

A standard assumption of structural models of default is that firms' assets evolve exogenously. In this paper, we examine the importance of accounting for investment options in models of credit risk. In the presence of financing and investment frictions, firm‐level variables that proxy for asset composition are significant determinants of credit spreads beyond leverage and asset volatility, because they capture the systematic risk of firms' assets. Cross‐sectional studies of credit spreads that fail to control for the interdependence of leverage and investment decisions are unlikely to be very informative. Such frictions also give rise to a realistic term structure of credit spreads in a production economy.     

From equity to default correlation with taxes

For fixed income investment, the preponderant risk is the clustering of defaults in the portfolio. Accurate prediction of such clustering depends on the knowledge of default correlation. We develop models with exogenous debt and endogenous debt to predict default correlations from equity correlations based on a self-consistent structural framework. We also examine how taxes affect the prediction of default correlations based on the two models. The empirical analysis shows that the corporate taxes tend to decrease default correlations, while personal taxes could increase or decrease default correlations. Our default correlation model with exogenous debt does a better job of predicting default correlations for high quality bonds, while the one with endogenous debt predicts more accurately for lower rated bonds. Our studies not only theoretically improve the modeling of default correlation in the structural setting but also shed new light on various aspects of default correlations and thereby help financial practitioners price credit derivatives more accurately and formulate more effective strategies to manage default risk of credit portfolios.     

Do Credit Spreads Reflect Stationary Leverage Ratios?

Most structural models of default preclude the firm from altering its capital structure. In practice, firms adjust outstanding debt levels in response to changes in firm value, thus generating mean-reverting leverage ratios. We propose a structural model of default with stochastic interest rates that captures this mean reversion. Our model generates credit spreads that are larger for low-leverage firms, and less sensitive to changes in firm value, both of which are more consistent with empirical findings than predictions of extant models. Further, the term structure of credit spreads can be upward sloping for speculative-grade debt, consistent with recent empirical findings.     

An Estimation Model for the Term Structure of Yield Spread

An estimation model for term structure of yield spread has become an extremely important subject to evaluate securities with default risk. By Duffie and Singleton model, yield spread was explained by two factors, namely collection rate and default probability. An estimation of the collection rate is given from historical earnings data, but estimation of default probability is known to be a remaining problem.There are some approaches to express default probability. One of them is to describe it through hazard process, and the other is to represent it by risk neutral transition probability matrix of credit-rating class. Some models that use Gaussian type hazard process or Vasicek type hazard process have already constructed.An advantage of evaluation using a rating transition probability matrix is that it is easy to obtain an image of movement of the credit-rating class. We do not need to show the calculation basis of the threshold or an assumption for distribution of prospective yield spread. But the model that uses the risk neutral transition probability matrix has not established yet, because of the computational difficulty required to estimate large number of the parameters.At first, for the purposes of this article, we will estimate the term structure of credit spreads results from the possibility of future defaults. It is assumed that credit risk is specified as a discrete-state Markov chain. And we construct a model which can be used to estimate the baseline transition matrix of the credit-rating class, risk-adjusting factors, industrial drift factors, corporate drift factors and recovery ratio, from yield spreads for individual bond. This enables us to compute the implied term structure from market data. We are capable of computing the implied term structure from market date by this process. Next, we will provide a valuation model for the term structure of yield spread.     

The search for relative value in bonds

Comparing asset swap spreads across bonds is a widely used tool for measuring relative value. This approach leads portfolio managers to increase their risk exposure in ways that are not transparent. Credit default swaps are utilized to demonstrate that viewing wide asset swaps as an indicator of relative value is a mirage. The paper documents the empirical regularities in the term structure of credit spreads and spread volatilities that make this result possible. In addition, we present empirical evidence of the imprint made on corporate bond returns by the widespread use of the asset swaps data.     

Measuring Credit Risk of Individual Corporate Bonds in US Energy Sector

In this paper, using the measures of the credit risk price spread (CRiPS) and the standardized credit risk price spread (S-CRiPS) proposed in Kariya’s (A CB (corporate bond) pricing model for deriving default probabilities and recovery rates. Eaton, IMS Collection Series: Festschrift for Professor Morris L., 2013) corporate bond model, we make a comprehensive empirical credit risk analysis on individual corporate bonds (CBs) in the US energy sector, where cross-sectional CB and government bond price data is used with bond attributes. Applying the principal component analysis method to the S-CRiPSs, we also categorize individual CBs into three different groups and study on their characteristics of S-CRiPS fluctuations of each group in association with bond attributes. Secondly, using the market credit rating scheme proposed by Kariya et al. (2014), we make credit-homogeneous groups of CBs and show that our rating scheme is empirically very timely and useful. Thirdly, we derive term structures of default probabilities for each homogeneous group, which reflect the investors’ views and perspectives on the future default probabilities or likelihoods implicitly implied by the CB prices for each credit-homogeneous group. Throughout this paper it is observed that our credit risk models and the associated measures for individual CBs work effectively and can timely provide the market credit information evaluated by investors.     

Tempered stable structural model in pricing credit spread and credit default swap

In this paper, we explore the features of a structural credit risk model wherein the firm value is driven by normal tempered stable (NTS) process belonging to the larger class of Lévy processes. For the purpose of comparability, the calibration to the term structure of a corporate bond credit spread is conducted under both NTS structural model and Merton structural model. We find that NTS structural model provides better fit for all credit ratings than Merton structural model. However, it is noticed that probabilities of default derived from the calibration of the term structure of a bond credit spread might be overestimated since the bond credit spread could contain non-default components such as illiquidity risk or asymmetric tax treatment. Hence, considering CDS spread as a reflection of the pure credit risk for the reference entity, we calibrate it in order to obtain more reasonable probability of default and obtain valid results in calibration of the market CDS spread with NTS structural model.     

Credit Risk and the Term Structure of Lease Rates: A Reduced Form Approach

Previous research either assumes default free leases or leases subject to default risk using a structural approach. However, structural credit risk models suffer from a common criticism that the firm’s asset value process is unobservable. We develop a reduced form credit risk model for leases that avoids making assumptions regarding unobservable asset valuation processes. Furthermore, we assume a correlated market and credit risk that provides us with a simple analytic formula for valuing defaultable lease contracts. Numerical analysis reveals that tenant credit risk can have a substantial impact on the term structure of leases. Finally, we use the model to demonstrate the implied lease term structure for a set of retail and financial firms in the Fall of 2000.