On Correlation and Default Clustering in Credit Markets

In light of the recent financial crisis, credit risk models and the way they are used have become the target of much criticism.¹ A common concern is that the basic statistical properties of market data are not the same in a crisis as they are during stable periods (Danielsson, 2002). Hence the same models that price sophisticated financial instruments during normal times are virtually useless during times of crisis.The insight that statistical models fail at times when credit markets need them the most may motivate the search for a general framework in which stress testing can be easily conducted.

In response to previous market downturns, the Basel Committee on Banking Supervision 2006 already requires banks to conduct stress tests to limit their risk exposure throughout the business cycle (Basel II). That notion was recently reinforced when the Federal Reserve urged major investment banks to stress test their balance sheets in order to make sure that they could survive under the sort of pressures that led to the near collapse of Bear Stearns (Quinn, 2008). But, according to a recent survey conducted by the Economist Intelligence Unit (2008), 85% of senior executives of financial services firms worldwide say that their stress-testing practices are not adequate and need to be improved.

This article develops a Markovian framework in the Heath, Jarrow, and Morton 1992 (hereafter HJM) paradigm for pricing credit derivatives on both single and multiple names. It is uniquely suitable for credit risk stress testing since it is both tractable and general at the same time. In particular, the models we develop have the following properties. Firstly, our framework, being Markovian, permits the riskless and risky credit spread curves to be analytically computed at any point in time, based on a limited collection of state variables. Secondly, the models fully incorporate the current riskless term structure information, as well as the full credit spread curve information for each firm. Thirdly, the term structure of interest rate and credit spread volatilities could be time homogeneous, level dependent, and initialized to closely match the observable term structures of volatilities. Fourthly, the actual structure for the volatilities of instantaneous interest rates and credit spreads can be selected from a large class and need not be restricted to the affine family of  and Kan (1996). Fifthly, the models allow for arbitrary correlations between riskless interest rates and credit spreads, as well as arbitrary correlations between credit spreads of different firms. Sixthly, the models permit shocks to the economy that cause interest rates to jump, as well as credit spreads of firms to change. Finally, we permit the default of some firms to cause jumps in the term structure of credit spreads of other surviving firms, a feature that allows defaults to cluster over time.

Duffie and Singleton (1999), Schönbucher 2000 , and others have previously shown how the HJM paradigm can be extended to include risky debt. Specifically, necessary restrictions on the dynamics of drift terms of forward rates and risky forward credit spreads have been identified that permit risky bonds to be priced in an arbitrage-free environment. Unfortunately, the resulting dynamics of all riskless forward rates and risky forward credit spreads are not in general Markov in a finite number of state variables. As a result, implementing these existing models, even via Monte Carlo simulation, is delicate and computationally intensive. The problem is compounded further if the derivative security that needs to be priced depends on the credit spreads of multiple names. In contrast, by imposing modest restrictions on forward volatilities and jumps, we are able to obtain low-dimensional Markovian representations for the term structures of riskless rates and credit spreads. The constraints will leave free the specification of the volatilities of instantaneous riskless rates and credit spreads for each name and only involve curtailing the volatilities of forward rates relative to their spot rates.

Most studies of dynamic term structure models have focused on cases where both the drift vector and the instantaneous covariance matrix are affine functions of the state variables. Duffie and Kan (1996) establish that these affine models allow for an exponential affine association between bond prices and the underlying state variables. This appealing property has made affine models the workhorse term structure models over the past decade.² However, the analytical tractability of affine term structure models comes at a cost. Specifically, the admissibility conditions entail a tradeoff where reasonable degrees of stochastic or level-dependent volatility can only be obtained at the expense of restricting correlations to perhaps unreasonable levels. Indeed, Dai and Singleton (2003) study this tradeoff and conclude that to obtain negative correlations between state variables, a feature particularly important if the state variables include the spot riskless rate and spot default intensity, the degree of heteroscedasticity in yields and/or spreads has to be dramatically curtailed.

Ahn, Dittmar, and Gallant 2002 extend affine to quadratic term structure models and show that the latter can permit flexibility in correlations while having a broader choice of yield heteroscedasticity. Using risk-free term structure data, Brandt and Chapman (2002) conclude that quadratic term structure models are superior to affine models in terms of fitting historical volatility. But, quadratic models also have some drawbacks, in that, unlike affine models, they are not that easy to estimate and they cannot readily handle regime shifts and jumps in the state variables. For the risky term structure, Berndt (2007) employs nonparametric specification tests developed in Hong and Li (2005) to evaluate one-factor reduced-form credit risk models. She shows that popular univariate affine model specifications are strongly rejected by the data, findings that also are consistent with the evidence in Pan and Singleton (2008). Given the limitations of the workhorse term structure models, it seems desirable to search for a larger class of specifications that complements the affine class of Duffie–Kan while at the same time retaining analytical tractability.

In this article, we establish families of models that permit analytical representations of riskless and risky bond prices. Perhaps somewhat surprisingly, our analytical representations of the yield curve and credit spread curves are affine functions of the state variables, even though our models are typically outside the Duffie–Kan framework. We accomplish this by curtailing volatility structures of forward rates and forward credit spreads relative to their spot rate counterparts without unnecessarily curtailing the spot rate volatilities. The number of state variables that result depends largely on the type of constraint imposed, with the most restrictive constraint leading to the smallest collection of state variables. For example, in a model where riskless rates are driven by one factor and credit spreads by a correlated second factor, the restriction requires forward rate volatilities relative to their spot rate volatilities to be decaying exponentially as a function of their maturities. A less restrictive model, however, could permit volatility humps in both the term structure of riskless volatilities, as well as the term structure of credit spread volatilities, properties that have been well documented in the literature and that cannot be accommodated in affine models of the same dimensionality (Dai and Singleton, 2003).

The advantage of our results is that we now have a large class of arbitrage-free riskless and credit spread term structure models to work with that allows us to entertain models with level-dependent volatilities that permit positive, as well as negative, correlations between interest rates and credit spreads. Equally exciting, the family of models we establish can be estimated in similar ways to the affine models. Indeed, the implementation issues are typically less difficult since the more interesting affine models often require numerical solutions to systems of Riccati differential equations (Duffie, Pan, and Singleton, 2000). Such requirements are absent when estimating our models.

When the credit spread dynamics and jumps are shut down, our models reduce to the multivariate extensions of Ritchken and Sankarasubramanian (1995), developed by Inui and Kijima (1998).³ With credit spreads, our HJM models become more interesting, especially if credit spreads are correlated with interest rates and if jumps are permitted. For such cases, we identify volatility restrictions that ensure that finite-state-variable models can be identified, which make implementation issues associated with the HJM paradigm easy to address.⁴

Using time-series data on term structures of both riskless rates and credit spreads, we illustrate how models within this family can be estimated using extended Kalman filters. We also provide a detailed road map for future empirical studies and show that, from a practical perspective, estimating models in our family is typically no more difficult than estimating models in the workhorse affine family of Duffie–Kan. We also provide two applications to illustrate the important role of correlations between interest rates and credit spreads. Firstly, we investigate options on risky debt. Secondly, we value insurance contracts that offer protection against default of a counterparty to an underlying derivatives position. For both applications, we find prices to be very sensitive to the interest rate–credit spread correlations.

After establishing a powerful theory for single-name credit derivatives, we extend our framework to allow stress tests to be conducted for credit contracts that depend on multiple names. In our models, we introduce default correlation in three ways. Firstly, we correlate the intensity processes of different firms. Secondly, we allow shocks to the economy to cause jumps not only to the riskless yield curve but also to the credit spread curves of individual firms. Thirdly, we permit the default of certain firms to affect the entire credit spread curve of other surviving firms, with the size of the impact depending on the surviving firm, the firm that defaulted, and the maturity of the forward credit spread. In this regard, our models are an extension of Jarrow and Yu (2001), and especially Yu (2007).⁵

With these additions, clustering of defaults is permissible. Further, unlike most copula models, where the risk-neutral probabilities of default are static, our models permit risk-neutral default probabilities to adjust dynamically in response to changing information.⁶ We provide two illustrative applications of the resulting models. The first extends our earlier example of pricing an insurance contract that protects against counterparty credit risk in an underlying derivatives contract to the case where the insurer itself may not be immune from default. Our numerical results highlight the necessity of including contagion effects in our models. The second application demonstrates that our family of models can be implemented for a large portfolio of names. Specifically, we price credit default swap (CDS) index tranches, where the underlying pool consists of a portfolio of one hundred firms, each credit spread curve of which is modeled by its own dynamics and appropriately initialized to its date 0 values. Interestingly, since information is used on all credits in the portfolio, we can easily explore how heterogeneity in the composition of the portfolio affects tranche values.

Our models permit events to occur that can trigger dramatic interest and/or credit shocks that may permeate through a large number of firms. They are therefore suited for stress testing alternative systemic scenarios. In light of the recent financial crisis, the proposed models offer a unique framework to study the impact of small probability events that cause large correlated losses throughout the economy.

The article proceeds as follows. In Section 1, we describe general HJM models of riskless and risky term structures. In Section 2, we present our main results that allow us to price derivatives contracts on single names using Markovian HJM models. Section 3 illustrates how model parameters can be estimated using extended Kalman filtering and highlights the importance of interest rate–credit spread correlations using two numerical examples. In Section 4, we extend the analysis to portfolios of credits where default of certain firms can impact the credit spread curves of other firms, provide applications, and perform a number of stress tests. Section 5 concludes.

1 HJM Models for Defaultable Bonds

Consider a collection of I firms. The default state of these firms is summarized by the process Y (t) = (Y₁(t), Y₂(t), . . . . , Y_I(t)), where Y_i(t) = 1 if firm i has defaulted by time t, and Y_i(t) = 0 otherwise. Set τ_i = inf{t|Y_i(t) = 1} and let X(t) be a vector of state variables that influence riskless yields and credit spreads of corporate debt. Given the state variables (X(t), Y (t)), the defaults of surviving firms over the next time increment are independent, time-inhomogeneous Poisson events. The risk-neutral default intensity for firm i at date t isη_i(t). In addition to the default intensity for each firm, we also require assumptions on recovery given default. Let ℓ_i(t) denote the risk-neutral expected fractional loss in market value if default of firm i were to occur at time t, conditional on the information available up to time t. Both η_i(t) and ℓ_i(t) could depend on the state variables (X(t), Y (t)). The instantaneous credit spread, λ_i(t), relates to the default intensity η_i(t) by λ_i(t) = η_i(t)ℓ_i(t). The modeling of the credit spreads can be delicate because it is here where interesting dependence between defaults arises, and care has to be taken to ensure that feedback effects between Y (t) and X(t) are properly taken into account.

If there are several firms, then the correlation between their credit spread innovations will presumably be determined by the nature of the operations and the capital structure of the firms. However, the credit spread of any specific firm at any point in time will not be influenced by defaults of any other firm. From a computational point of view then, the price of a zero-coupon bond of such a firm is not dependent on the path of credit spreads of other firms up to that date but is a function of the riskless yield curve up to that date, as well as of the dynamics of the firm-specific credit spreads.

2 Markovian Models for Defaultable Bonds

We now impose modest restrictions on volatilities and jumps that enable us to obtain low-dimensional Markovian representations for the term structures of riskless interest rates and credit spreads. The constraints will leave free the specification of the volatilities of instantaneous riskless rates and instantaneous credit spreads for each name. Instead, the constraints will involve the volatilities of forward rates relative to their spot rates.

With these volatility and impact restrictions, Markovian models with exponential affine riskless and risky bond prices can be obtained.

For the above model, the number of state variables drops to six. Four of these (ψ(t), ξ₁(t), ξ₂(t), and ξ₃(t)) are simply path statistics, that is, deterministic functions conditional on sample paths for r and λ_A. When interest rates and credit spreads are uncorrelated, the number of state variables drops to four as ξ₂(t) and ξ₃(t) are no longer needed. Finally, if the predictable functions, h_f (t) and h_A(t),  are constants, then all path statistics fall away, and the generalized Vasicek (1977) models for both riskless and risky bond prices obtains, with just the two state variables r(t) and λ_A(t)

The state variables for the riskless term structure in these simple models are Φ(t), where Φ(t) = {r(t), ψ(t)}. The state variables for the price of a bond issued by firm A are X_A(t), where X_A(t) = {Φ(t), ϒ_A(t)}. Here, ϒ_A(t) = {λ_A(t), ξ₁(t), ξ₂(t), ξ₃(t)} are the additional state variables for the credit spreads.

2.1 Are the volatility restrictions severe?

2.2 Relationship with Duffie–Kan affine models

The above analysis highlights the fact that our family of models is not inconsistent with Duffie and Kan but rather complements their results. Our models are built on different underlying stochastic processes, where the number of state variables is larger than the number of stochastic drivers, and the drift terms of the path statistics offset spot rate volatilities in a manner that allows bond yields to be affine in the states, even though the state variables themselves do not have to be affine processes. As a result, we have established a family of models that are very rich in structure yet are easy to implement.

3 Empirical Evidence and Sensitivity Analysis

This section describes a road map for empirical work for our proposed class of models and provides an estimation example for a specific model satisfying the Corollary to Proposition 2. We also perform a sensitivity analysis that highlights the importance of incorporating the interaction between credit spreads and interest rates into our model structure using two additional examples.

3.1 Empirical evidence

The majority of empirical studies on the term structure of riskless rates have been conducted within the affine and to a lesser degree the quadratic families.¹⁰ Much fewer studies have focused on the Markovian HJM family. Examples include the cross-sectional studies by Fan, Gupta, and Ritchken (2003, 2007) on Markovian HJM models for riskless rates. While the former study uses cap and swaption data and shows that Markovian models can explain the volatility skew in derivatives markets very well, the latter argues that a one-factor multi–state-variable Markovian HJM model can price caps and swaptions as well as four-factor models where the volatility structure is identified from a principal component analysis, as in Longstaff, Santa-Clara, and Schwartz (2001).

Studies that exploit information not only in the cross-section but also in the time series of bond yields necessarily rely on specifications under both the physical and risk-neutral measures and are more likely to empirically discriminate between models. There are a few studies of Markovian HJM models for riskless rates that are carried out along these dimensions. For example, De Jong and Santa-Clara (1999) find that their one-factor two-state HJM Markovian models outperform one-factor affine term structure models as proposed by Duffie and Kan (1996). Chiarella, Hung, and To (2009) develop an estimation framework that can be applied to the class of Markovian HJM models. They examine several multifactor models in international markets, with extremely promising results.

Empirical studies on the term structure of credit spreads have come under less scrutiny than the term structure of riskless rates. Theoretical option models, starting with Merton (1974), Longstaff and Schwartz (1995), and Jarrow, Lando, and Turnbull (1997), among others, permit credit spread curves to be increasing, decreasing, or hump shaped.¹¹ Given the increased liquidity over the past five years of the CDS market, where insurance against default is traded, firm-specific credit spread curves can now be easily identified. Further, similar to riskless rates, the term structure of credit spread volatilities can be analyzed.

In this section, we explain how the parameters of our family of models can be readily estimated using full information on both the time series and cross-section of riskless yields and credit spreads. The top panel of figure 1 shows the time series of weekly (Wednesday) riskless interest rates, for maturities ranging from one to ten years. The data are provided by Gurkaynak, Sack, and Wright (2006). Our sample covers the period from February 1, 2005, until May 15, 2009. The median five-year riskless rate during that period was 4.1%. The bottom panel shows the time series of credit spreads for two large U.S. firms, in this case, AMR and Lennar. Credit spreads are measured as at-market default swap rates, which are provided by Credit Market Analysis, Inc. (CMA). AMR Corporation is the parent company of American Airlines, the second largest airline in the world, and Lennar is one of the top two homebuilders in the United States. During the sample period, AMR and Lennar had a median S&P long-term credit rating of B and BBB, respectively, representing the low- and medium-credit-quality spectrum of the CDS market. The median five-year CDS rates were 10.8% (AMR) and eighty-eight basis points (Lennar).¹²

The plots clearly reveal significant time variation in the levels and shapes of the riskless yield and credit spread curves. Interest rates and credit spreads generally move in opposite directions, while we observe positive correlations between changes in credit spreads of the two firms. Interestingly, the correlations among changes in riskless yields of different maturities are lower than the correlations among credit spread changes of the same maturities for the two firms, indicating that the former are driven by a larger number of stochastic factors.

We use the data to illustrate the estimation procedure for a simple model in the family provided by the Corollary to Proposition 2, but the same procedure carries forward to more general models. For this case, the state variables are:

the form for M₀(Θ) and M₁(Θ) is determined by the Corollary to Proposition 2.

In our implementation, we first estimate the interest rate parameters θ_f, κ_f, σ_f, ν_f, and |${\sigma _{{\epsilon _f}}}$| from the term structure time-series data. To estimate the credit spread dynamics using firm-specific data, we follow Duffee (1999) and assume that the riskless parameters estimated in the first phase are the true parameters and that the values of r(t) and ψ(t) predicted by the Kalman filter recursion are their true values. To estimate the parameters for riskless yields, we include the one- through ten-year maturity yields into the measurement Equation (31). For credit spreads, we rely on the one-, three-, five-, seven-, and ten-year CDS rates. To better interpret our findings, we fix θ_f and θ_A at the long-run mean of one-year riskless yields and credit spreads, respectively.

Table 1 reports the parameter estimates and their Monte Carlo standard errors for riskless rates, credit spreads of AMR, and credit spreads of Lennar. All parameter estimates are significantly different from zero, except for the market prices of interest rate and credit risk. The fact that these prices are notoriously difficult to estimate (e.g., De Jong and Santa-Clara, 1999) suggests that alternative structures should be considered, perhaps with market prices of risk being affine in volatilities rather than proportional. The top panel of figure 2 compares the time series of fitted riskless yields with their observed counterparts. The fitted five-year yield closely tracks the actual five-year yield over the entire sample period. However, the fit of the one-year riskless yield is, on average, biased low with rather poor fits in the middle of the sample period. At the long horizon, the fitted ten-year yield (not shown) closely tracks its actual counterpart until the recent financial crisis. While the one-factor model for riskless rates does a reasonable job, not surprisingly, the fit is not excellent. This performance can be improved by incorporating the volatility hump in Equation (27) and/or by adding additional stochastic drivers.

The middle and bottom panels of figure 2 show the model-implied credit spreads for AMR and Lennar vis-à-vis their data counterparts. The fits appear to be fairly reasonable across maturities. Note that the five-year credit spreads for AMR fluctuate from a low of 3.35% in November 2006 to a high of over 56% in July 2008. Even before the recent financial crisis, AMR was plagued with volatile credit spreads, a fact that is well captured by the model. The extreme volatility during the financial crisis is also well picked up by the model. Disguised, perhaps by the scale, is the fact that the fit is not always excellent. Indeed, there are time periods where the percentage errors in five-year credit spreads are as large as 25%, suggesting that the one-factor model can be improved upon. In contrast to AMR, the credit spreads of Lennar were low and fairly stable prior to Spring 2007. Since then, credit spreads have surged. The plots clearly indicate that Lennar’s short-term credit spreads were more volatile than the longer-maturity spreads. For the most part, the model spreads closely track the actual values, but like AMR, there were occasions where errors were significant. In summary, the two examples suggest that improvements in the fit can possibly be obtained by adding additional structure to the volatilities of credit spreads, changing the structure for the market price of credit risk, and/or increasing the number of stochastic drivers.

3.2 The importance of interest rate–credit spread correlations

We illustrate the importance of incorporating the interaction between credit spreads and interest rates into our model structure with two examples.

3.2.1 Bond options.

We simulate the price of a three-year European call option on a five-year zero-coupon bond issued by a risky firm, say firm A. Figure 3 shows the percentage change in the value of the at-the-money option as the correlation between the riskless and the risky diffusive terms, ρ^A, moves away from zero.¹⁴ As ρ^A becomes negative, as is the usual case in U.S. corporate bond markets (e.g., Duffee, 1999), the price of the at-the-money option decreases. For our benchmark set of parameters, as ρ^A decreases from 0 to −0.9, option prices decline dramatically by more than 20%. The owner of the call option profits from low interest rates and low credit spreads at expiration, which implies that the value of the call option increases as ρ^A increases. Note that the sensitivity to ρ^A is stronger for out-of-the-money options. It diminishes as the option moves into the money.

In table 2, we compare the sensitivity of bond option prices to ρ^A to the sensitivity of the option prices to the jump intensity, η_f, and the impact factor, c_fA. We find that even when the jump intensity is increased to 0.5, while keeping c_fA at a rather high level of 2.5%, the impact of the shocks are relatively minor. We also document a lack of sensitivity of option prices to an increase in the impact factor,c_fA. Specifically, increasing c_fA to 0.1, while keeping the jump intensity constant at one hundred basis points, results in very little change in option prices. The lack of sensitivity of option prices to jump intensities and impact factors is primarily due to the fact that the initial credit spread curves are kept fixed. Therefore, increasing the frequency of jumps or their impact factors forces the risk-neutral dynamics to adjust in an offsetting direction so as to keep credit spreads curves in line with their initially observed values.

3.2.2 Insuring credit risk in derivatives contracts

The recent financial crisis has highlighted the importance of appropriately pricing credit risk in all financial transactions. Here, we focus on the pricing of counterparty credit risk in aderivatives contract. Specifically, we consider an insurance contract that provides protection on an over-the-counter derivative by ensuring that the instrument will be fully replacedupon the default of the derivative’s counterparty. For our numerical implementation, we assume that firm F enters a five-year floating-for-fixed interest rate swap with counterpartyB. Firm F is concerned that firm B will default before the maturity of the interest rate swap contract. It therefore purchases counterparty risk insurance from a third party, firmA, in the form of a five-year contract that stipulates that if firm B defaults within the next five years, the third-party protection seller A pays to firm F the market value of thefloating-for-fixed swap, as long as it is positive at the time of default. In return, F pays A a quarterly insurance premium until the end of the five-year term or until default of firm B,whichever occurs first. In this application, we ignore any potential default risk associated with the third-party supplier of the insurance (firm A), but later, we reconsider contractswhere the insurer may default. Insurance contracts of this form are quite common, and an over-the-counter market for such products, also called contingent credit default swaps (CCDS), has emerged.

In summary, our two examples stress the fact that credit-sensitive products can be highly sensitive to the diffusive correlation between interest rates and credit spreads. A unified framework that prices all kinds of credit derivatives therefore needs to be flexible enough to allow for interest rate–credit spread correlations.

4 Systemic Credit Shocks and Default Clustering

We now extend the analysis of the earlier section to more fully incorporate systemic risk features and clustering of defaults. The primary motivation for this extension is to permit meaningful stress tests to be conducted on a portfolio of credits.

So far, we incorporate systemic credit shocks by allowing a single market factor to induce an exogenous jump not only in riskless rates but also in each firm’s instantaneous credit spreads. In addition, our models build in correlated defaults by correlating the default intensities. But, conditional on the realization of the default intensities, default rates are independent across firms. This conditional independence may prevent us from generating the observed level of default clustering.¹⁵ The empirical evidence suggests that default contagion of some form is important. Collin-Dufresne, Goldstein, and Helwege (2003) and Jorion and Zhang (2007) argue that a major credit event at one firm may be associated with significant increases in spreads of other firms.¹⁶

We now extend our class of models to capture this feature by allowing the default event of certain firms to have a ripple effect on credit spreads of other firms. In particular, we distinguish between two types of firms: primary firms and secondary firms. The latter firms may be affected by the demise of a primary firm. For example, a secondary firm may carry a significant debt from a primary firm, or a significant portion of its sales may flow through a primary firm. So, when the primary firm defaults, the credit spreads of the secondary firm may jump up. Alternatively, the default of a primary firm could be good news for a secondary firm, in that it now may play a bigger role in the competitive market. In this case, an unanticipated default of its large competitor could result in a surprise downward shock to all credit spreads along the maturity spectrum of the secondary firm. Such unidirectional contagion models were first considered by Jarrow and Yu (2001) and Yu (2007).

The models developed here require additional parameters, some of which can only be estimated from historical data on a large cross-section of firms as they relate to rare events. However, our approach yields models that produce internally consistent prices that can then be used to assess the impact of systemic shocks created by highly correlated jumps in credit spread curves across firms. As such, the main application of the models presented below is geared toward revaluing credit-sensitive derivative products under different systemic risk scenarios. They provide an important tool for financial stress testing and risk management purposes.

Similarly, the primary firms’ diffusive terms are correlated according to |$E(d{z_{{A_i}}}(t)d{z'_{{A_j}}}(t)) = \sum\nolimits_{n \times n}^{{A_i}{A_j}} {{\rm{d}}t},$| where |${\left( {\sum {^{{A_i}{A_j}}} } \right)_{kl}} = \rho _{kl}^{{A_i}{A_j}}.$|

Proposition 3 below provides the linkage for credit spreads of a secondary firm to its date 0 credit spread curve through a well-defined collection of state variables.

Unlike primary firms, the prices of secondary firms’ bonds depend on the default status of primary firms, as well as the timing of the primary firms’ defaults. In addition, computingU_AiB(t) requires knowledge of firm A_i’s risk-neutral default intensity |${\eta _{{A_i}}}(t),$| which in turn requires knowledge of the state variables for the primaryfirm A_i.

To highlight the computational effort involved with our models for secondary firms, consider a simple specification with two stochastic drivers: one for interest rates (m=1) and one for each firm’s credit spreads (n=1). We show that bond prices are Markovian in at most 8 + 2m_B state variables, with no more than 3 + m_B stochastic drivers. Even in this simple framework, systemic credit shocks can be captured via the response of each firm’s credit spreads to shocks in the economy that cause jumps in both interest rates and credit spreads, as well as ripple effects on credit spreads of secondary firms caused by defaults of primary firms.

Notice that if exogenous jumps in instantaneous riskless rates and credit spreads are turned off and if default of a primary firm does not induce any contagion effects, then this Corollary isreduced to the Corollary to Proposition 2.

4.1 The importance of default contagion

In this section, we implement our model to price an array of multiname credit-sensitive products. Our primary goals are to highlight the importance of building contagion effects intoour model structure and to illustrate that these models can be used in larger applications involving multiple names. In our first example, we price the counterparty risk in a defaultinsurance contract. In our second application, we price tranches of synthetic CDS indices,effectively extending the two-firm setting of the first example to a much larger portfolio ofcredits.

4.1.1 Counterparty risk in insurance contracts.

Revisiting the CCDS example from Section 3.2.2, we now explicitly take into account the default risk of the protection seller (firm A) and compute the price of the counterparty risk in the insurance contract itself. This source of counterparty risk borne by the protection buyer, firm F, should not be confused with the counterparty risk in the underlying interest rate swap contract. Instead, it is the risk associated with default of the protection seller in the insurance contract, firm A, prior to default of the counterparty in the underlying interest rate swap contract, firm B.

To better reflect reality, we assume that the protection seller, A, is a primary firm of good credit quality and that the counterparty to the underlying swap contract, B, is a riskier secondary firm. We first compute the at-market CCDS rate when the default risk of the protection seller is ignored and then the reduction in at-market rates when the default risk of firm A is taken into account.¹⁷

Figure 4reports how the reduction in at-market CCDS rates due to the default risk of the protection seller, expressed as a fraction of the at-market CCDS rates when the default risk of the protection seller is ignored, varies as key correlation parameters change.¹⁸The correlation parameters under investigation areρ^A and ρ^B, ρ^AB, c_AB, and η_f. For the latter two, it is important to emphasize that the initial credit spread curves are kept fixed for each scenario to ensure that the focus is on correlation effects and not contaminated by changes to the level of the initial term structure of credit spreads.

In the top left panel of Figure 4, we find a significant negative relationship between the prices of insurance against default of the protection seller in the CCDS contract and diffusive correlations between interest rates and credit spreads, as measured by ρ^A and ρ^B. This is in line with the results from Section 3.2. Moreover, we find a positive impact of the parameter ρ^AB, which captures the diffusive correlation between the credit spreads of firms A and B, on the difference between at-market CCDS rates when the default risk of the protection seller is ignored and when it is priced. A positive correlation between the protection seller in the CCDS contract, A, and the counterparty to the underlying swap contract, B, implies that at times when firm A’s default risk is high (and hence the protection seller in the CCDS contract is likely to default), the default risk associated with the underlying swap contract is also high. Conversely, if the correlation is negative, then at times when firm A’s default risk is high, the default risk associated with the underlying swap contract is low. As a result, the reduction in CCDS rates due to the default risk of the protection seller increases as ρ^AB increases.

When compared to the diffusive correlation parameters, the results for the credit spread impact factor, c_AB, which measures the impact of default of the protection seller A on the credit spread curve of firm B, are striking. If c_AB is increased from zero to 0.1, the reduction in CCDS rates due to the default risk of the protection seller, expressed as a fraction of the CCDS rate when the default risk of the protection seller is ignored, increases from 2% to over 5%. An increase of the instantaneous credit spread by one thousand basis points translates into an increase of almost twenty basis points in the likelihood of default over the next week. The appropriate size of c_AB ultimately depends on the closeness of the relationship between the primary firm A and the secondary firm B. For example, c_AB is likely to be higher if both firms belong to the same sector and lower if they are close competitors. The final figure shows the impact of η_f. Although increasing the jump intensity parameter has a positive effect on the reduction in CCDS rates due to the default risk of the protection seller, the impact is of moderate size when compared to that of c_AB.

The relative importance of the credit spread impact factor c_AB increases as the volatility of credit spreads decreases. When credit spread volatility is low, the impact of the diffusive correlation between interest rates and credit spreads, ρ^A and ρ^B, and of the diffusive correlation of credit spreads across firms, ρ^AB, is significantly smaller. This highlights the importance of including credit spread impact factors in our models.

4.1.2 CDS index tranches.

To further emphasize the effectiveness of the credit spread impact factor c_AB in generating a wide range of prices for multiname credit derivatives, we extend the two-firm setting of the previous section to a much larger portfolio of credits.¹⁹ Specifically, we now investigate the sensitivity of CDS index tranche prices to c_AB. We mimic the setup of the benchmark five-year high-yield CDS index (ticker CDX.NA.HY, for details see www.markit.com) by considering a synthetic portfolio of one hundred five-year speculative-grade CDS contracts and price tranches on its loss distribution. CDX.NA.HY indices are sliced into five tranches: the equity tranche that incurs the first 0–10% of losses, junior mezzanine and senior mezzaninetranches that are responsible for the subsequent 10–15% and 15–25% of losses, and a senior and super-senior tranche that account for the next 25–35% and 35–100% of losses, respectively.

Figure 5 shows simulated tranche prices as a function of c_AB while holding everything else, including the initial credit spread curves of the underlying names, fixed. We assume that there are five primary firms and ninety-five secondary firms in the index. As the credit spread impact factor increases from 0 to 0.1, senior tranche prices increase from less than one hundred basis points to almost six hundred basis points due to an increase in the likelihood of a large number of defaults occurring within a relatively short time span. At the same time, the value of the junior mezzanine tranche decreases from 33.4% (at c_AB = 0) to 17.9% (at c_AB = 0. 1). This decrease in value is due to the fact that an increase in default contagion, while keeping the level of initial credit spread curves constant, indirectly increases the likelihood of only a few firms defaulting, the only scenario under which junior mezzanine tranche investors do not have to pay.

In summary, our results highlight the importance of including credit spread impact factors in our models that permit jumps to occur in the intensities of secondary firms when a primary firm defaults. The senior tranche investor in a CDS index has to make payments only if a relatively large number of defaults occurs within a rather short time frame. Similarly, the protection seller in the CCDS contract fails to make a promised payment only in the scenario where it defaults before the counterparty in the underlying swap contract, that is, in cases where the default of firm A is followed within a short time period by the default of firm B. To further emphasize the effectiveness of the credit spread impact factor c_AB in generating such default clustering, figure 6 shows sample paths for the timing of default events over a five-year period. Each sample path is associated with a particular value for c_AB, while everything else is kept the same. The plots indicate that for higher credit spread impact factors, defaults are more likely to occur in clusters and that they are spread out more evenly across time if no feedback effects from defaults of primary firms are allowed.

4.2 The importance of the initial credit spread curve distribution

For the correlation studies above, it has been crucial to keep initial credit spread curves fixed. But, since an appealing feature of our family of models is that they fully incorporate the current credit spread curve information, we investigate how sensitive multiname products are to the distribution, across firms, of initial credit spread curves. Our goal is to demonstrate the importance of taking into account the full credit spread curve information for each firm and their distribution across firms. The latter should be of particular concern when pricing basket credit derivatives.

Continuing with our CDS index example from Section 4.1.2, table 4 shows the simulated tranche prices of a five-year CDX.NA.HY mimicking index for different distributions of initial credit spread curves. In particular, we consider two simplified scenarios. Firstly, we investigate the case where the initial credit spread curve is flat for all firms but possibly at different levels. The credit spread level for each firm is simulated from a uniform distribution that is centered around, say, 5%. Secondly, we simulate tranche prices under the assumption that the initial credit spread curve is a linear function of time, with a fixed point of 5% at 2.5 years (half the contract term).

Our results indicate that the mezzanine and senior tranche prices increase significantly as the probability of higher initial forward credit spreads at the short end of the term structure increases. In the first example, the junior mezzanine (senior) tranche price increases by 2% (9%) when moving from a flat initial credit spread curve at 5% for all firms to a flat initial credit spread curve at a level that is uniformly distributed, across firms, between 0% and 10%.

The effect of changes in the slope of the initial credit spread curves is even more dramatic. For our second set of distributions, the results displayed in table 4 show that the junior mezzanine (senior) tranche price increases by 43% (27%) as the initial spread curve, for each firm, is tilted from being flat at 5% to being downward sloping from 8.75% (at zero-year maturity) to 1.25% (at five-year maturity). Junior mezzanine tranche prices decrease dramatically as initial spread curves are tilted from being flat to being upward sloping, whereas senior tranche prices remain fairly flat.

5 Conclusion

Our article’s contribution has been to establish a family of multifactor HJM term structure models for valuing interest- and credit-sensitive cash flows that permit an exponential affine representation of riskless and risky bond prices while offering significant flexibility in the choice of volatility structures. We accomplish this without restricting our processes to the class of affine jump diffusions (Duffie and Kan, 1996; Duffie, Pan, and Singleton, 2000). Rather, we only need to curtail the forward rate volatilities in relation to their spot rate volatility counterparts. Our class of resulting volatility structures complements the Duffie–Kan family by allowing a broad choice of specifications while still retaining an exponential affine representation.

To facilitate empirical studies, we provide an illustrative example of how estimation can proceed using an extended Kalman filter. We show that, from a practical perspective, estimating models in our family is typically no more difficult than estimating models in the workhorse affine family. For cases where the affine models require solving systems of Riccati equations, estimating models with the same number of stochastic drivers in our family may even be easier. It remains for future empirical research to establish models within our family that best describe the joint dynamics of interest rates and credit spreads.

This article also considers how failures of certain firms could affect the credit spread curves of other surviving firms. The complexity of such contagion models increases since the dimensionality of the system of equations needed to characterize the economy expands. The applications considered involve pricing multiple sources of counterparty credit risk in derivative transactions and credit derivatives written on multiple names, both of which would not be feasible with general HJM models. Interestingly, our models incorporate more information than most in pricing CDS index tranches and allow us to explore more precisely the impact of heterogeneity in the index composition.

Our family of models captures systemic credit shocks through common jumps and, more importantly, default contagion. The primary role of incorporating the latter into our models is to permit stress tests to be conducted on a large portfolio of credits. By revaluing derivative products like CDS index tranches under different assumptions on credit spread impact factors, we can control the degree of clustering of defaults. Moreover, our valuations under differing degrees of possible impact factors are all conducted in a consistent no-arbitrage framework. As the recent financial crisis reminded us, credit risks are not easily diversified, and the role of stress testing credit-sensitive portfolios cannot be understated.

A Proofs

B The Extended Kalman Filter

The standard Kalman filter recursion for the system (31)–(32) of measurement and transition equations begins with a candidate parameter vector Θ. It is used to compute the unconditional expectation and covariance matrix for X, which we denote by X(0|0) and P(0|0). Simplifying notation when possible, the steps in the recursion are:

1. Use X(t|t) and Θ to evaluate F₀(t), F₁(t), and Q(t).

2. Compute the one-period-ahead prediction and variance of X(t + Δt): |$X(t + \Delta t{\rm{|}}t) = {F_0}(t) + {F_1}(t)X(t{\rm{|}}t)$| and |$P(t + \Delta t{\rm{|}}t) = {F_1}(t) + P(t{\rm{|}}t){F_1}(t)' + Q(t).$|

3. Compute the one-period-ahead prediction and variance of y(t + Δt): |$y(t + \Delta t{\rm{|}}t) = {M_0} + {M'_1}X(t + \Delta t{\rm{|}}t)$| and |$V(t + 1{\rm{|}}t) = {M'_1}P(t + 1{\rm{|}}t){M_1} + R.$|

4. Compute the forecast error in |$y(t + \Delta t):e(t + 1) = y(t + 1) - y(t + 1|t).$|

5. Update the predictions: |$X(t + \Delta t|t + \Delta t) = X(t + \Delta t|t) + P(t + \Delta t|t){M_1}{V^{ - 1}}(t + \Delta t|t) \times e(t + \Delta t)$| and |$P(t + \Delta t|t + \Delta t) = P(t + \Delta t|t) - P(t + \Delta t|t){M_1}{V^{ - 1}}(t + \Delta t|t){M'_1} \times P(t + \Delta t|t).$|

References