The determinants of CDS spreads: evidence from the model space

Abstract

We apply Bayesian model averaging and a frequentistic model space analysis to assess the pricing determinants of credit default swaps (CDSs). Our study focuses on the complete model space of plausible models and thus supports ultimate robustness. Using a large dataset of CDS contracts we find that CDS price dynamics can be mainly explained by factors describing firms’ sensitivity to extreme market movements. More precisely, our results suggest that dynamic copula based measures of tail dependence incorporate most essential pricing information, making other potential determinants such as Merton-type factors or linear variables measuring the systematic market evolution negligible.

Appendices

Appendix

A Time-varying copula models

This section describes the time-varying copula models employed in our study as well as the estimation of tail dependence measures and goodness-of-fit tests.

1.1 A.1 Modeling extreme comovements

To capture the extreme comovement between financial time series, we rely on Copula-GARCH models (see Jondeau and Rockinger 2006) in our study. These models allow for some time-varying parameters in an autoregressive manner, conditional on the set of past information. To consider conditional distributions of some random variable $X_t$ given an information set $\mathscr {F}$, an extension of Sklar’s theorem (see Sklar 1959) is necessary. This extension to the conditional case is due to Patton (2006), who defines the conditional copula:

$$\begin{aligned}&\mathbf {F}_t (x \mid \mathscr {F}_{t-1} ) = \mathbf {C}_t(F_{1,t} (x_1 \mid \mathscr {F}_{t-1}), F_{2,t} \\&\quad (x_2 \mid \mathscr {F}_{t-1}), \ldots , F_{n,t} (x_n \mid \mathscr {F}_{t-1}) \mid \mathscr {F}_{t-1}), \ \ \ \ \forall x \in \mathbb R^n. \end{aligned}$$

Here, $X_i \mid \mathscr {F}_{t-1} \sim F_{i,t}$ and $\mathbf {C}_t$ denotes the conditional copula of $\mathbf {X}_t$ given $\mathscr {F}_{t-1}$.

1.1.1 A.1.1 Univariate modeling of log differences

As noted by Sklar (1959), the use of copulas allows to define multivariate models where the marginal distributions are not of the same type as the copula model. Thus, we can combine non-parametric estimation for the marginal distributions with parametric estimation of the copula (see Chen and Fan 2006; Chen et al. 2006). As our study focuses on the dependence structures between CDS spreads, we are not interested in the estimation of the marginals and only estimate the copula. Estimation methods for the copula assume identical independent distributed data (see Genest et al. 1995). However, stylized facts of financial time series state that model residuals are often skewed and fat tailed in addition to a leverage effect (see Cont 2001; Joe 2015). Focusing on CDS spread time series, Cont and Kan (2011) provide evidence that CDS spread changes are stationary and exhibit positive autocorrelation. Moreover, the authors find that the returns are described by conditional heteroscedasticity, two-sided heavy tails, serial dependence in extreme values, and sizable co-movements that are not necessarily linked to credit events. Hence, as returns of financial data in general and specifically CDS data are usually not i.i.d. and we can confirm the stylized facts for our dataset, we have to filter the data first.

Thus, we first apply our data to an AR(m)-GARCH(p,q) model, before we standardize the i.i.d. residuals from the filtration to uniform. We use a probability integral transform of the following form. Due to the filtration, $\epsilon _t$, $t=1, \ldots , T$ is a time series of i.i.d. variables. Under the assumption $\epsilon _t \sim F_i$, $t = 1, \ldots , T$, $u_{i,t} = F_i(\epsilon _t)$ is the probability integral transform of $\epsilon _t$ with $u_{i,t} \sim U[0,1]$, $t =1, \ldots , T$. Specifically, we employ the empirical cumulative distribution function (CDF) for the transformation:

$$\begin{aligned} \hat{F_i}(\epsilon ) = \frac{1}{T+1} \sum _{t=1}^T \mathbf {1}_{\epsilon _{i,t} \le \epsilon } \end{aligned}$$

where $\mathbf {1}$ denotes the indicator function. Finally, we can estimate the copula parameters using the uniform residuals with maximum likelihood estimation.

We do not summarize the estimation results for the univariate AR(m)-GARCH(p,q) model and copula model estimations to preserve space. The results are available from the authors upon request. We choose the AR lag m for each time series and the GARCH model according to the AIC value.

Figure 6 presents representative plots of autocorrelation functions of the CDS market changes prior to applying the AR-GARCH model and after filtering. Note that the residuals do not exhibit significant autocorrelations.

Fig. 6

Autocorrelation functions before and after applying the AR-GARCH filter. The figure shows representative plots of autocorrelation functions of the CDS market changes prior to applying the AR-GARCH model and after filtering

1.1.2 A.1.2 Time-varying copulas

To allow for possible time-varying dependency structures (see, e.g. Andersen et al. 2006; Bauwens et al. 2006) between assets we make use of time-varying specifications of the copula models in our study. We follow Patton (2006) and Creal et al. (2013) to specify the time dynamics of the copulas.

Time-varying copula models can be characterized by their time-varying parameters. As noted by Patton (2006) it is “difficult to know what might (or should) influence” the copula parameter to change unless the parameter has some kind of interpretation. Hence, we limit our study to copulas characterized by such a parameter. For example, for the time-varying t-copula, the correlation among the risk factors and the degrees of freedom are time-varying, while for the Symmetrized Joe-Clayton Copula the upper and lower tail dependence parameter is time-varying and for the Clayton copula Kendall’s tau is the time-varying parameter.

Patton (2006) defines the updating equation for the time-varying parameter of a bivariate model by

$$\begin{aligned} f_{t+1} = \omega - \frac{1}{m} A_1 \cdot \sum _{i=1}^m \mid u_{1,t-i+1} u_{2,t-i+1} \mid + B_1 f_t, \end{aligned}$$

where m is a positive integer that characterizes the smoothness of the function and $u_{1,t}$ and $u_{2,t}$ are the probability integral transforms of the univariate marginals. Let $\omega < 0$, $A_1 > 0$, and $1> B_1 > 0$. If the most recent values of $u_{1,t}$ and $u_{2,t}$ are close together, hinting at a stronger dependence, $f_{t+1}$ is likely to increase, while for recent values that are far apart $f_{t+1}$ more likely decreases. We employ the approach by Patton for the symmetric t and the SJC copula. Time dynamics are captured for the symmetric t-copula by a time-varying correlation parameter $\rho _t$ that follows the transformation $\rho _t = \frac{1- e^{-f_t}}{1+e^{-f_t}} = \text {tanh}\left( \frac{f_t}{2}\right)$. The modified logistic transformation is used to keep $\rho _t$ in $(-1,1)$ at all times.

Additionally, we allow for trends in the degrees of freedom (see also Christoffersen et al. 2014). We assume that the degree of freedom at time t is given by the exponential quadratic spline

$$\begin{aligned} \eta _{C,t} = \underline{\eta _C} + \delta _{C,0} \exp \Big ( \delta _{C,1} t + \sum _{j=1}^k \delta _{C,j+1} \max ( t-t_{j-1}, 0)^2 \Big ). \end{aligned}$$

$\underline{\eta _C}$ marks the lower bound for the degrees of freedom and $\delta _{C,0}, \ldots , \delta _{C,k+1}$ are scalar parameters that have to be estimated. We split the sample in k segments of equal length to obtain $\{ t_0 = 0, t_1, t_2, ldots , t_k = T \}$. For our estimations, we set $k=3$, which allows us to capture times of positive and times of negative trends.

By introducing time dependence in correlations and degrees of freedom, Christoffersen et al. (2014) allow for time variation in tail dependence that is distinct from time variation in correlations as tail dependence for the t-copula is characterized by both, correlation and degrees of freedom.

For the SJC copula we use $\Lambda (x) \equiv \frac{1}{1 + e^{-x}}$ as logistic transformation, which is used to keep the parameters $\tau ^U$ and $\tau ^L$ in (0, 1) at all times (see Patton 2006).

For the Clayton and the Gumbel copula we rely on the generalized autoregressive score (GAS) model, GAS (1, 1), introduced by Creal et al. (2013), which is similar to the model of Patton (2006) for $m=1$. Creal et al. (2013) provide a dynamic version of following their GAS specification. The density with regard to Creal et al. (2013) reads

$$\begin{aligned} c_C(u_1,u_2, \theta ) = 1 - 2 + u_1^{-\theta } + u_2^{-\theta }. \end{aligned}$$

For the Clayton copula the time-varying factor following the GAS equation reads

$$\begin{aligned} f_t= & {} \frac{1}{\theta ^2_t} \text {ln} \ c_t(\theta _t) - \Big ( \frac{1}{1 - \theta _t} - \text {ln} (u_{1,t}) + \frac{1}{1 - 2 \theta _t} - \text {ln} (u_{2,t}) \Big ) \\&+\, \Big ( \frac{1}{\theta _t} + 2\Big ) c_t (\theta _t)^{-1} \Big ( u_{1,t}^{-\theta _t} \text {ln}(u_{1,t}) + u_{2,t}^{-\theta _t} \text {ln}(u_{2,t}) \Big ). \end{aligned}$$

For the Gumbel copula the updating equation for the dynamic parameter is given by

$$\begin{aligned} f_{t+1} = \omega + \beta f_t + \alpha \cdot \frac{\frac{\partial }{\partial \theta } \log c_G(u_1,u_2 \mid \theta )}{\sqrt{\text{ E }_{t-1} \left[ (\frac{\partial }{\partial \theta } \log c_G(u_1,u_2 \mid \theta ) )^2\right] }} \end{aligned}$$

and the density reads

$$\begin{aligned}&c_G(u_1,u_2 \mid \theta ) = C_G(u_1,u_2, \theta ) \frac{((-\log u_1)(-\log u_2))^{\theta -1}}{((-\log u_1)^{\theta } + (-\log u_2)^{\theta } )^{2-1/\theta }} \Big (((-\log u_1)^{\theta }\\&\quad +\, (-\log u_2)^{\theta }) + \theta - 1 \Big ). \end{aligned}$$

The copula parameter of the Gumbel copula is required to be greater than one. Thus, we use the function $\theta _t = 1 + \exp {f_t}$ to ensure this.

1.2 A.2 Measures of tail dependence

A convenient way to measure for the upper tail dependence at time t is via the probability limit:

$$\begin{aligned} \tau ^U_{i,j,t} = \lim _{\zeta \rightarrow 1} \text{ P }[u_{i,t} \ge \zeta \mid u_{j,t} \ge \zeta ] = \lim _{\zeta \rightarrow 1} \frac{1-2 \zeta + C_t (\zeta , \zeta )}{1 - \zeta }. \end{aligned}$$

Similarly, for the lower tail dependence

$$\begin{aligned} \tau ^L_{i,j,t} = \lim _{\zeta \rightarrow 0} \text{ P }[u_{i,t} \le \zeta \mid u_{j,t} \le \zeta ] = \lim _{\zeta \rightarrow 0} \frac{C_t (\zeta , \zeta )}{\zeta }. \end{aligned}$$

As far as our copula models are concerned, the Gumbel and the Clayton copula posses upper and lower tail dependence that is not equal to zero, while dependence in the opposite tail is zero for both cases. For the rotated versions of both copula models, dependence in the tails is opposite. For both (or all four, together with the rotated versions) copula models, the tail dependence depends on the dynamic copula parameter. The SJC copula is characterized by positive tail dependence in the lower and the upper tail, and these measures may be different. The tail dependence is given by the (time-varying) copula parameters. Lastly, the (symmetric) t-copula has the property of non-negative tail dependence for both tails. However, the lower and the upper tail dependence is identical for the symmetric t-copula. For all copulas included in our study, closed form equations for the tail dependence are established in the literature.

1.3 A.3 Tests of goodness of fit

This section briefly presents several goodness of fit tests that we employ to estimate the fit of each copula model and identify one model as the model with the best fit. Relying on the works of Genest et al. (2009), Kole et al. (2006) and Berg (2009), we first use the Kolmogorov–Smirnov test and the Cramer-von Mises test to test the goodness of fit of our copula models. Both of these tests rely on the empirical copula serving as a nonparametric estimate of the true conditional copula. However, as our estimated copula is time-varying, we cannot rely on the two tests directly. Instead, we follow Diebold et al. (1999), Genest et al. (2009) and Rémillard (2010) and employ the Rosenblatt transformation (Rosenblatt 1952) before using the Kolmogorov–Smirnov and the Cramer–von Misses tests.

The Rosenblatt transformation of the original data produces a vector of identical independent uniform distributed variables on the interval [0, 1]. In the bivariate case, the transform is given by

$$\begin{aligned} v_{1,t}= & {} u_{1,} \ \ \forall t, \\ v_{2,t}= & {} \mathbf {C}_{2 \mid 1, t} (u_{2,t} \mid u_{1,t} , \theta ). \end{aligned}$$

Note that the ordering of the variables affects the transformation. Employing the Rosenblatt transformation, the Kolmogorov–Smirnov and the Cramer–von Misses tests are as follows:

$$\begin{aligned} \hat{\mathbf {C}}_T (\mathbf {v})= & {} \frac{1}{T} \sum _{t=1}^T \prod ^n_{i=1} \mathbf {1}_{V_{i,t}\le v_i}, \\ \mathbf {C} (\mathbf {V}_t,\hat{\theta _T})= & {} V_{1,t} \cdot V_{2,t},\\ \text {KS}= & {} \max _t \mid \mathbf {C} (\mathbf {V}_t, \hat{\theta }_T ) - \hat{\mathbf {C}}_T (\mathbf {V}_t) \mid , \\ \text {CvM}= & {} \sum _{t=1}^T \Big (\mathbf {C} (\mathbf {V}_t, \hat{\theta }_T ) - \hat{\mathbf {C}}_T (\mathbf {V}_t) \Big )^2. \end{aligned}$$

Note that these gof-tests are designed to test the fit of the copula model over the entire domain. For example, the Kolmogorov–Smirnov distances are sensitive to deviations in the center of the distribution. However, as our study focuses on tail dependence, we additionally estimate a non-parametric dynamic estimator of upper tail dependence. More precisely, we estimate the log estimator proposed by Frahm et al. (2005) using rolling window estimation. Note that we again perform the Rosenblatt transformation before estimating the log estimator of tail dependence. We then calculate the Integrated Anderson–Darling distances between the non-parametric tail dependence coefficients and the tail dependence coefficients we infer from the fitted copula models via

$$\begin{aligned} D_{j,AD} = \sum _{t_i=1}^T \sum _{t_k=1}^T \frac{\Big ( \hat{\lambda }^U ( \frac{t_i}{T},\frac{t_k}{T} ) - \lambda _j^U (\frac{t_i}{T},\frac{t_k}{T}, \hat{\theta }) \Big )^2}{\lambda _j^U (\frac{t_i}{T},\frac{t_k}{T},\hat{\theta }) \cdot \left( 1- \lambda _j^U (\frac{t_i}{T},\frac{t_k}{T},\hat{\theta }) \right) }, \end{aligned}$$

where $\lambda ^U$ denotes the upper tail dependence coefficient, $\hat{\lambda }^U$ denotes the empirical tail dependence coefficient, $\hat{\theta }$ the estimated copula parameter(s), and the copula models are evaluated at every point on the lattice

$$\begin{aligned} L = \Big [ \Big ( \frac{t_i}{T}, \frac{t_k}{T} \Big ), t_i = 1, ldots , T, t_k = 1, ldots , T \Big ]. \end{aligned}$$

Lastly, we estimate the fit of each copula model based on the AD-distance to Pickands dependency function evaluated at 0.5.

B Correlation analysis

In this section, we examine simple correlations of the regressors. Table 14 presents the largest correlations of regressors with CDS spreads, while Table 15 presents pairwise correlations between all variables included in the BMA regression setup. We observe the highest correlations between the first principal component, which is a weighted average of the individual CDS spreads measuring commonality, and the spreads, followed by the market and the spread. The higher correlation between Comm and CDS spreads can be attributed to the Commonality extracted by the pca. Thus, CDS spreads that experience less commonality with the other are attributed a lower weight. Also, correlations with overall market movements (Russell) and financial stress are high. Moreover, we observe high correlations between the CDS spreads and the tail dependence measured with the t-copula (0.52). We attribute this high correlation at least in part to the correlation being part of the tail dependence estimate.

Further correlations can be seen from Table 15. Most strikingly, high correlations between the tail dependence measures are estimated with different copula models. Note that the t-copula measure is least correlated with other copulas, but most correlated with market and commonality. Moreover, the other copula measures are not highly correlated with any of the other variables, which can be interpreted as a sign of uniqueness of the copula measures. The commonality factor is highly correlated with the financial distress indicator. Also, equity returns do not show any large correlations with other variables. Notable correlations are only formed with equity volatility and, not surprisingly, the change in firm value.

We run additional correlation analysis for our liquidity subsample period. We find that the liquidity measure is very unique as we do not observe any correlations above .3 (in absolute values). Consequently, we do not impose any exclusion restrictions on the liquidity measure.

Table 14 Largest correlations with CDS spreads

Table 15 Correlation matrix of the regressors and exclusion restrictions (highlighted)

Fig. 7

LRM of all regressors in the elastic net estimation. The figure shows the LRM of all regressors in the regularization and variable selection via the elastic net method

Table 16 LRMs of regressors in the elastic net estimation

C Variable selection via the elastic net

The elastic net regression (Zou and Hastie 2005; Hastie et al. 2008) is a “shrinkage” or regularization method that minimizes the sum of squared residuals subject to a penalty term that penalizes the aggregate size of the coefficients in the model. The penalty term is defined as a linear combination of a L2 (quadratic) and L1 (manhattan) norms. Formally, elastic net regression minimizes

$$\begin{aligned} \hat{\beta }_{ENet}=(1+\lambda _{2}) \text {arg min}_{\beta }||y-X\beta ||+\lambda [(1-\alpha )||\beta ||_{2}^{2} + \alpha ||\beta ||_{1} ], \end{aligned}$$

(12)

where $\alpha$ is a hyperparameter with values between 0 and 1 and controls how much L2 or L1 penalization is used. The special case $\alpha =0$ correpsonds to the ridge regression (Hoerl and Kennard 1970) and $\alpha =1$ to the lasso regression (Tibshirani 1996). The second hyper-parameter $\lambda$ controls the “aggressiveness” of penalization for a given $\alpha$. The usual approach to optimizing $\lambda$ is through cross-validation, i.e. by minimizing the cross-validated mean squared error of prediction. Since the optimal $\lambda$ depends upon $\alpha$, usually a combined grid-search over different combinations of $\lambda$ and $\alpha$ is performed. The elastic net fits a model in which all potential predictors are contained which is possible due to the additional constraints or regularization on the coefficient estimates compared to OLS. The introduction of the regularization/penalty term induces an bias to the estimated coefficients, however, the mean squared prediction error is decreased compared to OLS estimation. In Figure 7 and Table 16 the results from the application of the elastic net method to our dataset are shown. The optimal $\lambda$ in our application is found to be 0.012 and the optimal $\alpha$ is 0.72.