Modeling Corporate CDS Spreads Using Markov Switching Regressions

3.1 Credit Risk Measures

A single-name credit default swap (CDS) is a contract in which two parties are involved. On the one hand, the protection buyer pays periodic fixed payments to the counterparty until the CDS expires due to a credit event or maturity. On the other hand, the protection seller, who receives the fixed payments and, as the credit event occurs, must buy the bonds owned by the counterparty at their face value. The fixed spread represents the compensation for the insurance in case of a credit event. Conversely, multi-name CDSs, which consider multiple entities, are contracts that include CDS indexes, basket products, and CDS tranches. In this case, the credit event of a single reference entity does not terminate the contract. So, a CDS is an insurance contract against the risk of default of the underlying reference entity. The total value of the bonds that can be sold is known as the notional principal, and the settlement can be physical if the protection buyer delivers the bonds to the protection seller, or cash settlement in which just the difference between the face value and the value of the bonds at the time of default is paid. A key aspect is the aforementioned credit event, usually defined as either bankruptcy or restructuring of debt.

Its origin dates back to the mid-90s. In 1994, Morgan created CDSs to reduce credit risk exposure extending its loan capability. The CDS was written with the European Bank of Reconstruction and Development (EBRD), and the deal aimed to allow the bank to offer a line of credit of USD 5 billion to Exxon, maintaining balance sheet flexibility (Vogel, Bannier, and Heidorn 2013). Over the years, CDSs have become more and more complex instruments and saw a steady increase in volumes and notional outstanding, reaching the peak of USD 60 trillion before the subprime crisis.

In 2008, the insurance giant AIG was one of the main CDS sellers. The use of those derivatives at that time was to concentrate risk rather than disperse it, pushing AIG on the brink of collapse as the subprime bubble burst (see for example Krishnamurthy 2010).

Given their primary role in the 2007–2009 crisis and the fact that they are traded only on over-the-counter (OTC) markets, causing a lack of transparency, has forced the regulator to standardize the market, introducing central counterparties (CCPs) to erase the counterparty risk (Aldasoro 2018). These changes were introduced for European markets in 2009 with the small-bang protocol, including two main features: the standardized coupon rates and the quoting conventions.

This paper focuses on the European iTraxx corporate index (labeled ΔiTraxx in Equations and Tables). Each considered index has a 5-year maturity because deemed as the most liquid in the market. The Markit iTraxx is a family of Asian, European, and Emerging Market credit default swap indexes. The iTraxx group was formed by the merger in 2004 of JP Morgan and Morgan Stanley Trac-X indexes and the ones created by Deutsche Bank, ABN Amro, and IBoxx (i.e. iBoxx CDS indexes). In November 2007, Markit acquired both CDX and iTraxx, and by 2011 Markit owned and managed most of the CDS indexes. They aim to transfer risk more efficiently rather than using multiple single-name CDSs. The Markit iTraxx Europe Main index^[1] trades with maturities of 3, 5, 7, and 10 years. It is an equally-weighted index composed of the 125 most liquid listed companies, with a weight of 0.8 % for each constituent firm’s CDS (icmagroup.org). Every six months the index is updated, rolling out the firms that either defaulted/merged or did not respect the liquidity parameter, and including new ones.^[2]

3.2 Explanatory Variables

We review the explanatory variables to be used to predict the CDS spread. Several multi-factor analyses from a firm-specific (microeconomic) and market-specific (macroeconomic) perspective have been used to study the determinants of credit default swap (CDS) spreads. In this paper, we take a macro perspective since we work with aggregate indices. Table 1 lists a short selection of studies following a macro approach and the market-specific factors considered, skipping firm-specific variables.

In our analysis, the first three variables are derived from the theory related to structural models: the firm’s debt-to-equity ratio, the firm’s operation volatility, and the risk-free interest rate (Merton 1974). The proxy for the debt-to-equity component is represented by returns of three different equally-weighted portfolios composed of the stocks of the firms included in the iTraxx indexes (iTraxx Main index, iTraxx Financial Seniors index, and iTraxx Non-financials index). There are some cases in which stock price series are not available. To solve that problem, we drop them from the dataset, decreasing the number of stocks included in that portfolio (ΔStockMain, ΔStockNon-financials, ΔStockFinancialSeniors). The single firm’s volatility is proxied by aggregate volatility. Hence, we include the VStoxx index, composed of the implied volatility of options with Euro Stoxx 50 as the underlying asset (ΔV). As a risk-free rate, we use the Euro swap rates with maturities from 1 to 30 years and compute the first two factors – level and slope – from the principal component analysis (PCA) applied to the different assets (PC1, PC2).

Following Byström (2006), Byström et al. (2005) and Avino and Nneji (2014), we also consider the lagged value of the iTraxx spreads. As suggested by Annaert et al. (2013) and Collin-Dufresne, Goldstein, and Martin (2001), we also include liquidity and macroeconomic indicators. As liquidity, we follow Collin-Dufresne, Goldstein, and Martin (2001) and apply the difference between the 10-year Euro swap rate and the 10-year Bund yield (Δliq). The first macroeconomic indicator is Brent oil returns (ΔBrent). Oil shocks are a common macroeconomic indicator analysts use to assess the economy’s health. Sabkha, de Peretti, and Hmaied (2019) find that in high volatility regimes, sovereign credit volatility becomes highly sensitive to these shocks. The second macroeconomic variable is the Baltic Dry Index (ΔBDI), which is a measure of economic activity available at a daily frequency. The index was created in 1985 and is measured by the intersection between the demand for shipping capacity and the supply of dry bulk carriers. It refers mainly to the shipping of dry raw materials (steel, concrete, food, and so on), so it does not consider oil across oceans. It is seen by financial analysts as an efficient indicator of economic activity. Lin, Chang, and Hsiao (2019) analyze the BDI spillover effects, finding that the BDI drives movements of equity, commodity, and currency markets.

We use daily observations between the 22nd of November 2013 and the 1st of September 2020 for a total of 1704 observations. The dependent and independent variables are considered in differential terms to obtain stationary time series.^[3]

Before the estimation results, some preliminary statistics are calculated and collected in Table 2. All variables, except interest rate ones, have high volatility, kurtosis, and skewness measures, which differ from the Normal distribution references. Minimum and maximum values are also spread over large intervals. These stylized facts call for the use of nonlinear and heteroscedastic models.

4.1 A Markov-Switching Regression Model

Let y _t , t = 1, …, T be a sequence of observations for a CDS measure (dependent variable) and x _t  = (x_1t, x_2t, …, x _nt )′, the observations for the set of n covariates discussed in the previous section. We consider the following time-varying regression model.

where β₀ (s _t ) is the time-varying intercept; β _j  (s _t ), l = 1, …, n, the time-varying coefficients and σ² (s _t ) the time-varying volatility. We assume the parameters are driven by a hidden Markov-chain process s _t , with K-states and constant transition probabilities P ( s t = j | s t − 1 = i ) = p i j , with i, j ∈ {1, …, K}.

Let us introduce the allocation variable ξ k t = I ( s t = k ) indicating the regime to which the current observation y _t belongs to, where I ( x ) is the indicator function that takes value 1 if x = 0, and 0 otherwise. We assume the time-varying parameters are.

Define z t ′ = ξ t ′ ⊗ 1 , x t ′ with ξ _t  = (ξ_1t, …, ξ _Kt )′, x _t  = (x_1t, …, x _nt )′, and β = β 1 ′ , … , β K ′ ′ with β _k  = (β_0k, β_1k, …, β _nk ). The allocation variables are used to write the random-coefficient regression model in equation (1) as a linear regression model.

with heteroskedastic effects γ t = ξ t ′ σ 2 with σ 2 = σ 1 2 , … , σ K 2 ′ (Frühwirth-Schnatter 2006).

The inference for latent variable models is a difficult task since the likelihood function is not tractable. Thus, we choose a Bayesian inference approach that allows us to make more accessible the inference task thanks to data augmentation (see Tanner et al. 1987) and Monte Carlo simulation methods. Another advantage of the Bayesian approach is that it naturally accounts for parameter uncertainty in the analysis. In our data-augmentation framework, the joint posterior distribution of parameters and latent variables given the observations is

4.2 Prior Specification

The description of the model is completed by specifying the prior distributions π( θ ) for the parameters θ  = (β′, σ ²′, p)′ where p = (p_1⋅, …, p_K⋅)′, p_k⋅ = (p_k1, …, p _kK ). For all the regime-specific volatilities σ k 2 and transition probabilities p_k⋅ we assume Gamma and Dirichlet prior distributions, respectively, that are

The set of regime-identification constrains A = σ 2 ∈ R + K , s . t . σ 1 2 < σ 2 2 < … < σ K 2 allows us to interpret the first and Kth state of the chain as low and high volatility states, respectively.

In the case of the regression coefficients β _k , the main model assumes a Normal distribution

where the hyperparameters ν _β and ϒ _β refer to the prior mean and variance–covariance matrix, respectively. In this paper, we also consider alternative priors, Laplace and Horseshoe prior, to induce variable selection (1 level prior), and shrinkage toward a common mean (2 level priors).

The different priors are presented in Table 3. The Normal prior with only 1-level is equivalent to (4), where the hyperparameters are assumed to be fixed. Under this prior structure, the same level of ridge shrinkage is applied to all the coefficients in all states. On the other hand, different prior variances are allowed under the Laplace (Bayesian LASSO) and the Horseshoe prior with 1-level, which induces a variable selection shrinkage across explanatory variables and states. In other words, these two prior distributions are characterized by a higher density close to zero and thicker tails, which a priori corresponds to the case of only a few relevant coefficients (Carvalho, Polson, and Scott 2010; Park et al. 2008). The global shrinkage is governed by the parameter λ², whose prior is inverse gamma (Gamma distribution G ( r , δ ) with expected value rδ) for the Horseshoe (Laplace) case.

In the third column of Table 7, the same prior distributions are considered, but with 2 levels extending Yau and Holmes (2011) to Markov-switching models. In particular, the prior mean ν is not fixed anymore, but a vector to be estimated, which can be interpreted as the average effect of the explanatory variables across states, α . The prior distribution on α is assumed to be normal. As in 1-level, Laplace and Horseshoe prior allow for different shrinkage levels, but in this case, towards the average effect across states. Therefore, while, in the first column, the coefficients are shrunk toward zero identifying the most significant explanatory marginal effects, the second column prior structures aim at identifying coefficients deviating more from α and potentially leading the heterogeneity between states. It is important to notice that in 2-level priors, there is no global shrinkage parameter but a shrinkage by explanatory variable, λ p 2 , p = 0 , … , n .

4.3 Posterior Approximation

Samples from the joint posterior distribution of the parameters and the allocation variables are obtained by sampling iteratively from the full conditional distributions of β, σ ², p_k. and ξ _1:T. The full conditional distribution of β is the following Gaussian distribution.

where μ ̄ β = ϒ ̄ β ∑ t = 1 T z t γ t − 1 y t + ϒ β − 1 ν β and ϒ ̄ β = ∑ t = 1 T z t γ t − 1 z t ′ + ϒ β − 1 − 1 . This also applies for the alternative priors in Table 3, see Appendix A for the additional steps of these priors. The full conditional distribution of σ ² is a product of gamma distribution truncated on the set of identifying restrictions

where u t = y t − β 0 k − x t ′ β k , a ̄ k = a k + T k / 2 and b ̄ k = b k + ∑ t ∈ T k u k t 2 / 2 with T k = Card ( T k ) . The transition probabilities p_k⋅ has the following Dirichlet full conditional distribution,

where p_−k = (p_1⋅, …, p_k−1⋅, p_k+1⋅, …, p_K⋅)′ and N k j = ∑ t = 1 T I ( s t − j ) I ( s t − 1 − k ) counts the number of transitions of the chain from the state k to the state j.

The joint posterior distribution of the hidden Markov process is obtained by iterating the prediction and updating steps for t = 1, …, T,

and evaluating the smoothing probabilities t = T, T − 1, …, 1,

where p( ξ _t  =  ι _j | ξ _t−1 =  ι _i ) = p _ij , with ι _m the mth column of the identity matrix and p (y _t |x _t , ξ _t ) is the density of y _t given s _t  = k, that is the density N β k ′ x t , σ k 2 evaluated at y _t (see Hamilton 1994, ch. 22). The smoothing probabilities are used in the forward-filtering backward sampling (FFBS) algorithm to sample jointly the allocation variables (see Frühwirth-Schnatter 2006, ch. 11–13).

5.1 Preliminary Analysis

As an initial step of our analysis, we update the linear regression model of Alexander et al. (2008) with the enlarged set of variables described in the previous section and the updated sample from October 2011 to April 2020. We apply Bayesian inference with weak diffuse priors. We compute the marginal likelihood and compare it to the MSRs. However, since we use weak diffuse priors, the results are qualitatively similar to OLS estimation.

The first row of Table 4 reports the estimates of the linear regression. Seven over eight regressors differ significantly from zero, and only the Baltic index does not. Precisely, the VStoxx index has a positive sign, as theory suggests: higher uncertainty increases CDS spreads. The equity index is also significant with the predicted sign as in Collin-Dufresne, Goldstein, and Martin (2001) and Alexander et al. (2008). Therefore, upside movements in stock returns cause a downside movement in CDS spreads. Oil shocks have been studied only considering sovereign CDS spreads, revealing their significance in explaining sovereign CDS changes during turmoil periods (Sabkha, de Peretti, and Hmaied 2019). They can be seen as a world economic health indicator, and we find that they also have a relevant explanatory power for corporate CDS spreads, affecting them negatively. Changes in corporate bond liquidity are significant. During the time considered, several monetary policies have been undertaken by the ECB to tackle both the impact of the sovereign debt crisis (i.e. APP) and the Coronavirus impact. Such policies have pushed the interest rates below 0 %, and this may be influenced by the investors’ risk-aversion, reallocating their investments toward stock markets.

The first principal component extracted from the Euro swap curve impacts the corporate CDS spreads significantly and with the expected sign, meaning that upward shifts of the term structure imply a decrease in CDS spreads. The result is consistent with the evidence provided by Alexander et al. (2008). The second principal component is also significant, as opposed to the original analysis of Alexander et al. (2008). Therefore, changes in the yield curve’s slope positively affect CDS spreads. The sensitivity of CDS spreads to changes in interest rates can also be seen empirically as in Figure 1 pagina §. In fact, after the announcement of each TLTRO, that is policy actions to invert the slope, it follows a decline in the iTraxx main index. Finally, the first lag has a negative impact on CDS spreads.

Figure 1:

This figure represents the corporate iTraxx main index reaction to ECB announcements. In particular, from left to right, TLTRO 1, TLTRO 2, TLTRO 3, and the more recent announcement of policies to tackle the COVID-19 pandemic recession (ecb.europa.eu).

To end this section, following Alexander et al. (2008), we apply a rolling Chow test to check the parameter’s stability. Results indicate strong instability in the estimated parameters, and we interpret this as evidence of a non-linear relationship. Therefore, in the next section, the regression model will be extended by considering non-linearities and a regime-dependent behavior of CDS spreads.

5.2 Markov-Switching Regression Analysis of the CDS Spreads Determinants

Table 4 reports the estimates of the Markov switching models with 2, 3, and 4 regimes. Compared to the linear model, zero is included in the 95 % credible intervals in several parameter posteriors suggesting that only a few covariates are relevant. In several cases, signs of the coefficients differ across regimes. VStoxx has a positive sign in the first regime and a negative sign in the second regime in the model with two regimes or a negative in the third regime of the 3-regime model and the fourth regime of the 4-regime model. Similar sign changes can be observed for the Brent, BDI, and the two PC variables. In the case of liquidity, it is not significant in any of the regimes or models. Therefore, the Markov-switching models reduce the explanatory variables’ contribution drastically, and instability plays a more important role.

Before continuing the analysis of parameter values and regime outputs, we compare the four specifications to see which shall be preferred. The last two columns in Table 4 show that the 4-regimes specification has the highest (log) marginal likelihood value and the lowest Deviance information criterion (DIC).^[4] The marginal likelihood is based on a recursive approximation of the complete likelihood using the Hamilton filter as in Kadhem, Hewson, and Kaimi (2016)). The DIC penalizes complex models based on the estimation of the number of parameters. Despite this, our larger model is preferred. The difference is sizeable with the linear model, and the values of the 2 and 3-regimes models are also inferior. Therefore, we describe results for the Markov switching 4-regimes model and refer to Figures 6, 7, 8 and 9 in Appendix B.1 for results on the 2 and 3-regimes models.

Returning to parameter estimates, implied volatility and stock prices are significant with the expected sign in regime 1, the regime with lower volatility (the mean posterior estimate of σ² is equal to 0.830 which is lower than the value of 2.614 for the linear regression model). In the second regime, the one with moderate volatility (the estimate of σ² is equal to 1.076), the implied volatility, and the term structure slope are significant. Both posterior mean values are higher than in the linear regression model and the other two regimes. In the third regime, the one with moderate-high volatility (the estimate of σ² is equal to 1.679), the Brent and the term structure slope have a significant positive impact. In the fourth regime, stock and the level of the slope curve are significant, both with a negative coefficient.

No covariate is consistently significant in all four regimes. The implied volatility is relevant only during periods of low uncertainty (regimes 1 and 2) with a positive coefficient, while stock prices are significant in the extreme scenarios of high and low volatility, in both cases having a negative effect on CDS spreads. The effect of the slope of the term structure (PC2) is only significant in two of the four regimes (regimes 2 and 3), but also with different signs (negative and positive, respectively). Hence, these results indicate that macroeconomic variables can explain part of the CDS spread variations, as suggested by Annaert et al. (2013), Collin-Dufresne, Goldstein, and Martin (2001), but more importantly, monitoring the changes in regimes is essential to have a better understanding of the role of each of these indicators, given that regime transitions can even imply a change in the relationship with CDS spreads.

Volatility estimates in regimes 1, 2, and three are lower than the ones for the linear regression model, whereas the estimate in the fourth regime is almost twice as large. See also Figure 2 for the full posterior distribution.

Figure 2:

Estimates of the regime-specific variance σ k 2 (dashed lines) and their posterior distributions (solid lines) in the 4-regimes model fitted to ΔiTraxx. From top-left to bottom-right, the posterior distribution in regimes from 1 to 4.

Figure 3 shows the estimated hidden states (top) and the log-volatility process (bottom). The first regime with lower volatility is the most probable over time, in particular in the years 2017–2018. The other three regimes are less persistent, given that they capture exceptional events. In some cases, the second regime anticipates or follows the third regime. The COVID-19 pandemic is mainly associated with the fourth regime, and also other periods, such as the beginning of 2016, Volatility estimates in regimes 1, 2, and three are lower than the ones for the linear regression model, whereas the estimate in the fourth regime is almost characterized by high volatility in the series. So, the fourth regime is necessary to fit the extremely high volatility that few significant regressors cannot capture alone. The red line in the bottom plot reports the estimated conditional volatility in the 4-regime model, that is V ( y t | s t , θ ) ̂ , which is constant within each regime. The Bayesian approach allows for incorporating the parameter uncertainty and the explanatory variables in the volatility process. Comparing the conditional volatility with the integrated volatility processes V ( y t ) ̂ (bottom, black line), we find that explanatory variables have an important role in explaining the volatility changes within each regime.

Figure 3:

Top: Time series of the ΔiTraxx (black solid) and the estimated maximum a posteriori (MAP) hidden regime by date for the 4-regime model (red stepwise). Bottom: Unconditional (black line) and conditional (red line) log-volatility process.

Finally, we investigate the impact of the COVID-19 pandemic by ending the data in December 2019 and dropping the pandemic period from the sample. Our results confirm the best model is the 4-regime MSR (see Panel E of Table 4). The results confirm that stock returns and volatility are significant, whereas Brent and term structure factors become relevant only when including the pandemic sample period.

5.3 Sensitivity to Alternative Prior Specifications

Alternative priors are applied to the regression coefficients to identify the most important determinants of the CDS main index and those covariates effects that significantly differ across regimes. However, also the variance parameters have an essential role in the regime heterogeneity. This is evidenced by Table 5 where two special cases of Model (1) are considered: (a) β_p1 =  …  = β _pK for p = 0, …, n (MSR, only variance), and (b) σ 1 2 = … = σ K 2 (MSR, only coefficients). Both special cases perform better than the Linear Regression Model (LRM), indicating that the two of them contribute to the regime heterogeneity, particularly the former. Indeed, the constant coefficients under case (b), still differ from LRM (see Table 8).

The most important coefficients are identified using Laplace and Horseshoe prior with 1 and 2 levels in Table 7. Under these two priors with 1-level, the most significant determinants are the positive effect of the VStoxx index (ΔV _t ) during periods of low volatility (regimes 1 and 2) and the equity index (ΔStock _t ), which consistently decrease CDS spreads in all regimes. Compared to Table 4, ΔBrent _t , ΔBDI _t and PC2 _t are not relevant anymore. During periods of high volatility (regime 3), the proxy of risk-free rate, PC1, becomes relevant, having a negative effect. In the case of 2-level priors, the estimates of the prior variances of the coefficients ϒ _β in Table 9 indicate that the factors that differ the most between regimes are the liquidity Δliq _t in regime three and equity ΔStock _t in regime 1.