Can CDS indexes signal future turmoils in the stock market? A Markov switching perspective

Abstract

Single-name Credit Default Swaps (CDS) are considered the main providers of direct information related with a reference entity’s creditworthiness and, for this reason, they have often been the core of news on the current financial crisis. The academic research has focused mainly on the capacity of CDS in anticipating agencies’ official rating changes and—in this respect—on their superior signalling power, compared to bond and stock markets. The aim of this work is, instead, to investigate the ability of fluctuations in CDS indexes in anticipating the occurrence of stock market crises. Our goal is to show that CDS indexes may provide investors and institutions with early warning signals of financial distresses in the stock market. We make use of a Markov switching model with states characterized by increasing levels of volatility and compare the times in which the first switch in a high volatility state occurs, respectively, in CDS and stock market index quotes. The data set consists of daily closing quotes for 5 years CDS and stock market index prices, covering the time period from 2004 to 2010. In order to capture possible geographic differences in CDS index capacity of foreseeing stock market distresses, data referring to two different regions, Europe and United States, are analyzed.

Appendix: The MCMC algorithm

In order to generate realizations from the posterior joint distribution of all the parameters, we alternate the following moves at each sweep of the MCMC algorithm:

• (a) generate a new realization of the transition matrix ${\varvec{\varLambda }}$;

• (b) generate a new realization of the vector of state variables ${{\varvec{s}}}$;

• (c) generate a new realization of the hyperparameter $\zeta$;

• (d) generate a new realization of the vector of variances ${\varvec{\sigma }}^{2}$;

• (e) generate a new realization of the hyperparameter $\kappa$;

• (f) generate a new realization of the vector of means ${\varvec{\mu }}$.

These six moves are fairly standard and all performed through Gibbs sampling.

Move (a) follows Robert et al. (1993). The $i$-th row of ${\varvec{\varLambda }}$ is sampled from a Dirichlet distribution $D(\delta _1+n_{i1}, \ldots , \delta _k+n_{ik})$, where $n_{ij}=\sum _{t=1}^{T-1}I\{s_t=i,s_{t+1}=j \}$ is the number of transitions from regime $i$ to regime $j$, with $I\{\cdot \}$ being the indicator function.

In (b), the standard solution for updating the state variable would be to sample $s_{1},\ldots , s_{T}$, one at a time from $t=1$ to $t=T$, drawing new values from their full conditional distribution:

$$\begin{aligned} p(s_{t}=i|\cdots )\propto \lambda _{s_{t-1}i}\phi (y_{t};\mu _{i},\sigma _{i}^{2})\lambda _{is_{t+1}} \end{aligned}$$

where ‘$\cdots$’ denotes ‘all other variables’; for $t=1$, the first factor is replaced by the stationary probability $\pi _{i}$ and, for $t=T$, the last factor is replaced by 1. However, to obtain a faster mixing algorithm, we preferred to sample ${{\varvec{s}}}$ directly from $p({{\varvec{s}}}|{{\varvec{y}}},{\varvec{\varLambda }})$, using a stochastic version of the forward-backward recursion (see Scott 2002).

In (c) we update $\zeta$ by a Gibbs move, sampling from its full conditional:

$$\begin{aligned} \zeta |\cdots \sim \text{ Ga }\left( f+k\eta , h+\sum _{i=1}^k \sigma _i^{-2}\right) . \end{aligned}$$

Before considering the updating of ${\varvec{\sigma }}^{2}$ in (d), we comment briefly on the issue of labeling the regimes. The whole model is, in fact, invariant to the permutation of the labels $i=1,2,\ldots , k$. In our application, it seems natural that the different regimes are identified by different levels in the volatility (see Sect. 5). Thus, we adopt a unique labeling in which the $\sigma _{i}^{2}$ are in increasing numerical order. As a consequence, the joint prior distribution of the $\sigma _{i}^{2}$ is $k!$ times the product of the individual inverse gamma densities, restricted to the set $\sigma _{1}^{2}<\sigma _{2}^{2}<\cdots <\sigma _{k}^{2}$. The $\sigma _{i}$ can be updated by means of Gibbs sampler, drawing them independently from the distribution

$$\begin{aligned} \sigma _{i}^{-2}|\cdots \sim \text{ Ga }\left( \eta +\frac{1}{2} (n_{i}+1),\zeta +\frac{1}{2}\sum _{t:s_{t}=i}(y_{t}-\mu _{i})^{2}+\frac{1}{ 2\kappa }(\mu _{i}-\xi )^{2}\right) , \end{aligned}$$

where $n_{i}=\#\{t:s_{t}=i\}$ is the number of observations currently allocated to the $i$-th regime. In order to preserve the ordering constraints on the $\sigma _{i}^{2}$, the move is accepted provided the ordering is unchanged and rejected otherwise.

In (e) we update $\kappa$ by a Gibbs move, sampling $\kappa ^{-1}$ from its full conditional:

$$\begin{aligned} \kappa ^{-1}|\cdots \sim \text{ Ga }\left( q+\frac{k}{2}, r+\frac{1}{2} \sum _{i=1}^k \frac{(\mu _i-\xi )^2}{\sigma _i^{2}}\right) . \end{aligned}$$

Finally, in (f) we update $\mu _{i}$ independently using a Gibbs move, sampling from its full conditional distribution:

$$\begin{aligned} \mu _{i}|\cdots \sim N\left( \frac{\kappa \sum _{t:s_{t}=i}y_{t}+\xi }{ 1+\kappa n_{i}},\frac{\sigma _{i}^{2}\kappa }{1+\kappa n_{i}}\right) . \end{aligned}$$