Measuring contagion risk in high volatility state among Taiwanese major banks

Abstract

This paper studies the structural tail dependence and contagion risk especially in high volatility state between domestic (Taiwanese) and foreign banks. Aptly the two-state threshold copula GARCH provides the threshold regression and copulas to classify the actual volatility index into a high or in a low state and estimate the structural tail dependences using Kendall taus to probe the co-movement among banks. In high volatility state, the average Kendall taus and value at risk as well as expected shortfall are about two times larger than in low volatility state. The asymmetric jumps of Kendall taus appear more frequent in the subprime crisis whereas the symmetric trends of Kendall taus appear higher in Greek debt crisis. Among three copula models in low volatility state, Gaussian and Student-t copula models have established a more significant estimate than the Clayton copula model. However, in high volatility state, Clayton copula model could still produce an acceptable estimate. Empirically, using Clayton copula in high volatility state has demonstrated clearly intensive tail jumps capable to distinguish the contagion risk.

Appendices

Appendix 1

The elliptical and archimedean copulas

The linear copula functions which are the type of elliptical copulas and the nonlinear copula functions which are the family of Archimedean copulas are useful to describe the structural tail dependence.

• (1) Elliptical copulas

  The type of elliptical copulas including Gaussian and Student-t copula have featured linear correlation and symmetric shape in copula function.

  For the bi-variate Gaussian cumulative distribution, a unique bi-variate Gaussian copula distribution function can be found using Eq. (16) as

  $$C_{N} (u_{1} ,u_{2} ;\rho ) = \int_{\infty }^{{\varPhi^{ - 1} (u_{1} )}} {\int_{\infty }^{{\varPhi^{ - 1} (u_{2} )}} {\frac{1}{{2\pi \sqrt {(1 - \rho^{2} } )}}} } \exp \left[ { - \frac{{x^{2} - 2\rho xy + y^{2} }}{{2(1 - \rho^{2} )}}} \right]dxdy$$

  (16)

  where ρ is the Pearson correlation coefficient which denotes the linear correlation between random variable x and y and $\varPhi^{ - 1}$ denotes the inverse of univariate cumulative standard normal distribution.

  Similarly, for the bi-variate Student-t cumulative distribution, a unique bi-variate Student-t copula distribution function can be found using Eq. (17) as

  $$C_{St} (u_{1} ,u_{2} ;\nu ,\rho ) = \int_{\infty }^{{t_{\nu }^{ - 1} (u_{1} )}} {\int_{\infty }^{{t_{\nu }^{ - 1} (u_{2} )}} {\frac{1}{{2\pi \sqrt {(1 - \rho^{2} } )}}} } \left\{ {1 + \frac{{x^{2} - 2\rho xy + y^{2} }}{{\nu (1 - \rho^{2} )}}} \right\}^{{ - \frac{v + 2}{2}}} dxdy$$

  (17)

  where $t_{v}^{ - 1}$ is the inverse of univariate cumulative Student-t distribution and v is the degree of freedom.

• (2) Archimedean copulas

  If the structural dependence is not linear, it is not fit to the elliptical copulas. Thus, the use of nonlinear copula to explain the nonlinear structural dependence is necessary. The family of Archimedean copulas is nonlinear in structure and has three copulas: Clayton (1978), Frank (1979), and Gumbel (1960). Among them, The Clayton copula that features intensive density to the right tail of returns (rising together) having the characteristic of lower left tail dependence and upper right tail independence is more pertinent to model downside contagion risk and the equation is

  $$C_{Cl} (u_{1} ,u_{2} ;\theta ) = \left\{ {u_{1}^{ - \theta } + u_{2}^{ - \theta } - 1} \right\}^{{ - \frac{1}{\theta }}} ,\theta \in (0,\infty )$$

  (18)

  where $\theta$ is the parameter of Clayton copula.

Appendix 2

Two-stage maximum likelihood estimation

The copula function $\tilde{c}( \cdot )$ and each marginal function $f_{i} ( \cdot )$ contain the unknown parameters required to estimate. Suppose that $\phi_{i}$ is the parameter vector in the ith marginal density function $f_{i} ( \cdot )$ as in Eq. (8) and $\psi$ is the parameter vector of copula density function $\tilde{c}( \cdot )$. For a structural tail dependence between ith bank stock returns with state j representing the low (j = 1) or high (j = 2) volatility state at time t, the equation of conditional multivariate log-likelihood function for paired $\varepsilon_{i,t}^{(j)}$ with bank i = 1, 2 is

$$\ln L(\psi^{(j)} ,\phi^{(j)} ){ = }\sum\limits_{t = 1}^{{n_{j} }} {\ln \tilde{c}^{(j)} (u_{1,t}^{(j)} (\phi_{1}^{(j)} ),u_{2,t}^{(j)} (\phi_{2}^{(j)} )} |\varepsilon_{1,t - 1}^{(j)} ,\varepsilon_{2,t - 1}^{(j)} ;\psi^{(j)} ) + \sum\limits_{t = 1}^{{n_{j} }} {\sum\limits_{i = 1}^{2} {\ln f_{i}^{(j)} (\varepsilon_{i,t}^{(j)} |\varepsilon_{i,t - 1}^{(j)} ;\phi_{i}^{(j)} )} }$$

(19)

where $\phi_{i}^{(j)}$ stands for $[\phi _{1}^{{(j)}} ,\phi _{2}^{{(j)}} ]^{\prime}$ and $n_{j}$ denotes the number of observations in state j.

However, due to the nonlinear property of copula function, the use of two-stage MLE estimation becomes necessary to optimize the log likelihood of copula GARCH effectively. Patton (2001) provides a useful two-stage procedure to solve for optimal $\phi_{i}^{(j)}$ and $\psi_{{}}^{(j)}$ as follows:

The first stage is to solve for $\phi_{i}^{(j)}$ written as

$$\hat{\phi }_{{}}^{(j)} = \arg \mathop {\hbox{max} }\limits_{{\phi^{(j)} }} \sum\limits_{t = 1}^{{n_{j} }} {\sum\limits_{i = 1}^{2} {\ln f_{i}^{(j)} (\varepsilon_{i,t}^{(j)} |\varepsilon_{i,t - 1}^{(j)} ;\phi_{i}^{(j)} )} }$$

(20)

The second stage uses the optimal $\hat{\phi }_{i}^{(j)}$ to solve for $\psi_{{}}^{(j)}$ written as

$$\hat{\psi }^{(j)} = \mathop {\arg \hbox{max} }\limits_{{\psi^{(j)} }} \ln L(\psi^{(j)} ,\hat{\phi }^{(j)} ) { = }\sum\limits_{t = 1}^{{n_{j} }} {\ln \tilde{c}^{(j)} (u_{1,t}^{(j)} (\hat{\phi }_{1}^{(j)} ),u_{2,t}^{(j)} (\hat{\phi }_{2}^{(j)} )} |\varepsilon_{1,t - 1}^{(j)} ,\varepsilon_{2,t - 1}^{(j)} ;\psi^{(j)} )$$

(21)