GAS Copula models on who’s systemically important in South Africa: Banks or Insurers?

Abstract

This paper makes use of the generalized autoregressive score (GAS) Copula model to estimate the Conditional Value at Risk (CoVaR) measure of systemic risk. The proposed measure of systemic risk considers the score of the conditional density as the main driver of time-varying dynamics of tail dependence among financial institutions. Not only does the GAS Copula-based CoVaR enable us to monitor the amount of systemic risk posed by different financial institutions at a specific date, it also allows for the forecasting of systemic risk over time. Our results based on a sample of daily equity returns collected from January 2000 to July 2017 surprisingly show that in South Africa, insurers are the most systemically risky compared to banks and other financial sectors. Moreover, we make use of flexible GAS Copulas in order to approximate complex dependence structures. To validate the robustness of our results over time, we divide our sample period into two sub samples, namely the pre-crisis period (January 2000 to June 2007) and the post-crisis period (January 2010 to July 2017). We obtain similar results in the pre-crisis period. However, in the post-crisis period banks are found to be the biggest threat to system-wide stability.

Appendices

Appendix A: Dependence models: Copulas

An n-dimensional Copula is a function C from $\left[ {0,1} \right]^{n} \to \left[ {0,1} \right]$ with the following properties:

• 1) $C\left( {u_{1} , \ldots ,u_{n} } \right)$ is decreasing in each component $u_{i}$$\in \left[ {0,1} \right]\, i = 1, \ldots ,n$

• 2) $C\left( {1, \ldots 1,u_{i} ,1 \ldots 1} \right) = u_{i}$ for $u_{i}$$\in \left[ {0,1} \right] i = 1, \ldots ,n$

Property (1) implies that Copula is a multivariate cumulative density function. Property (2) suggests that the Copula has uniform margins. It is apparent that the Copula is nothing but a multivariate distribution. However, Copula has its own power when dealing with multivariate distributions. This power is derived from Sklar’s theorem. For any random variable $X_{1} , \ldots , X_{n}$ with joint CDF $F\left( {x_{1} , \ldots , x_{n} } \right) = P(X_{1} \le x_{1} , \ldots , X_{n} \le x_{n}$ and marginal CDF’s $F_{i} \left( {x_{i} } \right) = P\left( {X_{i} \le x_{i} } \right)$ for $i = 1 \ldots n$, there exists a Copula such that

$$F\left( {x_{1} , \ldots , x_{n} } \right) = C\left( {F_{1} \left( {x_{1} } \right), \ldots F_{n} \left( {x_{n} } \right)} \right) = C\left( {u_{1} , \ldots ,u_{n} } \right),\;x_{1} , \ldots , x_{n} \in {\mathbb{R}}$$

If $F_{i }$ are continuous for $i = 1, \ldots ,n$ then C is unique. If C is a Copula and $F_{1} , \ldots , F_{n}$ are univariate distribution functions, the function F defined above is a multivariate distribution function with margins $F_{1} , \ldots , F_{n}$.

1.1 Gaussian Copula

Given $= (u_{1} ,u_{2} )^{T} \in [0,1]^{2}$, the Gaussian Copula can be formally written as follows

$$C_{G} \left( {u_{1} ,u_{2} ,\rho } \right) = \varPhi_{k,\varSigma } (\varPhi^{ - 1} \left( {u_{1} } \right)\varPhi^{ - 1} \left( {u_{2} )} \right)$$

where $\varSigma$ is 2*2 correlation matrix, $\varPhi ()$ is the standard normal distribution function, and $\varPhi^{ - 1} ()$ is the inverse of the $\varPhi ()$. This Copula does not cater for tail dependence.

1.2 Archimedean Copulas

Archimedean Copulas are developed by specifying a generator function. Given $\varphi$, a generator function with $(\varphi )^{ - 1}$ is completely monotonic; then, a bivariate Archimedean Copula can be specified as follows:

$$C\left( {u_{1} ,u_{2} ,} \right) = \varphi^{ - 1} \left( {\varphi \left( {u_{1} } \right) + \varphi \left( {u_{2} } \right)} \right)$$

Different forms of Archimedean Copulas exist as results of diverse generator functions and the dependence they take into account.

1.3 Clayton Copula

The Clayton Copula has the following generator function $\varphi = ( - \log \left( t \right))^{\theta }$ where $\theta \in \left[ {1, + \infty } \right]$, $\theta$ is the Copula parameter. The Clayton Copula only caters for lower tail dependence and in the bivariate case it is given by the following formula:

$$C_{C} \left( {u_{1} ,u_{2} ,\theta } \right) = (u_{1}^{ - \theta } + u_{2}^{ - \theta } - 1)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \theta }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\theta $}}}}$$

The coefficient for lower tail dependence is given by $\lambda_{L} = 2^{{ - {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \theta }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\theta $}}}}$

1.4 Rotated Clayton

The rotated Clayton Copula accounts for upper tail dependence, and its bivariate Copula function is given as follow:

$$C_{\text{RC}} \left( {u_{1} ,u_{2} ,\theta } \right) = u_{1} + u_{2} + C_{C} \left( {1 - u_{1} ,1 - u_{2} ;\theta } \right)$$

1.5 Gumbel Copula

This Copula accounts for upper tail dependence only and has the following generator function, $\varphi = \frac{1}{{\theta \left( {t^{\theta } - 1} \right)}}$ with $\theta \in \left[ { - 1, + \infty } \right]$. The Gumbel Copula is mathematically represented as follows:

$$C_{G} \left( {u_{1} ,u_{2} ,\theta } \right) = e^{{\left( { - [( - \log u_{1} )^{\theta } + ( - \log u_{2} )^{\theta } ]^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \theta }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\theta $}}}} } \right)}}$$

The coefficient for upper tail dependence is given by $\lambda_{u} = 2 - 2^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \theta }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\theta $}}}}$

1.6 Rotated Gumbel Copula

The rotated Gumbel Copula accounts for lower tail dependence. Its Copula function is represented as follows:

$$C_{RG} \left( {u_{1} ,u_{2} ,\theta } \right) = u_{1} + u_{2} + C_{G} \left( {1 - u_{1} ,1 - u_{2} ;\theta } \right)$$

1.7 Mixture Copulas

In addition to the Copulas presented above, one can consider a mixture Copula, which is simply the convex combination between different Copula functions. This makes it possible to obtain any dependence structure desired. A mixture of K Copulas is defined as

$$C_{M} \left( {u_{1} ,u_{2} ,\theta } \right) = \mathop \sum \limits_{i = 1}^{2} w_{i} C_{i} \left( {u_{1} ,u_{2} ,\theta_{i} } \right)$$

where $\mathop \sum \limits_{i = 1}^{K} w_{i} = 1$, $w_{i}$ is the weight for the ith Copula and lies between 0 and 1.

Appendix B

The figures are a graphical illustration of the evolution of the returns of the financial institutions used in this study (see Tables 5, 6).

Table 5 List of financial institutions

Table 6 Descriptive statistics

Appendix C

See Table 7.

Table 7 Estimation results for marginal models

Appendix D

See Table 8.

Table 8 Estimation results for different dynamic Copula models

Appendix E

A Copula Illustration: Comparisons of the dependence parameter estimates for the different GAS Copula models

These graphs depict the evolution of both the upper and lower tail dependence structures between selected financial institutions and the financial system.

A Copula Illustration: Comparisons of the dependence parameter estimates for the different GAS Copula models

These graphs depict the evolution of both the upper and lower tail dependence structures between selected financial institutions and the financial system.

A Copula Illustration: Comparisons of the dependence parameter estimates for the different GAS Copula models

These graphs depict the evolution of both the upper and lower tail dependence structures between selected financial institutions and the financial system.

A Copula Illustration: Comparisons of the dependence parameter estimates for the different GAS Copula models

These graphs depict the evolution of both the upper and lower tail dependence structures between selected financial institutions and the financial system.