Domestic Systemically Important Banks: A Quantitative Analysis for the Chinese Banking System

Table 2

Commonly used Copula functions.

Copula	𝐶(𝑢1,𝑢2,…,𝑢𝑁)	Note [17]
Gaussian Copula	𝐶𝑢1,𝑢2,…,𝑢𝑁;𝜌=Φ𝜌Φ^−1𝑢1,Φ^−1𝑢2,…,Φ^−1𝑢𝑁	𝜌 is a symmetric, positive definite matrix with diag 𝜌=𝟏;  Φ𝜌 is the standardized multivariate normal distribution with correlation matrix 𝜌.
T-Copula	𝐶𝑢1,𝑢2,…,𝑢𝑁;𝜌,𝑘=𝑡𝜌,𝑘𝑡𝑘^−1𝑢1,𝑡𝑘^−1𝑢2,…,𝑡𝑘^−1𝑢𝑁	𝜌 is a symmetric, positive definite matrix with diag 𝜌=𝟏;  𝑡𝜌,𝑘 is the standardized multivariate Student’s distribution with 𝑘 degrees of freedom and correlation matrix 𝜌.
Archimedean family of Copulas	𝐶𝑢1,𝑢2,…,𝑢𝑁=𝜑^−1[𝜑(𝑢1)+𝜑(𝑢2)+⋯+𝜑(𝑢𝑁)],if^𝑁𝑖=1𝜑(𝑢𝑖)≤𝜑(0)0,otherwise	𝜑(𝑢) is the generator of the Copula. It is a function that satisfies 𝜑(1)=0,𝜑^′(𝑢)<0, and 𝜑^′′(𝑢)>0, for all 0≤𝑢≤1.