Principal Component Copulas for Capital Modelling and Systemic Risk

We introduce a class of copulas that we call Principal Component Copulas (PCCs). This class combines the strong points of copula-based techniques with principal component analysis (PCA), which results in flexibility when modelling tail dependence along the most important directions in high-dimensional data. We obtain theoretical results for PCCs that are important for practical applications. In particular, we derive tractable expressions for the high-dimensional copula density, which can be represented in terms of characteristic functions. We also develop algorithms to perform Maximum Likelihood and Generalized Method of Moment estimation in high-dimensions and show very good performance in simulation experiments. Finally, we apply the copula to the international stock market to study systemic risk. We find that PCCs lead to excellent performance on measures of systemic risk due to their ability to distinguish between parallel and orthogonal movements in the global market, which have a different impact on systemic risk and diversification. As a result, we consider the PCC promising for capital models, which financial institutions use to protect themselves against systemic risk.