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-based models, 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 in order 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 market movements, which increase systemic risk, and orthogonal movements, which reduce systemic risk. As a result, we consider the PCC promising for internal capital models, which financial institutions use to protect themselves against systemic risk.

翻译：本文提出一类称为主成分Copula（PCC）的相依结构模型。该类模型融合了基于Copula的技术与基于主成分的模型的优势，能够灵活刻画高维数据中沿最重要方向上的尾部相依性。我们获得了对实际应用至关重要的PCC理论结果，特别是推导出了高维Copula密度的可处理表达式，该表达式可通过特征函数表示。我们还开发了在高维场景下进行极大似然估计与广义矩估计的算法，并在模拟实验中展现出优异性能。最后，我们将该Copula应用于国际股票市场以研究系统性风险。研究发现，由于PCC能够区分加剧系统性风险的平行市场波动与降低系统性风险的正交波动，其在系统性风险度量方面表现卓越。因此，我们认为PCC在金融机构用于抵御系统性风险的内部资本模型中具有广阔应用前景。