Operational risk is challenging to quantify because of the broad range of categories (fraud, technological issues, natural disasters) and the heavy-tailed nature of realized losses. Operational risk modeling requires quantifying how these broad loss categories are related. We focus on the issue of loss frequencies having different time scales (e.g., daily, yearly, monthly basis), specifically on estimating the statistics of losses on arbitrary time horizons. We present a frequency model where mathematical techniques can be feasibly applied to analytically calculate the mean, variance, and co-variances that are accurate compared to more time-consuming Monte Carlo simulations. We show that the analytic calculations of cumulative loss statistics in an arbitrary time window are feasible here and would otherwise be intractable due to temporal correlations. Our work has potential value because these statistics are crucial for approximating correlations of losses via copulas. We systematically vary all model parameters to demonstrate the accuracy of our methods for calculating all first and second order statistics of aggregate loss distributions. Finally, using combined data from a consortium of institutions, we show that different time horizons can lead to a large range of loss statistics that can significantly affect calculations of capital requirements.

翻译：操作风险因其涵盖的损失类别广泛（欺诈、技术问题、自然灾害）以及实际损失具有重尾特征而难以量化。操作风险建模需要量化这些广泛损失类别之间的关联关系。本文聚焦于损失频率存在不同时间尺度（例如日度、年度、月度）的问题，具体研究如何估计任意时间跨度上的损失统计量。我们提出了一种频率模型，在该模型中可便捷地运用数学技术解析计算均值、方差和协方差，其精度优于更为耗时的蒙特卡洛模拟方法。我们证明，在任意时间窗口内对累积损失统计量进行解析计算在此模型中是可行的，而由于时间相关性，这些计算在其它情况下将难以处理。我们的工作具有潜在价值，因为此类统计量对于通过连接函数近似损失相关性至关重要。通过系统变化所有模型参数，我们验证了所提方法在计算聚合损失分布所有一阶和二阶统计量时的准确性。最后，利用来自多家机构联合数据集，我们证明不同时间尺度可能导致损失统计量出现巨大差异，从而显著影响资本要求的计算。