Numerical calculations of risk measures and risk contributions in credit risk models amount to the evaluation of various forms of quantiles, tail probabilities and tail expectations of the portfolio loss distribution. Though the moment generating function of the loss distribution in the CreditRisk+ model is available in analytic closed form, efficient, accurate and reliable computation of risk measures (Value-at-Risk and Expected Shortfall) and risk contributions for the CreditRisk+ model poses technical challenges. We propose various numerical algorithms for risk measures and risk contributions calculations of the enhanced CreditRisk+ model under the common background vector framework using the Johnson curve fitting method, saddlepoint approximation method, importance sampling in Monte Carlo simulation and check function formulation. Our numerical studies on stylized credit portfolios and benchmark industrial credit portfolios reveal that the Johnson curve fitting approach works very well for credit portfolios with a large number of obligors, demonstrating high level of numerical reliability and computational efficiency. Once we implement the systematic procedure of finding the saddlepoint within an approximate domain, the saddlepoint approximation schemes provide efficient calculation and accurate numerical results. The importance sampling in Monte Carlo simulation methods are easy to implement, but they compete less favorably in accuracy and efficiency with other numerical algorithms. The less commonly used check function formulation is limited to risk measures calculations. It competes favorably in accuracy and reliability, but an extra optimization algorithm is required.

中文翻译：

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通用信用风险+模型的有效风险度量计算

信用风险模型中风险度量和风险贡献的数值计算相当于对组合损失分布的各种形式的分位数、尾部概率和尾部预期的评估。虽然损失分布的矩生成函数信用风险+模型以分析封闭形式提供，有效、准确和可靠地计算风险度量（风险价值和预期短缺）和风险贡献信用风险+模型带来了技术挑战。我们提出了各种数值算法，用于风险度量和增强的风险贡献计算信用风险+常用背景向量框架下的模型采用约翰逊曲线拟合法、鞍点逼近法、蒙特卡罗模拟中的重要性采样和校验函数公式。我们对程式化信贷组合和基准工业信贷组合的数值研究表明，约翰逊曲线拟合方法非常适用于具有大量债务人的信贷组合，证明了高水平的数值可靠性和计算效率。一旦我们实施了在近似域内找到鞍点的系统过程，鞍点近似方案就可以提供有效的计算和准确的数值结果。蒙特卡罗模拟方法中的重要性采样很容易实现，但与其他数值算法相比，它们在准确性和效率方面的竞争力较差。不太常用的检查函数公式仅限于风险度量计算。它在准确性和可靠性方面具有优势，但需要额外的优化算法。