Efficient Risk Measures Calculations For Generalized Creditrisk+ Models

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.