Efficient Risk Measures Calculations for Generalized CreditRisk+ Models

Abstract

Numerical calculations of risk measures in credit risk models amount to evaluation of various forms of tail expectations of portfolio loss distribution. Though the moment generating function of the loss distribution in CreditRisk+ model is available in analytic closed form, efficient, accurate and reliable computation of risk measures (VaR and Expected Shortfall) and risk contributions for the CreditRisk+ model pose technical challenge. 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, saddle-point approximation method, curve fitting 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. The importance sampling in Monte Carlo simulation methods are easy to implement, but they compete less favorably in accuracy and reliability 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 external industrial optimization software is required.