Calculation of Higher Moments in CreditRisk+ with Applications

by Michael B. Gordy of the Federal Reserve Board

September 21, 2001

Abstract: CreditRisk+ is an influential and widely implemented model of portfolio credit risk. As a close variant of models long used for insurance risk, it retains the analytical tractability for which the insurance models were designed. Value-at-risk can be obtained via a recurrence-rule algorithm, so Monte Carlo simulation can be avoided. Little recognized, however, is that the algorithm is fragile. Under empirically realistic conditions, numerical error can accumulate in the execution of the recurrence rule and produce wildly inaccurate results for value-at-risk.

This paper provides new tools for users of CreditRisk+ based on the cumulant generating function ("cgf") of the portfolio loss distribution. Direct solution for the moments of the loss distribution from the cgf is almost instantaneous and is computationally robust. Thus, the moments provide a convenient, quick and independent diagnostic on the implementation and execution of the standard solution algorithm. Better still, with the cgf in hand we have an alternative to the standard algorithm. I show how tail percentiles of the loss distribution can be calculated quickly and easily by saddlepoint approximation. On a large and varied sample of simulated test portfolios, I find a natural complementarity between the two algorithms: Saddlepoint approximation is accurate and robust in those situations for which the standard algorithm performs least well, and is less accurate in those situations for which the standard algorithm is fast and reliable.

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