Don’t fall from the saddle: The importance of higher moments of credit loss distributions

The original Panjer recursion of the CreditRisk+ model is said to be unstable and therefore to yield inaccurate results of the tail distribution of credit portfolios. A much-hailed solution for the flaws of the Panjer recursion is the saddlepoint approximation method. In this paper we show that the saddlepoint approximation is an accurate and robust tool only for credit portfolios with low skewness and kurtosis of the loss distribution. However, often credit portfolios are heterogeneous with large skewness and kurtosis. We show that for such portfolios the commonly applied saddlepoint approximations (the Lugannani-Rice and the Barndorff-Nielsen formulas) are not reliable. We explain it by the dependence of the high-order standardized cumulants and the relative error on the saddlepoint. The more the cumulants and the relative error fluctuate for variations in the value of the saddlepoint (from 0 to the upper bound) the less accurate the saddlepoint approximation is. Hence, the saddlepoint approximations are not a universal substitute to the Panjer recursion algorithm. We also provide users of CR+ with a set of diagnostics to identify beforehand when the saddlepoint approximations are prone to failure.