We are grateful to Dr Mâcé Mesters and Dr Sacha van Weeren from Rabobank for drawing our attention to this issue and for their helpful comments. We would also like to thank the participants at the first AMAMEF conference for valuable feedback. 94 X. Huang et al

We propose a new framework for modeling systematic risk in Loss-Given-Default (LGD) in the context of credit portfolio losses. The class of models is very flexible and accommodates well skewness and heteroscedastic errors. The quantities in the models have simple economic interpretation. Inference of models in this framework can be unified. Moreover, it allows efficient numerical procedures, such as the normal approximation and the saddlepoint approximation, to calculate the portfolio loss distribution, Value at Risk (VaR) and Expected Shortfall (ES).

Abstract. We derive two types of saddlepoint approximations for expectations in the form of E[(X − K)+ ], where X is the sum of n independent random variables and K is a known constant. We establish error convergence rates for both types of approximations in the independently and identically distributed case. The approximations are further extended to cover the case of lattice variables. An application of the saddlepoint approximations to CDO pricing is presented. (2) We present explicit saddlepoint approximations for the log-return model considered in [14] and [15]. With our formulas, only one saddlepoint needs to be computed, whereas the measure change approach employed in [14] and [15] requires the calculation of two saddlepoints. (3) We have also provided the corresponding saddlepoint approximations for lattice variables. The lattice case is largely ignored in the literature so far, even in applications where lattice variables are highly relevant, for example, the pricing of CDOs.Our main contribution is the second type of saddlepoint approximations. They are derived following the approach in [11] and [5] where the Lugannani-Rice formula for tail probabilities was derived. The higher order version of the approximations distinguishes itself from all

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We present a dedicated algorithm for the nonnegative factorization of a correlation matrix from an application in financial engineering. We look for a low-rank approximation. The origin of the problem is discussed in some detail. Next to the description of the algorithm, we prove, by means of a counter example, that an exact nonnegative decomposition of a general positive semidefinite matrix is not always available.