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**Credit Risk Modeling through the use of an Extended and Numerically Stable Version of CreditRisk+ and a Merton Model**
by Ludovic Dubrana of AXA Polska S.A. June 2010 **Abstract:** Accessing and managing credit risk has been a major area of interest and concern for both academics, practitioners and regulators, particularly in the afterwards of the recent 2008 financial crisis. Moreover, the effective management of credit risk is a challenge faced by any banking and insurance companies, and a critical success factor for a strong embedded Enterprise Risk Management. In this paper, we propose a model that mimics actuarial mathematics in order to describe credit risk in a portfolio including the modeling of both credit default risk and credit migration risk The credit default risk model can be described as an extended version of the CreditRisk+ model2 using both the Panjer recursion and the Fourier transform through an idiosyncratic risk factor and various systematic risk factors in order to adequately derive a probability density function for losses. The generated distribution is equivalent to a convolution of a large number of Gamma-Poisson mixtures. The main innovation is to separately derive a loss distribution for each risk factor with the Panjer recursion allowing to analyze the idiosyncratic or systematic risk supported by any risk factor and then, to use the attractive feature of complex numbers, via the Fourier transform, to determine an appropriate global loss distribution for credit default risk. The algorithm is based on both actuarial sciences and numerical mathematics and turns out to be particularly useful for analyzing very large credit portfolios by accurately capturing the tail and the body of the loss distribution for credit default risk. The main advantage is to ensure a numerically stable computation for credit default risk avoiding the well-know round-off error that accumulates in the original version of the generalized CreditRisk+ model. From our knowledge, this mixed algorithm was not published so far. In addition of both the Value-at-Risk and Expected Shortfall measurement, we computed the Value-at-Risk contribution per exposure that can be of major interest for business making decision and introduced a Markow process to quantify credit migration risk based on an extended Merton model. The attractive feature of both models is that only a few inputs are required to perform well and no assumptions are made on the default event so that the model can be easily extended to some risk categories such as operational risk where no attempts are made on the causes of default or migration for quantifying an economic capital.
**Keywords:** credit risk, credit default risk, credit migration risk, credit portfolio modeling, CreditRisk+, Markov process, Merton model, transition matrix model, Value-at-Risk, Expected Shortfall, Value-at-Risk contribution, Panjer recursion, Fourier transform, operational risk.
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