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Analytical Credit VaR with Stochastic Probabilities of Default and Recoveries

by Antonio Castagna of iason Ltd.,
Fabio Mercurio of Bloomberg & iason Ltd., and
Paola Mosconi of iason Ltd.

June 1, 2009

Abstract: We extend the model presented in Bonollo et al. [3] by introducing a multi-scenario framework that allows for a richer and more realistic specification, including non-static (stochastic) probabilities of default and losses given default. Though more complex from a computational point of view, the model with scenarios is still tractable analytically, yielding results in closed form expressions. The approximated value at risk has been calculated by generalizing the procedure exposed in [3] for the single scenario case, in the presence of granularity in the exposures, sector concentration and contagion. The outcome is not simply a weighted sum of the VaRs in the individual scenarios, but results in a more involved function of the single scenarios' parameters. The theoretical model description is complemented with an in-depth numerical analysis.

JEL Classification: C63, G11, G38.

Keywords: Basel II, second pillar, credit VaR, analytical formula, contagion risk, sectoral risk, stochastic default probability, stochastic recovery, scenario.

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