Comparing the Value at Risk Performance of the CreditRisk⁺ and its Enhancement: A Large Deviations Approach

Abstract

The standard CreditRisk⁺ (CR⁺) is a well-known default-mode credit risk model. An extension to the CR⁺ that introduces correlation through a two-stage hierarchy of randomness has been discussed by Deshpande and Iyer (Central Eur J Oper Res 17(2):219–228, 2009) and more recently by Sowers (2010). It is termed the 2-stage CreditRisk⁺ (2-CR⁺) in the former. Unlike the standard CR⁺, the 2-CR⁺ model is formulated to allow correlation between sectoral default rates through dependence on a common set of macroeconomic variables. Furthermore the default rates for a 2-CR⁺ are distributed according to a general univariate distribution which is in stark contrast to the uniformly Gamma distributed sectoral default rates in the CR⁺. We would then like to understand the behaviour of these two models with regards to their computed Value at Risk (VaR) as the number of sectors and macroeconomic variables approaches infinity. In particular we would like to ask whether the 2-CR⁺ produces higher VaR than the CR⁺ and if so then for which type of credit portfolio. Utilizing the theory of Large deviations, we provide a methodology for comparing the Value at risk performance of these two competing models by computing certain associated rare event probabilities. In particular we show that the 2-Stage CR⁺ definitely produces higher VaR than the CR⁺ for a particular class of a credit portfolio which we term as a “balanced” credit portfolio. We support this statistical risk analysis through numerical examples.