Optimisation of conditional-VaR in an actuarial model for credit risk assuming a student copula dependence structure

In this paper we present a model for the valuation of the risk of credit portfolios. It uses both traditional tools of credit risk valuations and more recent ones like copula functions and Conditional VaR theory. The model we propose is based on some key assumptions we here summarise: first of all, the risk of default is modelled using the time-until-default of an exposure; moreover the hazard rates are random variables whose values follow gamma distributions coherently with CreditRisk+ proposed by Credit Suisse and others; recovery rates themselves are supposed to be stochastic (following a Beta distribution). The main aspect of our proposal is the introduction of credit migration in the context of an intensity-based model with copula function dependence structure (we use a Student copula to model correlations between the obligors). This permits to quantify the loss distribution of the portfolio and to calculate some useful indexes of risk for the probability distribution of the values of the portfolio: expectation, variance, alpha-VaR, and, following Rockafellar & Uryasev, the alpha-conditional VaR (alpha-CVaR) of the portfolio itself. The final aim of the model is to present a more flexible and realistic approach to valuation and management of the risk of credit portfolios. Infact, in comparison with the traditional approaches, we remove some restrictive assumptions and try to generalize the valuation scheme (i.e. CreditMetrics considers constant hazard rates while CreditRisk+ takes into account constant recovery rates with no credit migrations). We conclude the article with a large numerical example in order to test the model