In this paper we present a model for the valuation of the risk of credit portfolios. It usesboth traditional tools of credit risk valuations and more recent ones like copula functions and ConditionalVaR 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; moreoverthe hazard rates are random variables whose values follow gamma distributions coherentlywith CreditRisk+ proposed by Credit Suisse and others; recovery rates themselves are supposed tobe stochastic (following a Beta distribution).The main aspect of our proposal is the introduction of credit migration in the context ofan intensity-based model with copula function dependence structure (we use a Student copula tomodel correlations between the obligors). This permits to quantify the loss distribution of the portfolioand to calculate some useful indexes of risk for the probability distribution of the values ofthe portfolio: expectation, variance, α − VaR, and, following Rockafellar & Uryasev, theα − conditional VaR (α − CVaR) of the portfolio itself.The final aim of the model is to present a more flexible and realistic approach to valuationand 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 recoveryrates with no credit migrations).We conclude the article with a large numerical example in order to test the model.

### OPTIMISATION OF CONDITIONAL-VAR IN AN ACTUARIAL MODEL FOR CREDIT RISK ASSUMING A STUDENT COPULA DEPENDENCE STRUCTURE

#### Abstract

In this paper we present a model for the valuation of the risk of credit portfolios. It usesboth traditional tools of credit risk valuations and more recent ones like copula functions and ConditionalVaR 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; moreoverthe hazard rates are random variables whose values follow gamma distributions coherentlywith CreditRisk+ proposed by Credit Suisse and others; recovery rates themselves are supposed tobe stochastic (following a Beta distribution).The main aspect of our proposal is the introduction of credit migration in the context ofan intensity-based model with copula function dependence structure (we use a Student copula tomodel correlations between the obligors). This permits to quantify the loss distribution of the portfolioand to calculate some useful indexes of risk for the probability distribution of the values ofthe portfolio: expectation, variance, α − VaR, and, following Rockafellar & Uryasev, theα − conditional VaR (α − CVaR) of the portfolio itself.The final aim of the model is to present a more flexible and realistic approach to valuationand 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 recoveryrates with no credit migrations).We conclude the article with a large numerical example in order to test the model.
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2007
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11386/4819096`
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