Sector Analysis

Generally, the CreditRisk + assumes that the default probability (PD) of each debtor is random and its variability depends on some factors, present in the portfolio, named risk factors. The debtors may be divided into sectors, i.e. sets of debtors that are affected by the influence of a particular (systemic) risk factor. Any factor influences the sector through the total PD, which is modeled as a random variable Λ[s], with mean λ[s] and standard deviation σ[s]. If we have k sectors, the Credit-Risk + assumes that Λ[s] ∼ Gamma(αs, βs), so that the default number in each sector s follows a Gamma-Poisson distribution with parameter Λ[s]. Furthermore, it assumes that the variables Λ[1], …, Λ[k] are two by two independent. Analogously to the procedure adopted in Chapter 15, first, we compute the PGF of the default amount in the sector s. Second, we derive the probability distribution of the default events in s. Third, we obtain the PGF for the portfolio loss.