By C.Bluhm, L.Overbeck & C.Wagner
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Extra info for An Introduction to Credit Risk Modeling
2. 2) conditional on Λ = λ the portfolio loss is again a Poisson distribution with intensity mλ, the probability of exactly k defaults equals ∞ P[L = k | Λ = λ] dF (λ) P[L = k] = (2. 18) 0 ∞ = e−mλ 0 mk λ k dF (λ) . k! Again, note that due to the unbounded support of the Poisson distribution the absolute loss L can exceed the number of “physically” possible defaults. As already mentioned at the beginning of this section, the probability of a multiple-defaults event is small for typical parametrizations.
Regarding random variables and probabilities we repeat our remark from the beginning of the previous chapter by saying that we always assume that an appropriate probability space (Ω, F, P) has been chosen, reflecting the “probabilistic environment” necessary to make the respective statement. Without loss of generality we will always assume a valuation horizon of one year. Let’s say we are looking at a credit portfolio with m counterparties. 1 we know the default probability pi corresponding to rating Ri .
As already mentioned at the beginning of this section, the probability of a multiple-defaults event is small for typical parametrizations. In the Poisson framework, the uniform default probability of borrowers in the portfolio is defined by ∞ p = P[Li ≥ 1] = P[Li ≥ 1 | Λ = λ] dF (λ) (2. 19) 0 ∞ = (1 − e−λ ) dF (λ) . 0 The counterpart of Formula (2. 16) is Corr[Li , Lj ] = V[Λ] V[Λ] + E[Λ] (i = j). (2. 20) Formula (2. 20) is especially intuitive if seen in the context of dispersion, where the dispersion of a distribution is its variance to mean ratio DX = V[X] E[X] for any random variable X.