By C.Bluhm, L.Overbeck & C.Wagner
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Additional info for An Introduction to Credit Risk Modeling
In such a case, F is a Dirac measure εp , concentrated in p, and the absolute portfolio loss L follows a binomial distribution with default probability p. The other extreme case regarding (2. 10), Corr[Li , Lj ] = 1, implies a “rigid” behaviour of single losses in the portfolio: Either all counterparties default or all counterparties survive simultaneously. The corresponding distribution F of P is then a Bernoulli distribution, such that P = 1 with probability p and P = 0 with probability 1 − p.
Crouhy, Galai and Mark , Gordy , Frey and McNeil , and Hickman and Koyluoglu , just to mention a few references. For the sequel we make a notational convention. Bernoulli random variables will always be denoted by L, whereas Poisson variables will be denoted by L . In the following section we first look at the Bernoulli1 model, but then also turn to the case of Poissonian default variables. 3 we briefly compare both approaches. , Li = 1 with probability pi . 0 with probability 1 − pi that the Bernoulli model benefits from the convenient property that the mixture of Bernoulli variables again yields a Bernoulli-type random variable.
11) where pi again denotes the default probability of obligor i. Note that (2. 11) allows for multiple defaults of a single obligor. The likelihood of the event that obligor i defaults more than once is given by P[Li ≥ 2] = 1 − e−λi (1 + λi ) , which is typically a small number. 5 basispoints. 01 we can expect that only 1 out of 20,000 scenarios is not applicable because of a multiple default. 5 basispoints is a relatively high number. The intensity λi is typically quite close to the default probability pi , due to pi = P[Li ≥ 1] = 1 − e−λi ≈ λi (2.