tailieunhanh - an introduction to credit risk modeling phần 3

Thứ hai, cuốn sách của chúng tôi là nhằm mục đích hữu ích để quản lý rủi ro đang tìm kiếm một cách tiếp cận định lượng rủi ro tín dụng. Sau này ý định ở một bên, cuốn sách của chúng tôi được viết bằng một giảng Ghi chú phong cách rất | assumed to be conditionally independent. The joint distribution of the Li s is given by Í X - l1 - lm P L1 11 . L m 4 0 e-mA . . dF - . Because see the beginning of Section conditional on A - the portfolio loss is again a Poisson distribution with intensity m- the probability of exactly k defaults equals P L k i P L k A - dF - 2. 18 0 r e- m-. dF - . 0 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. In the Poisson framework the uniform default probability of borrowers in the portfolio is defined by p P Li 1 i P Li 1 A - dF - 2. 19 0 tt 1 - e-A dF - . 0 The counterpart of Formula 2. 16 is Corr Li Lj V A E A 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 V X Dx E X for any random variable X. 2. 21 The dispersion of the Poisson distribution is equal to 1. Therefore the Poisson distribution is kind of a benchmark when deciding about overdispersion DX 1 respectively underdispersion DX 1 . In 2003 CRC Press LLC general nondegenerate8 Poisson mixtures are overdispersed due to 2. 15 . This is a very important property of Poisson mixtures because before using such a model for credit risk measurement one has to make sure that overdispersion can be observed in the data underlying the calibration of the model. Formula 2. 20 can be interpreted by saying that the correlation between the number of defaults of different counterparties increases with the dispersion of the random intensity A. For proving this statement we write Formula 2. 20 in the form Corr Li Lj Da i j . 2. 22 Da 1 From 2. 22 it follows that an increase in dispersion increases the mixture effect which in turn strengthens the dependence between obligor s defaults.