an introduction to credit risk modeling phần 4

nhiều phản ánh từ khóa "giới thiệu" đã được sử dụng trong tiêu đề của cuốn sách. Chúng tôi do đó tránh xây dựng trên các chi tiết kỹ thuật không thực sự cần thiết cho sự hiểu biết ý tưởng cơ bản. Ở phía bên kia, chúng tôi tiếp tục trình bày toán học chính xác và bao gồm một số chứng cứ cũng như tài liệu tham khảo rất nhiều cho độc giả quan tâm đến | account that we are in a Bernoulli framework E L m Y jr WiE Li Y N N - - v1 p Y tr V1-Q such that Proposition guarantees that L m p Y N N-1 p -ỰỘY _ -1 - Q _ almost surely. 2. 54 m - So for portfolios with a sufficiently large portfolio size m satisfying Assumption the percentage quote of defaulted loans for a given state of economy Y y is approximately equal to the conditional default probability p y . In the limit we obtain a portfolio loss variable p Y describing the fraction of defaulted obligors in an infinitely finegrained credit portfolio. We now want to derive the cumulative distribution function and the probability density of the limit loss variable p Y Y N 0 1 with p - as in 2. 54 . Denote the portfolio s percentage number of defaults in an infinitely fine-grained portfolio again assuming constant LGDs of 100 by L. We then have for every 0 x 1 P L x P p Y x 2. 55 P -Y n 1 x x l - Np Q N p . In the sequel we will denote this distribution function by Fp e x P L x x G 0 1 . 2. 56 The corresponding probability density can be derived by calculating the derivative of o x . x which is f M X p. 57 X exp -2 1 - 2g N 1 x 2 - 20. - qN 1 x N 1 p N 1 p 2 2003 CRC Press LLC p 30bps p FIGURE The probability density fp s for different combinations of p and Q note that the x-axes of the plots are differently scaled . 2003 CRC Press LLC N 1 x 2 - -1 n 1 p - y 1 - eN 1 x 2ọ Figure shows the loss densities fp Q for different values of p and Q. It could be guessed from Figure that regarding the extreme cases . p and Q some reasonable limit of fp e should exist. Indeed one can easily prove the following statement Proposition The density fp e admits four extreme cases induced by the extreme values of the parameters p and Q namely 1. Q 0 This is the correlation-free case with loss variables Li l ri Zi N-1 p B 1 p taking 2. 48 into account. In this case the absolute size-m portfolio loss z Li follows a binomial .