tailieunhanh - an introduction to credit risk modeling phần 5

pj đã được quan sát thấy. Chuỗi thời gian p1, ., P31, giải quyết các tần số mặc định lịch sử quan sát cho lớp đánh giá được lựa chọn trong những năm 1970 đến năm 2000, được cho bởi các hàng tương ứng trong Bảng . Trong khuôn khổ tham số của mô hình | Pj has been observed. The time series p1 . p31 addressing the historically observed default frequencies for the chosen rating class in the years 1970 up to 2000 is given by the respective row in Table . In the parametric framework of the CreditMetricsTM KMV uniform portfolio model it is assumed that for every year j some realization yj of a global factor Y drives the realized conditional default probability observed in year j. According to Equation 2. 49 we can write Pj p yj N N-1 p - VQyj V1 Qi i 1 . m where p denotes the true default probability of the chosen rating class and Q means the unknown asset correlation of the considered rating class which will be estimated in the following. The parameter p we do not know exactly but after a moment s reflection it will be clear that the observed historic mean default frequency p provides us with a good proxy of the true mean default rate. Just note that if Y1 . Yn are copies of the factor Y then the law of large numbers guarantees that 1 n E p Yj E p Y p . n j 1 Replacing the term on the left side by 1 n p T. pj n 4-1 j 1 we see that p should be reasonably close to the true default probability p. Now a similar argument applies to the sample variances because we naturally have 1 n 1 è p Yj põã 2 j 1 y p Y . where p Y 52 p Yj n. This shows that the sample variance 1 s2 n 1 n E pj p 2 j 1 25 Here we make the simplifying assumption that the economic cycle represented by Y1 . Yn is free of autocorrelation. In practice one would rather prefer to work with a process incorporating some intertemporal dependency . an AR 1 -process. 2003 CRC Press LLC should be a reasonable proxy for the true variance V p Y . Recalling Proposition we obtain V p Y N2 N-1 p N-1 p e -p2 2. 66 and this is all we need for estimating Q. Due to our discussion above we can replace the true variance V p Y by the sample variance Ơ2 and the true default probability p by the sample mean p. After replacing the unknown parameters p .