By C.Bluhm, L.Overbeck & C.Wagner
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Additional resources for An Introduction to Credit Risk Modeling
0, hereby presenting some significant changes and improvements. Returning to the subject of this section, we now discuss the factor models used in KMV’s Portfolio ManagerTM and CreditMetricsTM CreditManagerTM . Both models incorporate the idea that every firm admits a process of asset values, such that default or survival of the firm depends on the state of the asset values at a certain planning horizon. If the process has fallen below a certain critical threshold, called the default point of the firm in KMV terminology, then the company has defaulted.
Combining (1. 22) with (1. 24), we finally obtain r = βW (BΓ + δ) + ε . (1. , rm ) can conveniently be written by means of underlying factors. Note that for computational purposes Equation (1. 25) is the most convenient one, because the underlying factors are independent. In contrast, for an economic interpretation and for scenario analysis one would rather prefer Equation (1. 22), because the industry and country indices are easier to interpret than the global factors constructed by ©2003 CRC Press LLC PCA.
2003 CRC Press LLC we again obtain the portfolio loss L as a convolution of the single loss variables, but this time with first and second moments m m pi E[L] = i=1 and pi (1 − pi ) . V[L] = (2. 1) i=1 This follows from E[Li ] = pi , V[Li ] = pi (1 − pi ), and the additivity of expectations resp. variances5 . Now, it is well known that in probability theory independence makes things easy. For example the strong law of large numbers works well with independent variables and the central limit theorem in its most basic version lives from the assumption of independence.