An Introduction to Credit Risk Modeling by C.Bluhm, L.Overbeck & C.Wagner

By C.Bluhm, L.Overbeck & C.Wagner

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31) is true. By definition, we have ˜Φ ˜T E Φ ˜ W (B Γ ˜ ˜ + δ) ˜ + δ) = E W (B Γ T ˜ ˜ T WT ˜ + δ)(B ˜ + δ) = W E (B Γ Γ ˜ Γ) ˜ T +E δ˜ δ˜ T ˜Γ ˜ T B T + BE Γ ˜ δ˜ T + E δ(B = W BE Γ = 0 WT . = 0 The two expectations above vanish due to our orthogonality assump˜Γ ˜ T is a tions. This proves (1. 31). Note that in equation (1. , K). Therefore, the calculation of asset correlations according to (1. t. the global factors. KMV customers have access to this information and can use Equation (1. 31) for calculating asset correlations.

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 DX = V[X] 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).

Equation (1. , Φm ) , (1. , εm ) . t. , m), (1. , ΨK are country indices. , wi,K0 are called the industry weights 17 That 18 Note is, normalized in order to have mean zero and variance one. that in the sequel we write vectors as column vectors. g. 9. in [21]. , wi,K are called the country weights of counterparty i. , m). k=K0 +1 In vector notation, (1. 20) combined with (1. 21) can be written as r = βW Ψ + ε , (1. , ΨK ) means the vector of industry and country indices. This constitutes the second level of the Global Correlation ModelTM .

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