A Theoretical Introduction to Numer. Analysis by V. Ryaben'kll, S. Tsynkov

By V. Ryaben'kll, S. Tsynkov

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M). (1. 28) Therefore, Equation (1. 27) combined with (1. 28) yields Corr[˜ ri , r˜j ] = = Ri V[Φi ] Ri Rj ˜ i] V[Φ ˜j] V[Φ Rj ˜ iΦ ˜j E Φ V[Φj ] ˜ iΦ ˜j E Φ ˜ i ]. because by construction we have V[Φi ] = V[Φ ©2003 CRC Press LLC (1. 29) Based on Equation (1. 25) we can now easily compute asset correlations according to (1. 29). After standardization, (1. 25) changes to ˜ (B Γ ˜ +ε ˜ + δ) ˜, r˜ = βW (1. 30) ˜ ∈ Rm×m denotes the matrix obtained by scaling every diagonal where β element in β by 1/σi , and ˜ = 0, E Γ ˜ = 0, E ε E δ˜ = 0 .

The covariance between single losses obviously equals Cov[Li , Lj ] = E[Li Lj ] − E[Li ]E[Lj ] = Cov[Pi , Pj ] . (2. 5) Therefore, the default correlation in a Bernoulli mixture model is Corr[Li , Lj ] = Cov[Pi , Pj ] E[Pi ] (1 − E[Pi ]) E[Pj ] (1 − E[Pj ]) . (2. 6) Equation (2. 5) respectively Equation (2. 6) show that the dependence between losses in the portfolio is fully captured by the covariance structure of the multivariate distribution F of P . 4 presents some examples for a meaningful specification of F .

Note that the EL of the original portfolio usually can be calculated based on the information from the rating, exposure, and LGD distributions of the portfolio. 8). Therefore, one now has to make an assumption regarding an average default correlation ρ. 1). 8) by setting all default correlations ρij equal to ρ will provide an estimated value for the original portfolio’s UL. t. first and second moments. This distribution is then interpreted as the loss distribution of an equivalent portfolio which was selected by a moment matching procedure.

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