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

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Assume we have two firms A and B which are positively correlated. For example, let A be DaimlerChrysler and B stand for BMW. 5. In our example we could think of the automotive industry as an underlying factor having significant impact on the economic future of the companies A and B. Of course there are probably some more underlying factors driving the riskiness of A and B. For example, DaimlerChrysler is to a certain extent also influenced by a factor for Germany, the United States, and eventually by some factors incorporating Aero Space and Financial Companies.

Calculate the first moment (EL) and estimate the second moment (UL). 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.

M) V[Li ] = V E[Li |Λ] + E V[Li |Λ] (2. 15) = V[Λi ] + E[Λi ] . Again we have Cov[Li , Lj ] = Cov[Λi , Λj ], and the correlation between defaults is given by Corr[Li , Lj ] = Cov[Λi , Λj ] V[Λi ] + E[Λi ] V[Λj ] + E[Λj ] . (2. 16) In the same manner as in the Bernoulli model this shows that correlation is exclusively induced by means of the distribution function F of the random intensity vector Λ. 2 Uniform Default Intensity and Uniform Correlation Analogously to the Bernoulli model, one can introduce a Poisson uniform portfolio model by restriction to one uniform intensity and one uniform correlation among transactions in the portfolio.

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