By C.Bluhm, L.Overbeck & C.Wagner
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Additional resources for An Introduction to Credit Risk Modeling
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. 5 Correlation induced by an underlying factor with a country factor for Germany and probably also with some other factors.
Additionally the residuals εi are assumed to be uncorrelated16 . Therefore, the returns ri are exclusively correlated by means of their composite factors. This is the reason why Φi is thought of as the systematic part of ri , whereas εi due to its independence from all other involved variables can be seen as a random effect just relevant for counterparty i. Now, in regression theory one usually decomposes the variance of a variable in a systematic and a specific part. Taking variances on both sides of Equation (1.
1. 18) specif ic Because the variance of ri captures the risk of unexpected movements of the asset value of counterparty i, the decomposition (1. 18) can be seen as a splitting of total risk of firm i in a systematic and a specific risk. The former captures the variability of ri coming from the variability of the composite factor, which is βi2 V[Φi ]; the latter arises from the variability of the residual variable, V[εi ]. Note that some people say idiosyncratic instead of specific. 15 Actually, although the KMV-Model in principal follows Merton’s model, it does not really work with Gaussian distributions but rather relies on an empirically calibrated framework; see Crosbie  and also Chapter 3.