  This paper generalizes Moody's correlated binomial default distribution for
homogeneous (exchangeable) credit portfolio, which is introduced by Witt, to
the case of inhomogeneous portfolios. As inhomogeneous portfolios, we consider
two cases. In the first case, we treat a portfolio whose assets have uniform
default correlation and non-uniform default probabilities. We obtain the
default probability distribution and study the effect of the inhomogeneity on
it. The second case corresponds to a portfolio with inhomogeneous default
correlation. Assets are categorized in several different sectors and the
inter-sector and intra-sector correlations are not the same. We construct the
joint default probabilities and obtain the default probability distribution. We
show that as the number of assets in each sector decreases, inter-sector
correlation becomes more important than intra-sector correlation. We study the
maximum values of the inter-sector default correlation. Our generalization
method can be applied to any correlated binomial default distribution model
which has explicit relations to the conditional default probabilities or
conditional default correlations, e.g. Credit Risk${}^{+}$, implied default
distributions. We also compare some popular CDO pricing models from the
viewpoint of the range of the implied tranche correlation.
