Measuring sectoral diversification in an asymptotic multi-factor   framework
  We investigate a multi-factor extension of the asymptotic single risk factor (ASRF) model that underlies the capital charges of the "Basel II Accord". In this extended model, it is still possible to derive closed-form solutions for the risk contributions to Value-at-Risk and Expected Shortfall. As an application of the risk contribution formulae we introduce a new concept for a diversification measure. The use of this new measure is illustrated by an example calculated with a two-factor model. The results with this model indicate that, thanks to dependence on not fully correlated systematic sectors, there can be a substantial reduction of risk contributions by sectoral diversification effects. 
Dirk Tasche