Risk contributions of lambda quantiles

Risk contributions of portfolios form an indispensable part risk adjusted performance measurement. The contribution a portfolio, e.g., in the Euler or Aumann-Shapley framework, is given by partial derivatives measure applied to portfolio return direction asset weights. For measures that are not positively homogeneous degree 1, however, known capital allocation principles do apply. We study class lambda quantile measures, includes well-known Value-at-Risk as special case, but for which no rule applicable. prove differentiability and derive explicit formulae quantiles with respect their composition, contribution. this purpose, we define on space compositions consider generic (also non-linear) operators. We further decomposition show compositions, homogeneity depends composition function. This result stark contrast positive properties defined random variables admit constant degree. introduce generalised version rule, compatible any non-linear portfolios. provide financial interpretations notion event-specific