  Capital allocation is a procedure for quantifying the contribution of each
source of risk to aggregated risk. The gradient allocation rule, also known as
the Euler principle, is a prevalent rule of capital allocation under which the
allocated capital captures the diversification benefit of the marginal risk as
a component of overall risk. This research concentrates on Expected Shortfall
(ES) as a regulatory standard and focuses on the gradient allocations of ES,
also called ES contributions (ESCs). We present the comprehensive treatment of
backtesting the tuple of ESCs in the framework of the traditional and
comparative backtests based on the concepts of joint identifiability and
multi-objective elicitability. For robust forecast evaluation against the
choice of scoring function, we also extend the Murphy diagram, a graphical tool
to check whether one forecast dominates another under a class of scoring
functions, to the case of ESCs. Finally, leveraging the recent concept of
multi-objective elicitability, we propose a novel semiparametric model for
forecasting dynamic ESCs based on a compositional regression model. In an
empirical analysis of stock returns we evaluate and compare a variety of models
for forecasting dynamic ESCs and demonstrate the outstanding performance of the
proposed model.
